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Price Theory Enocompassing. MIT 14.126, Fall 2001. Road Map. Traditional Doctrine: Convexity-Based Disentangling Three Big Ideas Convexity: Prices & Duality Order: Comparative Statics, Positive Feedbacks, Strategic Complements - PowerPoint PPT Presentation
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Price Theory Enocompassing
MIT 14126 Fall 2001
Road Map
Traditional Doctrine Convexity-Based Disentangling Three Big Ideas
Convexity Prices amp Duality Order Comparative Statics Positive Feedbacks
Strategic Complements Value Functions Differentiability and Characteri
zations Incentive Equivalence Theorems
Convexity and Traditional Price Theory
Convexity at Every Step
Global or local convexity conditions imply Existence of prices Comparative statics using second-order conditions Dual representations which lead tohellip
bull Hotellingrsquos lemmabull Shephardrsquos lemmabull Samuelson-LeChatelier principle
ldquoConvexityrdquo is at the core idea on which the whole analysis rests
Samuelson-LeChatelier Principle
Idea Long-run demand is ldquomore elasticrdquo than short-run demand
Formally the statement applies to smooth demand functions for sufficiently small price changes
Let p=(pxwr) be the current vector of output and input prices and let prsquo be the long-run price vector that determined the current choice of a fixed input say capital
Theorem If the demand for labor is differentiable at this point then
Varianrsquos Proof
Long- and short-run profit functions defined
Long-run profits are higher
So long-run demand ldquomust berdquo more elastic
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Road Map
Traditional Doctrine Convexity-Based Disentangling Three Big Ideas
Convexity Prices amp Duality Order Comparative Statics Positive Feedbacks
Strategic Complements Value Functions Differentiability and Characteri
zations Incentive Equivalence Theorems
Convexity and Traditional Price Theory
Convexity at Every Step
Global or local convexity conditions imply Existence of prices Comparative statics using second-order conditions Dual representations which lead tohellip
bull Hotellingrsquos lemmabull Shephardrsquos lemmabull Samuelson-LeChatelier principle
ldquoConvexityrdquo is at the core idea on which the whole analysis rests
Samuelson-LeChatelier Principle
Idea Long-run demand is ldquomore elasticrdquo than short-run demand
Formally the statement applies to smooth demand functions for sufficiently small price changes
Let p=(pxwr) be the current vector of output and input prices and let prsquo be the long-run price vector that determined the current choice of a fixed input say capital
Theorem If the demand for labor is differentiable at this point then
Varianrsquos Proof
Long- and short-run profit functions defined
Long-run profits are higher
So long-run demand ldquomust berdquo more elastic
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Convexity and Traditional Price Theory
Convexity at Every Step
Global or local convexity conditions imply Existence of prices Comparative statics using second-order conditions Dual representations which lead tohellip
bull Hotellingrsquos lemmabull Shephardrsquos lemmabull Samuelson-LeChatelier principle
ldquoConvexityrdquo is at the core idea on which the whole analysis rests
Samuelson-LeChatelier Principle
Idea Long-run demand is ldquomore elasticrdquo than short-run demand
Formally the statement applies to smooth demand functions for sufficiently small price changes
Let p=(pxwr) be the current vector of output and input prices and let prsquo be the long-run price vector that determined the current choice of a fixed input say capital
Theorem If the demand for labor is differentiable at this point then
Varianrsquos Proof
Long- and short-run profit functions defined
Long-run profits are higher
So long-run demand ldquomust berdquo more elastic
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Convexity at Every Step
Global or local convexity conditions imply Existence of prices Comparative statics using second-order conditions Dual representations which lead tohellip
bull Hotellingrsquos lemmabull Shephardrsquos lemmabull Samuelson-LeChatelier principle
ldquoConvexityrdquo is at the core idea on which the whole analysis rests
Samuelson-LeChatelier Principle
Idea Long-run demand is ldquomore elasticrdquo than short-run demand
Formally the statement applies to smooth demand functions for sufficiently small price changes
Let p=(pxwr) be the current vector of output and input prices and let prsquo be the long-run price vector that determined the current choice of a fixed input say capital
Theorem If the demand for labor is differentiable at this point then
Varianrsquos Proof
Long- and short-run profit functions defined
Long-run profits are higher
So long-run demand ldquomust berdquo more elastic
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Samuelson-LeChatelier Principle
Idea Long-run demand is ldquomore elasticrdquo than short-run demand
Formally the statement applies to smooth demand functions for sufficiently small price changes
Let p=(pxwr) be the current vector of output and input prices and let prsquo be the long-run price vector that determined the current choice of a fixed input say capital
Theorem If the demand for labor is differentiable at this point then
Varianrsquos Proof
Long- and short-run profit functions defined
Long-run profits are higher
So long-run demand ldquomust berdquo more elastic
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Varianrsquos Proof
Long- and short-run profit functions defined
Long-run profits are higher
So long-run demand ldquomust berdquo more elastic
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
A Robust ldquoCounterexamplerdquo
The production set consists of the convex hull of these three points with free disposal allowed
Fix the price of output at 9 and the price of capital at 3 and suppose the wage rises from w=2 to w=5 Demands are
Long run labor demand falls less than short-run labor demand Robustness Tweaking the numbers or ldquosmoothingrdquo the
production set does not alter this conclusion
Capital labor output
0 0 0
1 1 1
0 2 1
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
An Alternative Doctrine
Disentangling Three Ideas
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Separating the Elements
Convexity Proving existence of prices Dual representations of convex sets Dual representations of optima
Order Comparative statics Positive feedbacks (LeChatelier principle) Strategic complements
Envelopes Useful with dual functions Multi-stage optimaizations Characterizing information rents
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Invariance ChartConclusions about
maxxЄs f(xt)Transformations of
Choice variable
Supporting (ldquoLagrangianrdquo) prices exist
Linear (ldquoconvexity preservingrdquo)
Optimal choices increase in parameter
Order-preserving
Long-run optimum change is larger same direction
Order-preserving
Value function derivative formula Vrsquo(t) = f2(x(t)t)
One-to-one
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Convexity Alone
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Pure Applications of Convextity
Separating Hyperplane Theorem Existence of prices Existence of probabilities Existence of Dual Representations
bull Example Bondavera-Shapley Theorembull Example Linear programming duality
ldquoAllegedrdquo Applications of Duality Hotellingrsquos lemma Shephardrsquos lemma
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Separating Hyperplane Theorem
Theorem Let S be a non-empty closed convex set in and Then there exists such that
Proof Let y S be the nearest point in S to x Let
Argue that such a point y exists Argue that pmiddotxgt pmiddoty Argue that if and pmiddotzgtpmiddoty then for some sm
all positive t tz+(1-t)y is closer to x than y is
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Dual Characterizations
Corollary If S is a closed convex set then S is the intersection of the closed ldquohalf spacesrdquo containing it Defining
it must be true that
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Convexity and Quantification
The following conditions on a closed set S in is are equivalent S is convex For every x on the boundary of S there is
a supporting hyperplane for S through x For every concave objective function f th
ere is some λ such that the maximizers of f(x) subject to are maximizers of f(x)+λmiddotx subject to
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Order Alone
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
ldquoOrderrdquo Concepts amp Results
Order-related definitions Optimization problems
Comparative statics for separable objectives An improved LeChatelier principle Comparative statics with non-separable ldquotrade-offsrdquo
Equilibrium w Strategic Complements Dominance and equilibrium Comparative statics Adaptive learning LeChatelier principle for equilibrium
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Two Aspects of Complements
Constraints Activities are complementary if doing one enabl
es doing the otherhellip hellipor at least doesnrsquot prevent doing the other
bull This condition is described by sets that are sublattices hellipor at least doesnrsquot change the other from bei
ng profitable to being unprofitablebull This is described by payoffs satisfying a single crossing
condition
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Definitions ldquoLatticerdquo
Given a partially ordered set (xge) define The ldquojoinrdquo The ldquomeetrdquo
(xge) is a ldquolatticerdquo if
Example
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Definitions 2
(xge) is a ldquocomplete latticerdquo if for every non-empty subset S a greatest lower bound inf(S)and a least upper bound sup(S) exist in X
A function f XrarrR is ldquosupermodularrdquo if
A function f is ldquosubmodularrdquo if - f is supermodular
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Definitions 3
Given two subsets S T X ldquoS is as high as Trdquo written SgeT means
A function x is ldquoisotonerdquo (or ldquoweakly increasingrdquo) if
ldquoNondecreasingrdquo is not used becausehellip
A set S is a ldquosublatticerdquo if SgeS
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Sublattices
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Not Sublattices
Convexity order and topology are mostly independent concepts However in R these concepts coincide Topology S = compact set with boundary ab Convexity Order
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
ldquoPairwiserdquo Supermodularity
Theorem (Topkis) Let f The following are equivalent
f is supermodular For all nnem and x-nm the restriction f(x-nm) is su
permodular
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof of Pairwise Supermodularity
This direction follows from the definition Given xney suppose for notational simplicity th
at
Then
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
ldquoPairwiserdquo Sublattices Theorem (Topkis) Let S be a sublattice of Defin
e
Then
Remark Thus a sublattices can be expressed as a collection of constraints on pairs of arguments In particular undecomposable constraints like
can never describe in a sublattice
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof of Pairwise SublatticesIt is immediate that For the reverse
suppose Then and
Define For all
So satisfies for all i
QED
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Complementarity Complementaritysupermodularit
y has equivalent characterizations Higher marginal returns
Nonnegative mixed second differences
For smooth objectives non- negative mixed second derivatives
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Monotonictiy Theorem Theorem (Topkis) Let f XtimesR rarrR be a supermodular function a
nd define
If t ge trsquo and S(t) ge S(trsquo ) then x(t) ge x(trsquo ) Corollary Let XtimesR rarrR be a supermodular function and suppos
e S(t) is isotone Then for each t1 S(t) and x(t) are sublattices Proof of Corollary Trivially t ge t so S(t) ge S(t) and x(t) ge x(trsquo ) QED
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof of Monotonictiy Theorem Suppose that is supermodular and that Then So and If either any of these inequalities are strict then the
ir sum contradicts supermodularity
QED
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Necessity for Separable Objectives
Theorem (Milgrom) Let f be a supermodular function and suppose S is a sublattice
Let
Then the following are equivalent f is supermodular For all is isotone
Remarks This is a ldquorobust monotonicityrdquo theorem The function is ldquomodularrdquo
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof
Follows from Topkisrsquos theorem It suffices to show ldquopairwise supermodularity rdquoHence i
t is sufficient to show that supermodularity is necessary when N=2 We treat the case of two choice variables the treatment of a choice variable and parameter is similar
Let be unordered Fix
If thenis not a sublattice so
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Application Production Theory
Problem
Suppose that L is supermodular in the natural order for example L(lw)=wl
Then -L is supermodular when the order on is reversed L(w) is nonincreasing in the natural order
If f is upermodular then k(w) is also nonincreasing That is capital and labor are ldquoprice theory complementsrdquo
If f is supermodular with the reverse order then capital and labor are ldquoprice theory subsitutesrdquo
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Application Pricing Decisions
A monopolist facing demand D(pt) produces at unit cost c
p(ct) is always isotone in c It is also isotone in t if log(D(pt)) is supermodular in (pt) which is the same as being supermodular in (log(p)t) which means that increases in t make demand less elastic
nondecreaseing in t
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Application Auction Theory
A firmrsquos value of winning an item at price p is U(pt) where t is the firmrsquos type (Losing is normalized to zero) A bid of p wins with probability F(p)
Question Can we conclude that p(t) is nondecreasing without knowing F
Answer Yes if and only if log(U(pt) is supermodular
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Long v Short-Run Demand
Notation Let (wwrsquo) be the short-run demand for labor when the current wage is w and the wage determining fixed inputs is wrsquo
Setting w=wrsquo in gives the long run demands Samuelson-LeChatelier principle
which can be restated revealingly as
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Milgrom-Roberts Analysis
Remarks This analysis involves no assumptions about convexity dividibili
ty etc For smooth demands symmetry of the substitution matrix impli
es that locally one of the two cases above applies
Complements Substitutes
Wage Labor
Capital
-
+ +
Wage Labor
Capital
-
- -
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Improved LeChatelier Principle
Let H(xyt) be supermodular and S a sublattice Let (x(t)y(t))=maxargmaxH(xyt)
Let x(ttrsquo)=maxarg max H(xy(trsquo)t)
Theorem (Milgrom amp Roberts) x is isotone in both arguments In particular if tgttrsquo then
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof
By the Topkis Monotonicity Theorem y(t)gey(trsquo) Applying the same theorem again for all ttrsquorsquogtt
rsquo
Setting t=trsquorsquo completes the proof
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Long v Short-Run Demand
Theorem Let wgtwrsquo Suppose capital and labor are complements ie f(kl) is supermodular in the natural order If demand is single-valued at w and wrsquo then
Theorem Let wgtwrsquo Suppose capital and labor are substitutes ie f(kl) is supermodular whencapital is given its reverse order If demand is single-valued at w and wrsquo then
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Non-separable Objectives
Consider an optimaization problem featuring ldquotrade-offsrdquo among effects x is the real-valued choice variable B(x) is the ldquobenefits production functionrdquo Optimal choice is
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Robust Monotonicity Theorem
Define Theorem Suppose πis continuously differentiable
and π2 is nowhere 0 Then
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Application Savings Decisions
By saving x one can consume F(x) in period 2
Define Analysis If MRSxy increases with x then optimal savings are i
sotone in wealth
This is the same condition as found in price theory when F is restricted to be linear Here F is unrestricted
Also applies to Koopmans consumption-savings model
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Introduction toSupermodular Games
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Formulation
N players (infinite is okay) Stratege sets Xn are complete sublattices
Payoff functions Un(x) are
Continuous ldquosupermodular with isotone differencesrdquo
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Bertrand Oligopoly Models
Linearsupermodular Oligopoly Demand Profit which is increasing in Xn
Log-supermodular Oligopoly
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Linear Cournot Duopoly Inverse Demand
Linear Cournot duopoly (but not more general oligopoly) is supermodular if one playerrsquos strategy set is given the reverse of its usual order
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Analysis of Supermodular Games
Extremal Best Reply Functions
By Topkisrsquos Theorem these are isotone functions
Lemma is strictly dominated by Proof If then
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Rationalizability amp Equilibrium
Theorem (Milgrom amp Roberts) The smallest rationalizable strategies for the players are given by
Similarly thelargest rationalizable strategies for the players are given by
Both are Nash equilibrium profiles
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof
Notice that is an isotone bounded sequence so its limit z exists
By continuity of payoffs its limit is a fixed point of b and hence a Nash equilibrium
Any strategy less than zn is less than some and hence is deleted during iterated deletion of dominated strategies
QED
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Comparative Statics Theorem (Milgrom amp Roberts) Consider a family of
supermodular games with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular are isotone
Proof By Topkisrsquos theorem bt(x) is isotone in t Hence if tgttrsquo
and similarly for QED
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Adaptive Learning
Player nrsquos behavior is called ldquoconsistent with adaptive learningrdquo is for every date t there is some date trsquo after dominated in the game in which others are restricted to play only strategiesthey have played since date t
Theorem (Milgrom amp Roberts) In a finite strategy game if every playerrsquos behavior is consistent with adaptive learning then all eventually play only rationalizable strategies
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Equilibrium LeChatelier Principle
Formulation Consider a parameterized family of supermodular games
with payoffs parameterized by t Suppose that for all n x-n Un(xnx-nt) is supermodular in (xnt)
Fixing player 1rsquos strategy at in duces a supermodular game among the remaining players Let be the smallest Nash equilibrium in the induced game with
Theorem If tgttrsquo then If tlttrsquo then
hellipand a similar conclusion applies to the maximum equilibrium
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof
Observe (exercise) that
Suppose tgttrsquo By the comparative statics theorem z is isotone
so
Hence by the comparative statics theorem applied again y is isotone so
QED
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Envelope Functions Alone
Based on ldquoenvelope Theorems for Arbitrary Choice Sets ldquoby Paul Milgrom amp Ilya Segal
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
What are ldquoEnvelope Theoremsrdquo
Envelope theorems deal with the properties of the value function
Anwser questions abouthellip when v is differentiable directionally differentiable Lipsc
hitz or absolutely contiunous when v satisfies the ldquoenvelope formulardquo
ldquoTraditionalrdquo envelope theorems assume that set X is convex and the objective f(middott) is concave and differentiable
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Intuitive Argument
When x=x1x2x3hellip V is left-and right-differ
entiable everywhere If ft(xt) is constant on
then V is diferentiable at t
Envelope formulas apply forbull bull
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Envelope Derivative Formula
Theorem 1 Take and and suppose that ft(xt) exists If tlt1 and Vrsquo(t+) exists then Vrsquo(t+) ge ft(xt) If tgt0 and Vrsquo(t-) exists then Vrsquo(t-) le ft(xt) If t (01) and Vrsquo(t) exists then Vrsquo(t) = ft(xt)
Proof
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Absolute Continuity
Theorem 2(A) Suppose that f(xmiddot) is is differentiable (or just absolute
ly continuous) for all with derivative (or density) ft
There exists an integrable function b(t) such that for all and almost all
Then V is absolutely continuous with density satisfying
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Proof of Theorem 2(A)
Define
Then for trsquorsquogttrsquo
It suffices to prove the theorem for intervals because open intervals are a basis for the open sets QED
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Why do we need b(middot)
Let X=(01] and f(xt)=g(tx) where g is smooth and single-peaked with unique maximum at 1 V(0)=g(0) V(t)=g(1) V is discontinuous at 0 This example has no integrable bound b(t)
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Envelope Integral Formula
Theorem 2(B) Suppose that in addition to the assumptions of 2(A) the set of optimizers x(t) is non-empty for all t Then fro any selection
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Equi-differentiability Definition A family of functions
is ldquoequi-differentiablerdquoat if
If X is finite this is the same as simple differentiability
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Directional Differentiability
Theorem 3 If (i) is equi-differentiable at t0 (ii) x(t) is non-empty for all t and (iii) sup then for any selection V is left andright-differentiable at and the derivatives satisfy
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Role of ldquoEqui-differentiabilityrdquo
Simple differentiability (rather than equi-differentiability) is not enough for V to have left- and right-dirivatives
Let g(t)=t sinlog(t) f(xt)=g(t) if tgtexp(-π2-2πx) f(xt)=-t otherwise
Then-V(t)=g(t)
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Continuous Problems
Theorem 4 Suppose X is a non-empty compact space f is upper semi-continuous on X and ft is continuous in (xt) Then
V is directionally differentiable
In particular Vrsquo(t+)geVrsquo(t-) V is differentiable at t if any of the following hold
bull V is concave (because Vrsquo(t+)leVrsquo(t-))bull T is a maximum of V(middot) (because Vrsquo(t+)leVrsquo(t-)bull x(t) is singleton (because Vrsquo(t+)=Vrsquo(t-))
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Contrast to a ldquoTraditionalrdquo approach
In some approaches the differentiability of x is used in the argument However V can be differentiable even when x is not This often happens for example in strictly convex problems
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Applications
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Hotellingrsquos Lemma Define
Theorem Suppose X is compact Then πrsquo(p) exists if and only if x(p) is a singleton and in that case πrsquo(p) =x(p)
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Shephardrsquos Lemma Define
Remark The variable x1 represents ldquooutputrdquo and the other variables represent inputs measured as negative numbers
Theorem Suppose X is compact Then exists if and only if x(p) is a singleton and in that case = x(p)
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Multi-Stage Maximization Stage 1 choose investment tge 0 Stage 2 choose action vector xisinXneempty Assume 1048708
f(xt) is equidifferentiable in t and tgt0 1048708 f(xt) is usc in x and X is compact
Conclusion the value function V(t) is differentiable at tand Vrsquo(t)=0 1048708 Proof Apply theorem 4
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Mechanism Design Y=set of outcomes Agentrsquos type is t utility is f(xt) M=message space hMrarrY is outcome funct
ion X=h(M) is set of ldquoaccessible outcomesrdquo Assume that each type has an optimal choic
e
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Analysis Corollary 1 Suppose that the agentrsquos utility functi
on f(xt) is differentiable and absolutely continuous is t for all x isin Y and that sup is integrable on [01] Then the agentrsquos equilibrium utility V in any mechanism implementing a given choice rule x must satisfy the following integral condition
This had previously been shown only with (sometimes ldquoweakrdquo) additional conditions
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Mechanism Design Applications Models in which payoffs are so
Theorems Green-Laffont Theorem
bull Uniqueness of Dominant Strategy Mechanisms Holmstrom-Williams Theorem
bull Bayesian Revenue Equivalence Myerson-Satterhwait Theorem
bull Necessity of Bargaining Inefficiency Jehiel-Moldovanu Theorem
bull Impossibility of Efficiency with Value Interdependencies
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Green-Laffont Theorem ldquoUniqueness of dominant strategy implementatio
nrdquo Theorem (Holmstromrsquos variation) Suppose that
M is direct mechanism to implement the efficient outcome in dominant strategies
the type space is smoothly path-connected Then
the payment function for player j in mechanism M is equal to the payment function of the Vickrey-Clarke-Groves pivot mechanism plus some function gj(V-j) (which depends only on the other playerrsquos types)
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Green-Laffont Theorem Given any value vector v let vj(t)|t isin [01] be a smooth path
connecting some fixed value vj to vj = vj (1) By the Envelope Theorem applied to the path parameter t
So xj is fully determined by the functions p and fj
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Holmstrom-Williamsrsquo Theorem
Theorem Any mechanism that Bayes-Nash implements efficient outcomes on a smoothly path-connected type space entails the same expected payments as the Vickrey mechanism plus some bidder-specific constant
Proof Let vj(s)|s isin [01] be a path from some fixed value vector to any other value vector By the Envelope Theorem
Hence xj(v) is uniquely determined by Uj(0) It is equal to Uj (0) plus the expected payment in the Vickrey mechanism
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Two-Person Bargaining
Assume there is a buyer with value v distributed on [01] there is a buyer with value v distributed on [01]
The Vickrey-Clarke-Groves mechanism has each party report its value entails p(vc)=1 if vgtc and p(vc)=0 otherwise payments are
bull if p(vc)=0 on paymentsbull if p(vc)=1 buyer pays c and the seller receives v
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Myerson-Sattherthwaite Theorem
Expected profits are each bidder expects to receive the entire social surplus
Apply Holmstrom-Williams theorem Theorem (Myerson-Satterthwait) There is no me
chanism and Bayesian Nash equilibrium such that the mechanism implements for all v c and
UB(0)=Us(1)=0 (ldquovoluntary participation by worst typerdquo) E[UB(V)]+E[Us(C)leE[(V-C)1VgtC] (ldquobalanced expected budg
etrdquo)
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
Subtleties
Consider a model in which
Q Why doesnrsquot simply trading at price p=1 violate the theorem in this model
A Because it prescribes trade even when cgtv
