math.huji.ac.ilmath.huji.ac.il/~hart/papers/st-ne-p.pdf · Law: Right to Remain Silent U.S.: "You have the right to remain silent. Anything you say can and will be used against you

Evidence Games:Truth and Commitment

Sergiu Hart

This version: September 2016

SERGIU HART c© 2015 – p. 1

Evidence Games:Truth and Commitment

Sergiu HartCenter for the Study of Rationality

Dept of Economics Dept of MathematicsThe Hebrew University of Jerusalem

[email protected]://www.ma.huji.ac.il/hart


http://www.ma.huji.ac.il/hart

Joint work with

Ilan KremerMotty Perry

Hebrew University of JerusalemUniversity of Warwick


Papers


http://www.ma.huji.ac.il/hart/abs/st-ne.html

http://www.ma.huji.ac.il/hart/abs/st-ne-mixed.html

Papers

"Evidence Games: Truth and Commitment"Center for Rationality DP-684, May 2015Revised September 2016American Economic Review (forthcoming)

www.ma.huji.ac.il/hart/abs/st-ne.html




Papers

"Evidence Games: Truth and Commitment"Center for Rationality DP-684, May 2015Revised September 2016American Economic Review (forthcoming)

www.ma.huji.ac.il/hart/abs/st-ne.html

"Evidence Games with RandomizedRewards"

Center for Rationality 2016

www.ma.huji.ac.il/hart/abs/st-ne-mixed.html





Q: "Do you deserve a pay raise?"


Q: "Do you deserve a pay raise?"A: "Of course."



Q: "Are you guilty and deserve punishment?"



Q: "Are you guilty and deserve punishment?"A: "Of course not."




How can one obtain reliable information?





How can one determine the "right" reward, orpunishment?





How can one determine the "right" reward, orpunishment?

How can one "separate" and avoid"unraveling" (Akerlof 70)?


Evidence Games: General Setup



AGENT who is informed




PRINCIPAL who takes decision but isuninformed




PRINCIPAL who takes decision but isuninformed

Agent TRANSMITS information to Principal(costlessly)


Two Setups


Two Setups

SETUP 1: Principal decides after receivingAgent’s message


Two Setups


SETUP 2: Principal chooses a policy beforeAgent’s message


Two Setups



policy : a function that assigns a decisionof Principal to each message of Agent


Two Setups



policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)


Two Setups



policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)Principal is committed to the policy


Two Setups

GAME: Principal decides after receivingAgent’s message

MECHANISM: Principal chooses a policybefore Agent’s message

policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)Principal is committed to the policy


Main Result


Main Result

In EVIDENCE GAMES


Main Result

In EVIDENCE GAMES

the GAME EQUILIBRIUM outcome(obtained without commitment )


Main Result

In EVIDENCE GAMES


and the OPTIMAL MECHANISM outcome(obtained with commitment )


Main Result

In EVIDENCE GAMES



COINCIDE


Main Result: Equivalence

In EVIDENCE GAMES



COINCIDE


Evidence Games


Evidence Games

AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )


Evidence Games


PRINCIPAL decides on the reward


Evidence Games



PRINCIPAL wants the reward to be as closeas possible to the value


Evidence Games




AGENT wants the reward to be as high aspossible (same preference for all types )


Evidence Games




AGENT wants the reward to be as high aspossible (same preference for all types )

↑Differs from signalling, screening,

cheap-talk, ...SERGIU HART c© 2015 – p. 9

Evidence Games: Truth



Agent reveals:

"the truth, nothing but the truth"



Agent reveals:


NOT necessarily "the whole truth"



Agent reveals:


all the evidence that the agent reveals mustbe true (it is verifiable)




Agent reveals:




the agent does not have to reveal all theevidence that he has





"truth-leaning"



"truth-leaning"

revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability


Previous Work


Previous Work

UnravelingGrossman and O. Hart 1980Grossman 1981Milgrom 1981


Previous Work


Voluntary disclosureDye 1985Shin 2003, 2006, ...


Previous Work



MechanismGreen–Laffont 1986


Previous Work



MechanismGreen–Laffont 1986

Persuasion gamesGlazer and Rubinstein 2004, 2006


Evidence Games


Evidence Games

EVIDENCE GAMES model very commonsetups


Evidence Games


In EVIDENCE GAMES there is equivalencebetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)


Evidence Games


In EVIDENCE GAMES there is equivalencebetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)

The conditions of EVIDENCE GAMES areindispensable for this equivalence


Example 1


Example 1

Professor wants salary as high as possible


Example 1


Dean wants salary to be as close as possibleto the Professor’s value


Example 1



Professor’s evidence (verifiable):


Example 1



Professor’s evidence (verifiable):[t0] 50%: no evidence[t+] 25%: positive evidence[t−] 25%: negative evidence


Example 1



Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60

[t+] 25%: positive evidence → value = 90

[t−] 25%: negative evidence → value = 30


Example 1 t+ : 25% 90

t0 : 50% 60

t− : 25% 30







Example 1: Equilibrium t+ : 25% 90

t0 : 50% 60

t− : 25% 30



t0 : 50% 60

t− : 25% 30

GAME: (G1) Professor provides evidence(G2) then Dean sets salary



t0 : 50% 60

t− : 25% 30


EQUILIBRIUM



t0 : 50% 60

t− : 25% 30


EQUILIBRIUMProfessor:

t+ provides positive evidencet0, t− provide no evidence



t0 : 50% 60

t− : 25% 30




Dean:to positive evidence gives salary = 90

to negative evidence gives salary = 30



t0 : 50% 60

t− : 25% 30






to no evidence gives salary = 50= (50% · 60 + 25% · 30)/(50% + 25%)



t0 : 50% 60

t− : 25% 30


unique sequential EQUILIBRIUMProfessor:




to no evidence gives salary = 50= (50% · 60 + 25% · 30)/(50% + 25%)



t0 : 50% 60

t− : 25% 30



t0 : 50% 60

t− : 25% 30

30 60 90value:



t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+



t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

partial truth:



t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

partial truth:

prof says:



t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

partial truth:

prof says:

dean pays: 30 50 90


Example 1: Mechanism t+ : 25% 90

t0 : 50% 60

t− : 25% 30



t0 : 50% 60

t− : 25% 30

MECHANISM: (M1) Dean commits to salary policy(M2) then Professor provides evidence



t0 : 50% 60

t− : 25% 30


OPTIMAL MECHANISM



t0 : 50% 60

t− : 25% 30


OPTIMAL MECHANISM


to no evidence gives salary = 50

to negative evidence gives salary ≤ 50



t0 : 50% 60

t− : 25% 30


OPTIMAL MECHANISM




OPTIMAL MECHANISM = EQUILIBRIUMSERGIU HART c© 2015 – p. 17

Example 1: Explanation t+ : 25% 90

t0 : 50% 60

t− : 25% 30



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

the value of t0 is higher than the value of t−



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0


in MECHANISM:



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0


in MECHANISM:

the only way to separate t− from t0is to pay t− strictly more than to t0



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0


in MECHANISM:


this is not optimal



t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0


in MECHANISM:


this is not optimal

OPTIMAL MECHANISM does notseparate more than EQUILIBRIUM



t0 : 50% 60

t− : 25% 30




t0 : 50% 60

t− : 25% 30

30 60

t− t0

partial truth:



Example 2

y


Example 2





Example 2






Example 2






[t±] 5%: both evidences → value = 42


Example 2 t+ : 20% 102

t0 : 50% 60

t± : 5% 42

t− : 25% 30






[t±] 5%: both evidences → value = 42



t0 : 50% 60

t± : 5% 42

t− : 25% 30



t0 : 50% 60

t± : 5% 42

t− : 25% 30EQUILIBRIUM



t0 : 50% 60

t± : 5% 42


Professor:t+, t± provide positive evidencet0, t− provide no evidence



t0 : 50% 60

t± : 5% 42


Professor:t+, t± provide positive evidencet0, t− provide no evidence

Dean:to positive evidence gives salary = 90= (20% · 102 + 5% · 42)/25%

to no evidence gives salary = 50= (50% · 60 + 25% · 30)/75%


to both evidences gives salary = 42SERGIU HART c© 2015 – p. 21


t0 : 50% 60

t± : 5% 42

t− : 25% 30



t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:



t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±



t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

partial truth:



t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:



t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:

dean pays: 30 5042 90



t0 : 50% 60

t± : 5% 42

t− : 25% 30



t0 : 50% 60

t± : 5% 42

t− : 25% 30OPTIMAL MECHANISM



t0 : 50% 60

t± : 5% 42





to both evidences gives salary ≤ 90



t0 : 50% 60

t± : 5% 42





to both evidences gives salary ≤ 90

OPTIMAL MECHANISM = EQUILIBRIUM



t0 : 50% 60

t± : 5% 42

t− : 25% 30



t0 : 50% 60

t± : 5% 42

t− : 25% 30Another equilibrium



t0 : 50% 60

t± : 5% 42


Professor:always provides no evidence



t0 : 50% 60

t± : 5% 42



Dean:ignores all evidence and gives salary = 60= 50%·60+25%·30+20%·102+5%·42



t0 : 50% 60

t± : 5% 42




supported by the belief of the Dean whenreceiving the out-of-equilibrium positive evidencethat it mostly comes from t± rather than from t+



t0 : 50% 60

t± : 5% 42

t− : 25% 30Another equilibrium ("babbling" )






t0 : 50% 60

t± : 5% 42

t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS






t0 : 50% 60

t± : 5% 42




t0 : 50% 60

t± : 5% 42


30 42 60 102value:

t− t0 t+t±

partial truth:



t0 : 50% 60

t± : 5% 42


30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:



t0 : 50% 60

t± : 5% 42


30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:

dean pays: 30 6042 < 60



Example 3

y


Example 3



Example 3




Example 3



Dean wants salary close to Professor’s value


Example 3




Professor t0 wants salary high


Example 3





Professor t− wants salary close to 50


Example 3


Example 3

EQUILIBRIUM :

separation


Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30


Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)


Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)⇒ contradiction


Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)⇒ contradiction

⇒ no separation (babbling), x = 45


Example 3

EQUILIBRIUM : no separation , x = 45


Example 3


MECHANISM:


Example 3


MECHANISM:




Example 3


MECHANISM:


to no evidence gives salary = 71(separating: t− prefers 30 to 71)


Example 3


MECHANISM:



REQUIRES COMMITMENT :

after no evidence Dean prefers salary = 60


Example 3


MECHANISM:





MECHANISM is strictly better for the Dean thanEQUILIBRIUM


Example 3: Commitment Helps


MECHANISM:





MECHANISM is strictly better for the Dean thanEQUILIBRIUM
























↑NOT "evidence game"



Examples: Summary


Examples: Summary

Example 1 :equivalence (our result)


Examples: Summary


Example 2 :EQUILIBRIUM different fromOPTIMAL MECHANISM

⇒ "truth-leaning"


Examples: Summary


Example 2 :EQUILIBRIUM different fromOPTIMAL MECHANISM

⇒ "truth-leaning"

Example 3 :assumptions do not holdresult fails: commitment helps


Model


Model

AGENT (A)

PRINCIPAL (P ) (= "market")


Model

AGENT (A)


(finite) set of TYPES: T

the type t ∈ T is chosen according to aprobability distribution p ∈ ∆(T )


Model

AGENT (A)




the type t ∈ T is revealed to Agent and not toPrincipal


Model

AGENT (A)





Agent’s MESSAGE: s ∈ T


Model

AGENT (A)





Agent’s MESSAGE: s ∈ T

Principal’s DECISION: REWARD x ∈ R


Payoffs / Utilities


Payoffs / Utilities

UA and UP do not depend on the message s


Payoffs / Utilities


UA does not depend on the type t

UA(s, x; t) = x


Payoffs / Utilities



UA(s, x; t) = x

UP : "Canonical" example

ht(x) := UP (s, x; t) = −(x − v(t))2

v(t) = the "value" of type t (to PRINCIPAL )


Payoffs / Utilities



UA(s, x; t) = x

UP : "Canonical" example

ht(x) := UP (s, x; t) = −(x − v(t))2

v(t) = the "value" of type t (to PRINCIPAL )

General assumption:(SP) UP is single-peaked w.r.t. UA


Single Peakedness (SP)



For every distribution of types (belief) q ∈ ∆(T )

the principal’s expected utilityhq(x) =

∑t∈T qt ht(x)

is a single-peaked function of the reward x



For every distribution of types (belief) q ∈ ∆(T )

the principal’s expected utilityhq(x) =

∑t∈T qt ht(x)

is a single-peaked function of the reward x

⇔ There exists v(q) such thath′

q(x) > 0 for x < v(q)

h′q(x) = 0 for x = v(q)

h′q(x) < 0 for x > v(q)





Canonical example:ht(x) = −(x − v(t))2




v(q) = Eq[v(t)] =∑

t qt v(t)




v(q) = Eq[v(t)] =∑

t qt v(t)

More general:ht(x) is a differentiable strictly concavefunction of x, for each t




v(q) = Eq[v(t)] =∑

t qt v(t)

More general:ht(x) is a differentiable strictly concavefunction of x, for each t

(SP) is more general than concavity


Evidence and Truth


Evidence and Truth

Agent reveals:



Evidence and Truth

Agent reveals:




Evidence and Truth

Agent reveals:





Evidence and Truth

Agent reveals:






Evidence and Truth

Agent reveals:





⇒ Agent can pretend to be a type thathas less evidence


Evidence and Truth


Evidence and Truth

L(t) = the set of types that t can pretend to be= the set of possible messages of type t


Evidence and Truth


E = set of (verifiable) pieces of evidence


Evidence and Truth



Et ⊆ E = set of pieces of evidence that t has(and can provide)


Evidence and Truth




L(t) = {s ∈ T : Es ⊆ Et}


Evidence and Truth




L(t) = {s ∈ T : Es ⊆ Et}

(L1) Reflexivity : t ∈ L(t)


Evidence and Truth




L(t) = {s ∈ T : Es ⊆ Et}


(L2) Transitivity : If s ∈ L(t) and r ∈ L(s)then r ∈ L(t)


Evidence and Truth




L(t) = {s ∈ T : Es ⊆ Et}


(L2) Transitivity : If s ∈ L(t) and r ∈ L(s)then r ∈ L(t)

Assume only (L1) and (L2) (more general)SERGIU HART c© 2015 – p. 37

Type (summary)


Type (summary)

A type t has two characteristics:


Type (summary)


Value to the Principal(expressed by h(t) and its peak v(t))


Type (summary)



Evidence that he can provide(expressed by L(t))


Type (summary)



Evidence that he can provide(expressed by L(t))

No relation is assumedbetween Value and Evidence


Game


Game

(G1) Agent sends message s ∈ L(t) to Principal


Game


(G2) Then Principal sets reward x ∈ R


Game



STRATEGIES


Game



STRATEGIES

(Agent) σ(s|t) = probability that type t sendsmessage s in L(t)


Game



STRATEGIES

(Agent) σ(s|t) = probability that type t sendsmessage s in L(t)

(Principal) ρ(s) ∈ R = reward to message s


Equilibrium


Equilibrium

(σ, ρ) is a NASH EQUILIBRIUM if


Equilibrium


(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)


Equilibrium


(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

(P) σ̄(s) > 0 ⇒ ρ(s) = v(q(s))

whereσ̄(s) = total probability of message s

q(s) ∈ ∆(T ) = the posterior distributionon types conditional on message s


Equilibrium


(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

(P) σ̄(s) > 0 ⇒ ρ(s) = v(q(s))



OUTCOME: πt = maxs∈L(t) ρ(s) = ρ(σ(·|t))


Equilibrium


(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

(P) σ̄(s) > 0 ⇒ ρ(s) = v(q(s))



OUTCOME: πt = maxs∈L(t) ρ(s) = ρ(σ(·|t))

π = (πt)t∈T ∈ RT


Truth-Leaning


Truth-Leaning

Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability


Truth-Leaning


(T1) Revealing the whole truth is preferablewhen the reward is the same


Truth-Leaning


(T1) Revealing the whole truth is preferablewhen the reward is the same(lexicographic preference)


Truth-Leaning



(T2) The whole truth is revealed with infinitesimalpositive probability


Truth-Leaning



(T2) The whole truth is revealed with infinitesimalpositive probability(by mistake, or because the agent may benon-strategic, or ... [UK])


Truth-Leaning


Truth-Leaning

A Nash equilibrium is TRUTH-LEANING if itsatisfies:


Truth-Leaning


(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1


Truth-Leaning


(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1

(if message t is a best reply for type t then itis used for sure)


Truth-Leaning


(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1


(T2) σ̄(t) = 0 ⇒ ρ(t) = v(t)


Truth-Leaning


(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1


(T2) σ̄(t) = 0 ⇒ ρ(t) = v(t)

(if message t is not used then the rewardequals the value of type t; i.e., the belief isthat the [unexpected] message t comes fromtype t)


Truth-Leaning


Truth-Leaning

Truth-Leaning equilibria:


Truth-Leaning


coincide with the equilibria selected in the"voluntary disclosure" literature


Truth-Leaning



satisfy all the refinement conditions in theliterature


Truth-Leaning




eliminate "unreasonable" equilibria (such as"babbling" in Example 2)


Truth-Leaning




eliminate "unreasonable" equilibria (such as"babbling" in Example 2)

...


Mechanism


Mechanism

MECHANISM:


Mechanism

MECHANISM: Reward scheme ρ : T → R

(ρ(s) = reward to message s)


Mechanism



Agent’s payoff when type is t:

πt = maxs∈L(t)

ρ(s)


Mechanism




πt = maxs∈L(t)

ρ(s)

Outcome : π = (πt)t∈T ∈ RT


Mechanism




πt = maxs∈L(t)

ρ(s)

Outcome : π = (πt)t∈T ∈ RT

Principal’s payoff :

H(π) =∑

t∈T

pt ht(πt)


Incentive Compatibility



Outcome π = (πt)t∈T ∈ RT is generated by a

mechanism ρ




mechanism ρ

if and only if




mechanism ρ

if and only if

πt ≥ πs

for all s, t ∈ T with s ∈ L(t)




mechanism ρ

if and only if

πt ≥ πs


Immediate because L satisfies reflexivity (L1)and transitivity (L2)




mechanism ρ

if and only if

πt ≥ πs



"direct" mechanism: ρ(t) = πt




mechanism ρ

if and only if

πt ≥ πs



"direct" mechanism: ρ(t) = πt

Green–Laffont 86SERGIU HART c© 2015 – p. 45

Optimal Mechanism


Optimal Mechanism

OPTIMAL MECHANISM :


Optimal Mechanism

OPTIMAL MECHANISM :

Maximize H(π) =∑

t∈T pt ht(πt)


Optimal Mechanism

OPTIMAL MECHANISM :

Maximize H(π) =∑

t∈T pt ht(πt)

subject to (IC):

πt ≥ πs



Optimal Mechanism

OPTIMAL MECHANISM :

Maximize H(π) =∑

t∈T pt ht(πt)

subject to (IC):

πt ≥ πs


OPTIMAL MECHANISM = Maximum underIncentive Constraints


Main Result


Main Result

(i) There is a uniqueTRUTH-LEANING EQUILIBRIUM outcome.


Main Result


(ii) There is a uniqueOPTIMAL MECHANISM outcome.


Main Result



(iii) These two outcomes COINCIDE.





(iii) These two outcomes COINCIDE.





The equilibrium strategies need not be unique(happens only when the Agent is indifferent—andthen the Principal is also indifferent)





All the conditions are indispensable :




Truth structure: reflexivity





Truth structure: transitivity






Truth Leaning: whole truth slightly better







Truth Leaning: whole truth slightly possible








Agent’s utility: independent of type








Agent’s utility: independent of type

Principal’s utility: single-peaked with respectto Agent’s utility





EQUILIBRIUM (without commitment)yields the same as COMMITMENT




EQUILIBRIUM yields OPTIMAL SEPARATION(for the principal / "market")





EQUILIBRIUM yields PARETO EFFICIENCY(in the canonical setup)



UNDER INCENTIVE CONSTRAINTS



EQUILIBRIUM yields PARETO EFFICIENCY(in the canonical setup)


Applications


Applications: Finance

Disclosure by public firms




Disclosing false information is a criminal act





Withholding information is allowed in somecases





Withholding information is allowed in somecases, and is difficult (if not impossible) todetect





Withholding information is allowed in somecases, and is difficult (if not impossible) todetect

Impacts asset prices (e.g.: quarterly reports)





Our result:



Our result:Market’s behavior in EQUILIBRIUM yields theOPTIMAL SEPARATION between "good" and "bad"firms (even if mechanisms and commitmentswere possible)



Our result:Market’s behavior in EQUILIBRIUM yields theOPTIMAL SEPARATION between "good" and "bad"firms (even if mechanisms and commitmentswere possible)

The equilibria considered in this literature turnout to be the truth-leaning equilibria


Applications: Law


Applications: Law


Applications: Law

Commitments: constitutions, laws, legaldoctrine, precedents, ...


Applications: Law


Affect evidence provided in court


Applications: Law



Our result:The power of these commitments may not gobeyond selecting the truth-leaning equilibria


Applications: Law



Our result:The power of these commitments may not gobeyond selecting the truth-leaning equilibria– which are most natural here


Law


Law: Right to Remain Silent

U.S.: "You have the right to remain silent.Anything you say can and will be usedagainst you in a court of law ..."(Miranda Warning, following the 1966Miranda v. Arizona Supreme Court decision)




U.K.: "You do not have to say anything. But itmay harm your defence if you do not mentionwhen questioned something which you laterrely on in court. Anything you do say ..."(Criminal Justice and Public Order Act 1994)













(TRUTH-LEANING )





EQUILIBRIA entail (Bayesian) inferences




Our equivalence result implies that the sameinferences apply to OPTIMAL MECHANISMS





Therefore: Committing to not making adverseinferences ("RIGHT TO REMAIN SILENT") isNOT OPTIMAL






Instead: TRUTH-LEANING , which is OPTIMAL






Instead: TRUTH-LEANING , which is OPTIMAL(reinforce the advantages of truth-telling)


Medical Over-Treatment



Doctors and hospitals prefer to over-treat asthey are paid more when doing so



Doctors and hospitals prefer to over-treat asthey are paid more when doing so

Give doctors incentives to provide evidence


Equivalence Theorem: Intuition



In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:




s reveals his type (i.e., says s)





Else: s has a strictly better message (by (T1)),and then so does t (by transitivity)









v(t) < v(s)





v(t) < v(s)

No one pretends to be worth less than they are





v(t) < v(s)

v(t) < πt = πs ≤ v(s)





v(t) < v(s)

NOTE: These conclusions need not hold forequilibria that are not truth-leaning





v(t) < v(s)





v(t) < v(s)

⇒ t and s cannot be separated in any OPTIMALMECHANISM





v(t) < v(s)


- To separate: ρ(t) > ρ(s) (else t says s)





v(t) < v(s)


- To separate: ρ(t) > ρ(s) (else t says s)- Not optimal: decreasing ρ(t) or increasing ρ(s)brings rewards closer to values





v(t) < v(s)






v(t) < v(s)


CONCLUSION:OPTIMAL MECHANISM cannot separate

more than TRUTH-LEANING EQUILIBRIUM


Equivalence Theorem: Proof



0. Preliminaries



0. Preliminaries

1. Every TL-EQUILIBRIUM outcome equalsthe unique OPTIMAL MECHANISM outcome



0. Preliminaries

1. Every TL-EQUILIBRIUM outcome equalsthe unique OPTIMAL MECHANISM outcome

2. A TL-EQUILIBRIUM exists


Proof: 0. Preliminaries



"In betweenness" : v(t1) ≦ v(t2) implies

v(t1) ≦ v({t1, t2}) ≦ v(t2)




v(t1) ≦ v({t1, t2}) ≦ v(t2)

More generally: if q ∈ ∆(T ) is a weightedaverage of q1, q2, ..., qn ∈ ∆(T ) then

mini

v(qi) ≦ v(q) ≦ maxi

v(qi)




v(t1) ≦ v({t1, t2}) ≦ v(t2)

More generally: if q ∈ ∆(T ) is a weightedaverage of q1, q2, ..., qn ∈ ∆(T ) then

mini

v(qi) ≦ v(q) ≦ maxi

v(qi)

Proof : Follows from single-peakedness (SP)and differentiability





Let (σ, ρ) be a TL-EQUILIBRIUM with outcome π.Then:




message t is used in equilibrium:σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

message t is not used in equilibrium:σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)






Pf.If σ(t|t) = 0 then π(t) > ρ(t) = v(t) (by (T2)).If σ(t|t) > 0 then: σ(t|s) > 0 for s 6= t impliesπt = πs > v(s); but πt = v(q(t)) and soπt ≤ v(t) by in-betweeness. 2






Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).







Pf. v(s) ≥ ρ(s)

(s is used)







Pf. v(s) ≥ ρ(s) = πt

(s is optimal for t)







Pf. v(s) ≥ ρ(s) = πt > v(t)

(t is not used)







Pf. v(s) ≥ ρ(s) = πt > v(t) 2







Pf. v(s) ≥ ρ(s) = πt > v(t) 2

Note. May not hold for NON-TL-equilibria.SERGIU HART c© 2015 – p. 60

Proof: 1. equilibrium → mechanism



Let (σ, ρ) be TL-EQUILIBRIUM , with outcome π.




Special Case :There is a single message s that is used(i.e., σ(s|t) = 1 for all t).





⇒ πt = ρ(s) = v(T ) for all t






⇒ v(t) < v(T ) ≤ v(s) for all t 6= s






⇒ v(t) < v(T ) ≤ v(s) for all t 6= s

⇒ π is the unique OPTIMAL MECHANISM

Pf. π is optimal even if we keep only the(IC) constraints πt ≥ πs for all t 6= s,






⇒ v(t) < v(T ) ≤ v(s) for all t 6= s


Pf. π is optimal even if we keep only the(IC) constraints πt ≥ πs for all t 6= s,because v(t) < v(T ) ≤ v(s) for all t 6= s



Special Case :



Special Case :

v(t) v(t′) v(s)

t t′ s

L:



Special Case :

v(t) v(t′) v(s)

t t′ s

L:v(T )



Special Case :

v(t) v(t′) v(s)

t t′ s

L:v(T )

IC: πs ≤ πt

πs ≤ πt′





General Case.



General Case. For each message s that isused (i.e., σ̄(s) > 0)



General Case. For each message s that isused (i.e., σ̄(s) > 0) apply the Special Casewith:




set of types = Ts := {t : σ(s|t) > 0}(the types that use message s)




set of types = Ts := {t : σ(s|t) > 0}(the types that use message s)probability distribution = q(s)(the posterior given message s)





⇒ π restricted to Ts is the unique OPTIMALMECHANISM, for each s





⇒ π restricted to Ts is the unique OPTIMALMECHANISM, for each s



Proof: 2. existence of TL-equilibrium



For every ε > 0, let Γε be the perturbation of theGAME Γ:




UA = x + ε1s=t

(revealing the whole truth increases agent’spayoff by ε)




UA = x + ε1s=t

(revealing the whole truth increases agent’spayoff by ε)

σ(t|t) ≥ ε(probability of revealing the whole truth is atleast ε)





Proposition. Γε has a Nash equilibrium.




Proof. Standard use of Kakutani’s Fixed PointTheorem.





Proposition. A limit point (σ, ρ) of Nashequilibria (σε, ρε) of Γε yields aTL-EQUILIBRIUM (σ′, ρ) of Γ.






Proof.






Proof.σ(t|t) < 1 ⇒






Proof.σ(t|t) < 1 ⇒ σ(t|r) = 0 for all r







⇒ q(t) = 1t







⇒ q(t) = 1t

⇒ ρ(t) = v(t)







⇒ q(t) = 1t

⇒ ρ(t) = v(t)

If t ∈ BRA(t) then put σ′(t|t) = 1SERGIU HART c© 2015 – p. 65

Proof: 2’. mechanism → equilibrium

y



Let π be an OPTIMAL MECHANISM outcome.



Let π be an OPTIMAL MECHANISM outcome.

We will construct a TL-EQUILIBRIUM (σ, ρ) withoutcome π.





Recall that in a TL-EQUILIBRIUM we have

σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)




σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)

Principal’s strategy: put

ρ(t) = min{πt, v(t)} for each t




σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)

Agent’s strategy:




σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)

Agent’s strategy:

Let S := {t : v(t) ≥ πt} – these are themessages that will be used in equilibrium




σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)

Agent’s strategy:


If t ∈ S then put σ(t|t) = 1




σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)

Agent’s strategy:


If t ∈ S then put σ(t|t) = 1

If t /∈ S then put σ(t|t) = 0

We need to determine which messages(in S) type t /∈ S will choose



Agent’s strategy:

S := {t : v(t) ≥ πt} (messages used)

We need to determine which messages in Stypes t /∈ S will use



Agent’s strategy:



For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)



Agent’s strategy:




A simple case: S is a singleton



Agent’s strategy:




A simple case: S is a singleton

The general case: Partition T into disjointsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S


Hall’s Marriage Theorem



A set B of n boys, and a set G of n girls




Each boy b ∈ B knows a subset Gb ⊆ G ofgirls





Matching: one-to-one pairing of boys withgirls






Necessary condition for a matching to exist:







Every set of k boys knows at least k girls| ∪b∈C Gb| ≥ |C| for every C ⊆ B







Every set of k boys knows at least k girls| ∪b∈C Gb| ≥ |C| for every C ⊆ B

Theorem (Hall 1935). The condition is alsosufficient .


The Hull of Hall’s Theorem



Finite sets B and G



Finite sets B and G

For each b ∈ B a subset Gb ⊆ G



Finite sets B and G


Measures β on B and γ on G such that



Finite sets B and G


Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B



Finite sets B and G



Theorem . There exists a partitionof ∪b∈BGb into disjointsets (Fb)b∈B such that for every b ∈ B



Finite sets B and G




Fb ⊆ Gb



Finite sets B and G




Fb ⊆ Gb

γ(Fb) = β({b})



Finite sets B and G



Theorem . There exists a partitionof ∪b∈BGb into disjoint fractionalsets (Fb)b∈B such that for every b ∈ B

Fb ⊆ Gb

γ(Fb) = β({b})



Finite sets B and G



Theorem . There exists a partitionof ∪b∈BGb into disjoint fractionalsets (Fb)b∈B such that for every b ∈ B

Fb ⊆ Gb

γ(Fb) = β({b})

Proof. Hart–Kohlberg 74SERGIU HART c© 2015 – p. 70


Agent’s strategy:




Partition T into disjointsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S



Agent’s strategy:




Partition T into disjoint fractionalsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S



Agent’s strategy:




Partition T into disjoint fractionalsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S↔ the strategy σ



Glazer and Rubinstein (2004, 2006)



Messages : arbitrary



Messages : arbitrary ⇒ no "truth" structure




Result :{optimal mechanisms} ⊆ {equilibria}





Rewards : mixtures of two outcomes





Rewards : mixtures of two outcomes⇒ linear ht






Extended to concave ht: Sher (2011)






Extended to concave ht: Sher (2011)General condition:"G ENERALIZED SINGLE-PEAKEDNESS "






Extended to concave ht: Sher (2011)General condition:"G ENERALIZED SINGLE-PEAKEDNESS "⇔ "P RINCIPAL’S UNIFORM BEST"(includes convex ht, ...)


Summary


Summary



Summary


In EVIDENCE GAMES there is EQUIVALENCEbetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)


Summary



EQUILIBRIUM is CONSTRAINED EFFICIENT(in the canonical case)


Summary



EQUILIBRIUM is CONSTRAINED EFFICIENT(in the canonical case)

The conditions of EVIDENCE GAMES areindispensable for this EQUIVALENCE


And That Is The Whole Truth ...


And That Is The Whole Truth ...

c© Mick Stevens SERGIU HART c© 2015 – p. 75

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