Strategy-proof and Efficient Scheduling · Strategy-proofness: all players report preferences truthfully; Piecewise uniformity/dichotomy: the cake can be divided into acceptable and

Strategy-proof and Efficient Scheduling

Yuan Tian

Department of Economics

University of Chicago

August 30, 2013

Yuan Tian (University of Chicago) Scheduling August 30, 2013 1 / 57

Introduction

The problem of fair division (cake-cutting)

Division of a continuum of heterogeneous goods among players;

Why continuum?

Is convenient when modeling demand as time intervals;

Subsumes random assignments/matching of discrete goods;

Allows for continuous characteristics:

e.g. same good with different prices are treated as different goods;

Restricted preference domain:

Goods are either acceptable (needed) or unacceptable (unneeded);

No transfers allowed.


Introduction

Examples

Objects:

Player Children Students Employees State governments

Goods Cake Schools Tasks Federal funded projects

Time:

Player Airlines/Committees E-cars Readers

Goods Runway/Room occupancy Power outlet Reading time


Introduction

Contribution

Characterized a class of strategy-proof and Pareto efficient

mechanisms that accommodates fairness objectives such as

envy-freeness and arbitrary guaranteed shares of demand.


Introduction

Plan/approach and technique

1 Propose a social welfare function G (·) that is potentially truthfully

implementable in weakly dominant strategies;

2 Verify a non-inferiority condition that ensures the truthful

implementability of G (·);

3 Verify the existence of a feasible schedule (hence a direct mechanism)

supporting the allocation that maximizes G (·);

Main technique: monotone comparative statics of constrainedoptimization problems with lattice programming Literature review


Introduction

A quick demonstration

Player 1: (0, 1) always; Player 2: (0.5, l);

Choose x1 and x2 to maximize (2 ln x1 + ln x2) subject to

x1 ≤ 1, x2 ≤ l − 0.5, and x1 + x2 ≤ max l , 1 ;

The solutions (allocations, not payoffs):

x∗1 = 1.5− l , x∗2 = l − 0.5 l ≤ 5

6,

x∗1 =2

3· (max l , 1) , x∗2 =

1

3· (max l , 1) 5

6< l ≤ 1.5,

x∗1 = 1, x∗2 = l − 1 l > 1.5.


Introduction

Players’ payoffs

Suppose 2’s true type l is l∗ = 1.2. What is her payoff as a function of l?

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

0.25

0.5

0.75

1

1.25

Player 2's payoff

Player 1's payoff

Sum of players' payoffs


Introduction

Literature of cake-cutting

Continuous and heterogenous cake [0, 1] with utility 1 to any player.

Proportionality : among n players, each obtains utility of at least 1/n;

Envy-freeness: no player prefers any other player’s piece to her own;

Strategy-proofness: all players report preferences truthfully;

Piecewise uniformity/dichotomy : the cake can be divided into

acceptable and unacceptable pieces for any player.


Introduction

Literature of cake-cutting-continued

Mechanism Players Prop. EF SP Unif.

Cut-and-Choose 2 Yes Yes No No

Moving Knife n Yes No No No

Selfridge-Conway 3 Yes Yes No No

Chen et al. (2013) n Yes Yes Yes Yes

Brams and Taylor (1996), Robertson and Webb (1998), Procaccia (2013).


The Model

Routemap

1 Introduction

2 The Model

3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency

4 Discussion and Future Research


The Model

Readers and schedules

One book in a public library; Continuous time;

Fixed and finite number of readers: N ≡ 1, 2, · · · , n;

A schedule µ ≡ (µ1(·), µ2(·), · · · , µn(·)), where

µi (·) : R+ → 0, 1 , ∀i ∈ N;

A feasible schedule µ:

∑

i∈Nµi (r) ≤ 1, ∀r ∈ R+.

The set of feasible schedules is F .


The Model

Preferences and Pareto efficiency

Reader i ’s type: finite union of open intervals on R+, θi ∈ Θ;

Reader i ’s utility from a schedule µ:

ui (µ; θi ) ≡∫

θi

µi (r) dr ;

A schedule µ Pareto dominates another schedule µ′ if and only if

ui (µ; θi ) ≥ ui

(µ′; θi

), ∀i ∈ N

and the inequality is strict for at least one i .

A feasible schedule µ is Pareto efficient if and only if there does notexist another feasible µ′ that Pareto dominates µ.


The Model

Circulation mechanisms

Reader i ’s reported type: ti ∈ Θ;

Profile of all readers’ types: t ≡ (t1, t2, · · · , tn) ∈ Θn; t ≡ (ti , t−i );

A (offline) circulation mechanism is a function C : Θn → F ;

C (t) = µ ≡ (µ1(·), µ2(·), · · · , µn(·)) ⇐⇒ (Ci (t)) (·) = µi (·),

for all (t, i) ∈ Θn × N.


The Model

Strategy-proofness and Pareto efficiency

A mechanism C is strategy-proof if and only if

ui (C (θi , t−i ); θi ) ≥ ui (C (ti , t−i ); θi ) ,

for all (ti , t−i , θi , i) ∈(Θ×Θn−1 ×Θ× N

);

A strategy-proof mechanism is Pareto efficient if and only if

C (θ)

is a Pareto efficient schedule for all θ ∈ Θn.

Notation: ∀x ∈ Rn, x ≡ (x1, x2, · · · , xn).


The Results

Routemap

1 Introduction

2 The Model




The Results A constrained optimization problem

Routemap

1 Introduction

2 The Model





A constrained optimization problem: the components

An objective function G (x) ≡∑

i∈Ngi (xi ), where gi (·) : R+ → R is

continuous, strictly increasing, strictly concave for all i ∈ N;

∀t ∈ Θn and ∀S ⊆ N, let b(S ; t) ≡ λ(⋃

i∈S ti)—Lebesgue measure;

A budget:∑

i∈Sxi ≤ b(S ; t), referred to as binding when equal;

The budget set: B(t) ≡

x ∈ Rn+ :∑

i∈Sxi ≤ b(S ; t), ∀S ⊆ N

.

The optimum: x∗(t) ≡ arg maxx∈B(t)

G (x).



Interpretations

G (x) is a social welfare function: Nash collective utility;

(Bogomolnaia and Moulin (2004));

G (x) is Paretian: increase in any xi , ceteris paribus, increases G (x);

(Mas-Colell et al. (1995));

The budget set B(t) describes necessary conditions for the

existence of a feasible and Pareto efficient schedule;

Sufficiency will be established later. Feasibility



An example

Let t1 = (0, 12) ∪ (23, 35), t2 = (7, 17), and t3 = (13, 25);

(13 25)

(7 17)

(0 12) (23 35)

out of scale



An example-the optimum

g1(x1) = ln(x1), g2(x2) = 3 ln(x2), and g3(x3) = 2 ln(x3);

⇒ G (x) = ln(x1) + 3 ln(x2) + 2 ln(x3);

B(t) is given by

0 ≤ x1 ≤ 24, 0 ≤ x2 ≤ 10, 0 ≤ x3 ≤ 12; x1 + x2 + x3 ≤ 35;

x1 + x2 ≤ 29, x2 + x3 ≤ 18, x1 + x3 ≤ 34.

The optimum: x∗1 = 17, x∗2 = 10, and x∗3 = 8; Calculations


The Results A non-inferiority condition

Routemap

1 Introduction

2 The Model





A non-inferiority condition

Lemma

For any profile of reported types t, S ′ ⊆ N, and ∆S ′ > 0, let

B (t,∆S ′) ≡

x ∈ Rn+

∣∣∣∣∣∑

i∈S ′

xi ≤ b(S ′; t) + ∆S ′ ;

∑

i∈Sxi ≤ b(S ; t), ∀S ⊆ N and S 6= S ′

and x∗ ≡ arg maxx∈B(t,∆S′ )

G (x) and x∗ = arg maxx∈B(t)

G (x), then

x∗i ≥ x∗i ,∀i ∈ S ′.



Comments

A thought experiment on arbitrarily increasing the budget of a subset

of readers and see how their optimal reading times change;

The proof is based on the following observation: fix a vector x , if the

budgets for S1 ⊆ N and S2 ⊆ N are binding, then the budget for

S1 ∩ S2 is also binding. Calculations



Plan of proof of non-inferiority

Suppose, on the contrary, there is some i such that x∗i < x∗i ;

Name this group S2; Name the group S1 such that x∗i > x∗i ;

Look for a pair of vectors y∗, y∗ ∈ Rn+ such that y∗ + y∗ = x∗ + x∗,

and min x∗i , x∗i ≤ y∗i , y∗i ≤ max x∗i , x∗i , ∀i ∈ N

and x∗ 6= y∗ ∈ B and x∗ 6= y∗ ∈ B;



Plan of proof of non-inferiority: the goal

Concavity of G (x) implies that

G (x∗) ≥ G (y∗)⇒ G (y∗) ≥ G (x∗) .

Equivalently, look for two subsets S1 ⊆ S1 and S2 ⊆ S2 such that

∑

i∈Sx∗i = b(S)⇒

∣∣S ∩ S1

∣∣ =∣∣S ∩ S2

∣∣ ,∀S ⊂ S ′;

∑

i∈S

x∗i = b(S)⇒∣∣∣S ∩ S1

∣∣∣ =∣∣∣S ∩ S2

∣∣∣ ,∀S ⊂ S ′.



Sketch of proof of non-inferiority: notations

Define P ⊆ S1

P ≡

p ∈ S1

∣∣∣∣∣ ∃S ⊂ S ′ such that p ∈ S and∑

i∈Sx∗i = b(S)

;

S(p) ≡

S ⊂ S ′

∣∣∣∣∣ p ∈ S and∑

i∈Sx∗i = b(S)

;

The core of p at x∗: M(p) ≡⋂

S∈S(p)

S ,∀p ∈ P; Notice M(p) ∈ S(p);

Similarly for Q ⊆ S2, S(q), and M(q), ∀q ∈ Q.



Sketch of proof of non-inferiority: observations

O(p) ≡ M(p) ∩ S2 6= ∅, O(q) ≡ M(q) ∩ S1 6= ∅, ∀(p, q) ∈ P × Q;

S1 and S2: Black , P and Q: Green ;

O(p) and O(q): Blue , S1 and S2: Red ;

Figure : Groupings of readers

STRATEGY-PROOF AND EFFICIENT SCHEDULING 23

although S and S are only restricted to be proper subsets of S0, (29) holds trivially for S0 since S1 and S2621

have the same cardinality by (28). 622

Clearly, yi and y

i satisfy the conditions in (24) for all i and the inequalities are strict for at least one i if623

S1 6= ; 6= S2. To construct the the two subsets in (28) hence Lemma 4, I first show the following result.624

Lemma 5. If the budget for S is binding at x, then S \ S2 6= ;. Similarly, if the budget for S is binding625

at x, S \ S1 6= ;.626

Proof of Lemma 5. I will prove the first statement—the second follows identically. S \ S2 = ; implies that627

S \ S0 S1 [ Se 6= S0.

By Lemma 3, the budget for S \ S0 must also be binding at x. However, this implies628

b (S \ S0; t) =X

i2S\S0

xi <

X

i2S\S0

xi ) x /2 B(t,S0).

As Lemma 4 suggests, let629

P (

p 2 S1 : 9S S0 such that p 2 S andX

i2S

xi = b(S; t)

);

630

Q

8<:q 2 S2 : 9S S0 such that q 2 S and

X

i2S

xi = b(S; t)

9=; .

That is, P and Q are the set of readers in S1 and S2 such that there exists a group including these readers631

whose budgets are binding at their corresponding points. Furthermore, let632

S(p) (

S S0 : p 2 S andX

i2S

xi = b(S; t)

)and M(p)

\

S2S(p)

S, 8p 2 P ;

633

S(q)

8<:S S0 : q 2 S and

X

i2S

xi = b(S; t)

9=; and M(q)

\

S2S(q)

S, 8q 2 Q.

Let634

O(p) M(p) \ S2 6= ; if p 2 P 6= ;.635

O(q) M(q) \ S1 6= ; if q 2 Q 6= ;.M(p) (M(q)) is the intersection of all binding budgets at x (x) that include p (q). I shall refer to M(p)636

(M(q)) as the core of p (q) at x (x). The point reference may be dropped later since the hat-symbol637

should clarify when I am referring to x. The notion of the core is well-defined and nonempty (p 2 M(p),638

for example) as long as P and/or Q is nonempty. Two readers can share the same core. However, for each639

reader at each optimum, the core is uniquely defined since the intersection of any two distinct candidate640

cores is a proper subset of both, which denies either of the candidates of being a core.641

The following illustration presents an example of the various subgroups of readers.642

x1 · · · x

p · · · x|S1| x

|S1|+1 · · · x|S1|+|SE | x

|S1|+|SE |+1 · · · xq · · · x

|S0|

^ ^ ^ ^ ^ k k k _ _ _ _ _x

1 · · · xp · · · x

|S1| x|S1|+1 · · · x

|S1|+|SE | x|S1|+|SE |+1 · · · x

q · · · x|S0|

0BBBB@

1CCCCA



Algorithm O: initiation

S1 Without loss of generality, take i1 ∈ P in the first step and proceed to

the second step;

S2 In step two, take any i2 ∈ O(i1). If i2 /∈ Q, set S2 = i2 and

S1 = i1 and terminate. Otherwise, proceed to the next step;



Algorithm O: odd-numbered steps

In any odd-numbered step k ≥ 3, choose ik ∈ O(ik−1).

If ik /∈ P, terminate (“early termination” at k);

Set S1 = ik and S2 = ik−1.

Otherwise, if O(ik) ∩ i1, i2, · · · , ik−1 6= ∅,

⇒ l ≡ max l : il ∈ O(ik) ∩ i1, i2, · · · , ik−1 .

Terminate and save the ordered sequence(ik , il , il+1, · · · , ik−1

).

Otherwise, proceed to the next step.



Algorithm O: even-numbered steps

In any even-numbered step l ≥ 4, choose il ∈ O(il−1).

If il /∈ Q, terminate (“early termination” at l);

Set S1 = il−1 and S2 = il.

Otherwise, if O(il) ∩ i1, i2, · · · , il−1 6= ∅,

⇒ k ≡ max

l : il ∈ O(il) ∩ i1, i2, · · · , il−1.

Terminate and save the ordered sequence(ik , ik+1, · · · , il−1, il

).

Otherwise, proceed to the next step.



Algorithm O: after termination

After terminations that are not early,

Set S1 to be the set of all the odd-numbered terms in the saved

ordered sequence;

Set S2 the set of all the even-numbered terms, preserving the order.



Algorithm O: observations

It terminates in finitely many steps;

When it terminates, the saved ordered sequence contains equal

number of readers from S1 and S2;

Adjacent terms in the saved ordered sequence always appear in

binding constraints together;

For small enough ξ > 0 and S1 and S2 from Algorithm O,

y∗ = x∗ + ξ ·(1S1− 1S2

)∈ B;

y∗ = x∗ − ξ ·(1S1− 1S2

)∈ B.


The Results Feasibility and Pareto efficiency

Routemap

1 Introduction

2 The Model





Feasibility

Proposition (Existence of a feasible schedule)

Given any t ∈ Θn, for all x ∈ B(t), there exists a feasible schedule µ ∈ F

such that∫

ti

µi (r) dr = xi , (1)

where, recall, x ≡ (x1, x2, · · · , xn) and µ ≡ (µ1, µ2, · · · , µn).

Let F(x ; t) be the set of all such feasible schedules.



Proof of feasibility: the components

Given any t ≡ (t1, t2, · · · , tn),

Let Ti (ti ) be the set of all end points of ti ;

Let T (t) ≡⋃

i∈NTi (ti ) and m(t) ≡ |T (t)|; Let ε ∈ T (t);

Arrange the elements of T (t), indexed by j , in the increasing order;

T (t) =ε1(t), ε2(t), · · · , εm(t)(t)

.



Proof of feasibility: the network Γ(x ; t) Construction

Recall the example: t1 = (0, 12) ∪ (23, 35), t2 = (7, 17), and t3 = (13, 25);

⇒ T (t) = 0, 7, 12, 13, 17, 23, 25, 35 . Let x = (16, 5, 7)⇒ x ∈ B(t).

0∞

,,7 ∞ //

∞

))

i

16

&&

12∞

,,Ψ

7

==

5

88

1

44

4 //

6**

2

&&10

!!

13 ∞ //

∞

))

ii 5 // Ω

17∞

,,23 ∞ //

∞

<<

iii

7

88

25

∞

??



Max-flow-min-cut

Lemma

The maximum flow (and the minimum cut) of the network Γ(x ; t) is

∑

i∈Nxi .

A simple proof by contradiction since x ∈ B(t).



Proof of the max-flow by graph: a non-minimum cut

0∞

++7 ∞ //

∞

''

i

16

$$

12∞

++Ψ

7

@@

5

::

1

55

4 //

6))

2

$$10

13 ∞ //

∞

''

ii 5 // Ω.

17∞

++23 ∞ //

∞

>>

iii

7

::

25

∞

BB



Feasibility-the example

07

,,7

5

((

i

16

%%

12

Ψ

7

>>

5

99

1

44

4 //

6**

2

%%10

13

4

((

ii 5 // Ω

173

,,23

2

==

iii

7

99

25

7

@@

µ: i : (0, 7) ∪ (23, 32) ii : (7, 12) iii : (13, 20)



Mechanism G

Definition

For any G (x) satisfying the regularity conditions, let Mechanism G be

defined by

G(t) ∈ F (x∗(t); t) , ∀t ∈ Θn,

where, recall,

x∗(t) ≡ arg maxx∈B(t)

G (x).

F (x∗(t); t) is non-empty by the feasibility—mechanism G is well-defined.



Strategy-proofness and Pareto efficiency

Proposition

Mechanism G is strategy-proof and Pareto efficient.

Pareto efficiency is straight-forward to show: G (x) is a Paretian;

For strategy-proofness, suffices to show deviations to supersets and

subsets cannot be profitable;

Take ti 6⊂ θi and ti 6⊃ θi and θ′i ≡ ti ∩ θi 6= ∅. Take any t−i ,

ui (G (t) ; θi ) = ui

(G (t) ; θ′i

)≤ ui

(G(θ′i , t−i

); θ′i)

= ui

(G(θ′i , t−i

); θi)≤ ui (G (θi , t−i ) ; θi ) .



Strategy-proofness: deviations to subsets

Let B0 = B(θ). When ti ⊂ θi , let x(k)∗ ≡ arg maxx∈Bk

G (x), where

Bk ≡

x ∈ Rn

+

∣∣∣∣∣∣∑

i∈Sk′

xi ≤ b (Sk ′ ; t) , 1 ≤ k ′ ≤ k

and∑

i∈Sk′

xi ≤ b (Sk ′ ; θ) , k < k ′ ≤ 2n − 1

;

Let x∗ be the optimum when i reports truthfully. Clearly, x(0)∗ = x∗.

x(k)∗i ≤ x(k − 1)∗i ⇒ ui (G(t); θi ) ≤ ui (G(θ); θi ).



Strategy-proofness: deviations to supersets-1

Let x∗ be the optimum when i reports truthfully;

Arrange all subsets of N in a sequence, indexed by k , as follows:

1 Place all subsets that do not include i or subsets including i with

unbinding budgets at x∗ in the front of the sequence. Suppose there

are k such subsets;

2 The remaining subsets all have binding budgets at x∗ and include i ;

Among these, always place supersets before subsets.




Let B0 = B(θ) and x(0)∗ = x∗. Let M(i) be the core of i at x∗;

By construction, M(i) ⊆ Sk , ∀k > k;

Construct Bk by replacing only the budget b (Sk ; θ) in Bk−1 with

b (Sk ; t) for the k-th subset in the sequence constructed above;

Let x(k)∗ ≡ arg maxx∈Bk

G (x). Notice x(k)∗ = x(0)∗ = x∗, ∀k ≤ k ;




Let M ′(i) ≡ M(i)\ i.By non-inferiority,

x(k + 1)∗i ′ ≥ x(k)∗i ′ , ∀i ′ ∈ M ′(i) and k ≥ k .

ui (G(t); θi ) = λ

θi ∪

⋃

i ′∈M′(i)

ti ′

−

∑

i ′∈M′(i)

x(2n − 1)∗i ′

≤ λ

θi ∪

⋃

i ′∈M′(i)

ti ′

−

∑

i ′∈M′(i)

x∗i ′ = ui (G(θ); θi ).


Discussion and Future Research

Routemap

1 Introduction

2 The Model





Relations to Chen et al. (2013)

Corollary (Envy-freeness)

When gi (·) = g(·) for all i ∈ N, Mechanism G reduces to Mechanism 1 in

Chen et al. (2013), hence is proportional and envy-free.

Mechanism 1 in Chen et al. (2013)

Find and allocate ∪i∈S ti to S such that

S = arg minS⊆N

λ

(⋃

i∈Sti

)/|S |

;

Repeat with the remaining reading time and readers until all gone.



Guaranteed shares of demand

Corollary

Let G (x) =∑

i∈N αi ln(xi ) where αi > 0, ∀i and∑

i∈N αi = 1. Then

x∗i ≥ αi · λ (ti ) .

Modified Mechanism 1

Find and allocate ∪i∈S ti to S such that

S = arg minS⊆N

λ

(⋃

i∈Sti

)/(∑

i∈Sαi

);

Repeat with the remaining reading time and readers until all gone.



Future research

Online interval scheduling mechanisms;

Capacities and quotas;

Piecewise constant value densities.


Monotone comparative statics of constrained optimization

Paper Obj. fn. Budget set

Milgrom and Shannon (1994) Super-modular Strong set order

Antoniadou (2004) N.A. Direct value order

Quah (2007) S.M. & concave C-flexible order

Topkis (2001), Athey (2002), Vives (2001).


Let S0 = S1 ∩ S2 and S3 = S1 ∪ S2. First notice the simple fact that

⋃

i∈S1

ti

∩

⋃

i∈S2

ti

⊇

⋃

i∈S0

ti ,

b (S3; t) =b (S1; t) + b (S2; t)− λ

⋃

i∈S1

ti

∩

⋃

i∈S2

ti

≤b (S1; t) + b (S2; t)− b (S0; t)


∑

i∈S1

xi +∑

i∈S2

xi =∑

i∈S3

xi +∑

i∈S0

xi ≤ b (S3; t) +∑

i∈S0

xi

≤ b (S1; t) + b (S2; t)− b (S0; t) +∑

i∈S0

xi

≤ b (S1; t) + b (S2; t) ,

∑

i∈S1

xi +∑

i∈S2

xi = b (S1; t) + b (S2; t)

⇒∑

i∈S0

xi = b(S0; t).


Calculation of the example

Let η(S) be the Lagrange multiplier of S ⊆ N. The Lagrangian is

L = G (x) +∑

S⊆Nη(S) ·

(b(S ; t)−

∑

i∈Sxi

).

In the example, the optimal values of the Lagrange multipliers are

η∗(2) =1

20, η∗ (2, 3) =

13

68, and η∗ (1, 2, 3) =

1

17.

And η∗(S) = 0 for all other S , by the KKT conditions.


Proof of feasibility: the network

Definition

Define the following directed network Γ(x ; t):

create a source Ψ,

a node (called a bin node) for each bin (εj(t), 1 ≤ j ≤ m(t − 1)),

a node (called a reader node) for each reader (represented in

lowercase Roman numerals),

and a sink Ω.

Create the arcs as follows.


Definition of the network-continued

Connect the source to all bin nodes with arcs pointing to the binnodes—I will refer to such arcs as source-bin arcs;

Assign capacity (εj+1(t)− εj(t)) to the arc connecting the source tothe node corresponding to the bin with end points εj(t) and εj+1(t);

Connect the reader nodes to the sink with arcs pointing to the sink—Iwill refer to such arcs as reader-sink arcs;

Assign capacity xi to the arc connecting reader i ’s node to the sink;

Connect a bin node to a reader node with an arc pointing to thereader node—referred to as a bin-reader arc—if and only if the bin isa subset of the reader’s reported demand;

Assign infinite capacities to all bin-reader arcs.


References I

Antoniadou, E. (2004, June). Lattice programming and consumer theorypart ii: Comparative statics with many goods.http://ssrn.com/abstract=1370352.

Athey, S. (2002). Monotone comparative statics under uncertainty. TheQuarterly Journal of Economics 117(1), 187–223.

Bogomolnaia, A. and H. Moulin (2004). Random matching underdichotomous preferences. Econometrica 72(1), 257–279.

Brams, S. and A. Taylor (1996). Fair Division: From Cake-Cutting toDispute Resolution. Cambridge University Press.

Chen, Y., J. K. Lai, D. C. Parkes, and A. D. Procaccia (2013). Truth,justice, and cake cutting. Games and Economic Behavior 77(1), 284 –297.


References II

Mas-Colell, A., M. D. Whinston, J. R. Green, et al. (1995).Microeconomic theory, Volume 1. Oxford University Press New York.

Milgrom, P. and C. Shannon (1994). Monotone comparative statics.Econometrica 62(1), 157–180.

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Strategy-proof and Efficient Scheduling · Strategy-proofness: all players report preferences truthfully; Piecewise uniformity/dichotomy: the cake can be divided into acceptable and