math.huji.ac.ilmath.huji.ac.il/~hart/papers/st-ne-p.pdf · Law: Right to Remain Silent U.S.: "You...
Transcript of math.huji.ac.ilmath.huji.ac.il/~hart/papers/st-ne-p.pdf · Law: Right to Remain Silent U.S.: "You...
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Evidence Games:Truth and Commitment
Sergiu Hart
This version: September 2016
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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
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Joint work with
Ilan KremerMotty Perry
Hebrew University of JerusalemUniversity of Warwick
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Papers
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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
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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
SERGIU HART c© 2015 – p. 4
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SERGIU HART c© 2015 – p. 5
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Q: "Do you deserve a pay raise?"
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Q: "Do you deserve a pay raise?"A: "Of course."
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Q: "Do you deserve a pay raise?"A: "Of course."
Q: "Are you guilty and deserve punishment?"
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Q: "Do you deserve a pay raise?"A: "Of course."
Q: "Are you guilty and deserve punishment?"A: "Of course not."
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Q: "Do you deserve a pay raise?"A: "Of course."
Q: "Are you guilty and deserve punishment?"A: "Of course not."
How can one obtain reliable information?
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Q: "Do you deserve a pay raise?"A: "Of course."
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?
SERGIU HART c© 2015 – p. 5
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Q: "Do you deserve a pay raise?"A: "Of course."
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 "separate" and avoid"unraveling" (Akerlof 70)?
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Evidence Games: General Setup
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Evidence Games: General Setup
AGENT who is informed
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Evidence Games: General Setup
AGENT who is informed
PRINCIPAL who takes decision but isuninformed
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Evidence Games: General Setup
AGENT who is informed
PRINCIPAL who takes decision but isuninformed
Agent TRANSMITS information to Principal(costlessly)
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Two Setups
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Two Setups
SETUP 1: Principal decides after receivingAgent’s message
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Two Setups
SETUP 1: Principal decides after receivingAgent’s message
SETUP 2: Principal chooses a policy beforeAgent’s message
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Two Setups
SETUP 1: Principal decides after receivingAgent’s message
SETUP 2: Principal chooses a policy beforeAgent’s message
policy : a function that assigns a decisionof Principal to each message of Agent
SERGIU HART c© 2015 – p. 7
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Two Setups
SETUP 1: Principal decides after receivingAgent’s message
SETUP 2: Principal chooses a policy beforeAgent’s message
policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)
SERGIU HART c© 2015 – p. 7
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Two Setups
SETUP 1: Principal decides after receivingAgent’s message
SETUP 2: Principal chooses a policy beforeAgent’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
SERGIU HART c© 2015 – p. 7
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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
SERGIU HART c© 2015 – p. 7
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Main Result
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Main Result
In EVIDENCE GAMES
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Main Result
In EVIDENCE GAMES
the GAME EQUILIBRIUM outcome(obtained without commitment )
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Main Result
In EVIDENCE GAMES
the GAME EQUILIBRIUM outcome(obtained without commitment )
and the OPTIMAL MECHANISM outcome(obtained with commitment )
SERGIU HART c© 2015 – p. 8
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Main Result
In EVIDENCE GAMES
the GAME EQUILIBRIUM outcome(obtained without commitment )
and the OPTIMAL MECHANISM outcome(obtained with commitment )
COINCIDE
SERGIU HART c© 2015 – p. 8
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Main Result: Equivalence
In EVIDENCE GAMES
the GAME EQUILIBRIUM outcome(obtained without commitment )
and the OPTIMAL MECHANISM outcome(obtained with commitment )
COINCIDE
SERGIU HART c© 2015 – p. 8
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Evidence Games
SERGIU HART c© 2015 – p. 9
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Evidence Games
AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )
SERGIU HART c© 2015 – p. 9
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Evidence Games
AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )
PRINCIPAL decides on the reward
SERGIU HART c© 2015 – p. 9
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Evidence Games
AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )
PRINCIPAL decides on the reward
PRINCIPAL wants the reward to be as closeas possible to the value
SERGIU HART c© 2015 – p. 9
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Evidence Games
AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )
PRINCIPAL decides on the reward
PRINCIPAL wants the reward to be as closeas possible to the value
AGENT wants the reward to be as high aspossible (same preference for all types )
SERGIU HART c© 2015 – p. 9
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Evidence Games
AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )
PRINCIPAL decides on the reward
PRINCIPAL wants the reward to be as closeas possible to the value
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
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Evidence Games: Truth
SERGIU HART c© 2015 – p. 10
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Evidence Games: Truth
Agent reveals:
"the truth, nothing but the truth"
SERGIU HART c© 2015 – p. 10
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Evidence Games: Truth
Agent reveals:
"the truth, nothing but the truth"
NOT necessarily "the whole truth"
SERGIU HART c© 2015 – p. 10
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Evidence Games: Truth
Agent reveals:
"the truth, nothing but the truth"
all the evidence that the agent reveals mustbe true (it is verifiable)
NOT necessarily "the whole truth"
SERGIU HART c© 2015 – p. 10
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Evidence Games: Truth
Agent reveals:
"the truth, nothing but the truth"
all the evidence that the agent reveals mustbe true (it is verifiable)
NOT necessarily "the whole truth"
the agent does not have to reveal all theevidence that he has
SERGIU HART c© 2015 – p. 10
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Evidence Games: Truth
SERGIU HART c© 2015 – p. 11
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Evidence Games: Truth
"truth-leaning"
SERGIU HART c© 2015 – p. 11
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Evidence Games: Truth
"truth-leaning"
revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability
SERGIU HART c© 2015 – p. 11
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Previous Work
SERGIU HART c© 2015 – p. 12
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Previous Work
UnravelingGrossman and O. Hart 1980Grossman 1981Milgrom 1981
SERGIU HART c© 2015 – p. 12
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Previous Work
UnravelingGrossman and O. Hart 1980Grossman 1981Milgrom 1981
Voluntary disclosureDye 1985Shin 2003, 2006, ...
SERGIU HART c© 2015 – p. 12
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Previous Work
UnravelingGrossman and O. Hart 1980Grossman 1981Milgrom 1981
Voluntary disclosureDye 1985Shin 2003, 2006, ...
MechanismGreen–Laffont 1986
SERGIU HART c© 2015 – p. 12
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Previous Work
UnravelingGrossman and O. Hart 1980Grossman 1981Milgrom 1981
Voluntary disclosureDye 1985Shin 2003, 2006, ...
MechanismGreen–Laffont 1986
Persuasion gamesGlazer and Rubinstein 2004, 2006
SERGIU HART c© 2015 – p. 12
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Evidence Games
SERGIU HART c© 2015 – p. 13
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Evidence Games
EVIDENCE GAMES model very commonsetups
SERGIU HART c© 2015 – p. 13
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Evidence Games
EVIDENCE GAMES model very commonsetups
In EVIDENCE GAMES there is equivalencebetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)
SERGIU HART c© 2015 – p. 13
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Evidence Games
EVIDENCE GAMES model very commonsetups
In EVIDENCE GAMES there is equivalencebetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)
The conditions of EVIDENCE GAMES areindispensable for this equivalence
SERGIU HART c© 2015 – p. 13
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Example 1
SERGIU HART c© 2015 – p. 14
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Example 1
Professor wants salary as high as possible
SERGIU HART c© 2015 – p. 14
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Example 1
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
SERGIU HART c© 2015 – p. 14
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Example 1
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):
SERGIU HART c© 2015 – p. 14
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Example 1
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):[t0] 50%: no evidence[t+] 25%: positive evidence[t−] 25%: negative evidence
SERGIU HART c© 2015 – p. 14
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Example 1
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t+] 25%: positive evidence → value = 90
[t−] 25%: negative evidence → value = 30
SERGIU HART c© 2015 – p. 14
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Example 1 t+ : 25% 90
t0 : 50% 60
t− : 25% 30
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t+] 25%: positive evidence → value = 90
[t−] 25%: negative evidence → value = 30
SERGIU HART c© 2015 – p. 14
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
GAME: (G1) Professor provides evidence(G2) then Dean sets salary
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
GAME: (G1) Professor provides evidence(G2) then Dean sets salary
EQUILIBRIUM
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
GAME: (G1) Professor provides evidence(G2) then Dean sets salary
EQUILIBRIUMProfessor:
t+ provides positive evidencet0, t− provide no evidence
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
GAME: (G1) Professor provides evidence(G2) then Dean sets salary
EQUILIBRIUMProfessor:
t+ provides positive evidencet0, t− provide no evidence
Dean:to positive evidence gives salary = 90
to negative evidence gives salary = 30
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
GAME: (G1) Professor provides evidence(G2) then Dean sets salary
EQUILIBRIUMProfessor:
t+ provides positive evidencet0, t− provide no evidence
Dean:to positive evidence gives salary = 90
to negative evidence gives salary = 30
to no evidence gives salary = 50= (50% · 60 + 25% · 30)/(50% + 25%)
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
GAME: (G1) Professor provides evidence(G2) then Dean sets salary
unique sequential EQUILIBRIUMProfessor:
t+ provides positive evidencet0, t− provide no evidence
Dean:to positive evidence gives salary = 90
to negative evidence gives salary = 30
to no evidence gives salary = 50= (50% · 60 + 25% · 30)/(50% + 25%)
SERGIU HART c© 2015 – p. 15
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
SERGIU HART c© 2015 – p. 16
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
30 60 90value:
SERGIU HART c© 2015 – p. 16
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
30 60 90value:
t− t0 t+
SERGIU HART c© 2015 – p. 16
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
30 60 90value:
t− t0 t+
partial truth:
SERGIU HART c© 2015 – p. 16
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
30 60 90value:
t− t0 t+
partial truth:
prof says:
SERGIU HART c© 2015 – p. 16
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Example 1: Equilibrium t+ : 25% 90
t0 : 50% 60
t− : 25% 30
30 60 90value:
t− t0 t+
partial truth:
prof says:
dean pays: 30 50 90
SERGIU HART c© 2015 – p. 16
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Example 1: Mechanism t+ : 25% 90
t0 : 50% 60
t− : 25% 30
SERGIU HART c© 2015 – p. 17
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Example 1: Mechanism t+ : 25% 90
t0 : 50% 60
t− : 25% 30
MECHANISM: (M1) Dean commits to salary policy(M2) then Professor provides evidence
SERGIU HART c© 2015 – p. 17
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Example 1: Mechanism t+ : 25% 90
t0 : 50% 60
t− : 25% 30
MECHANISM: (M1) Dean commits to salary policy(M2) then Professor provides evidence
OPTIMAL MECHANISM
SERGIU HART c© 2015 – p. 17
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Example 1: Mechanism t+ : 25% 90
t0 : 50% 60
t− : 25% 30
MECHANISM: (M1) Dean commits to salary policy(M2) then Professor provides evidence
OPTIMAL MECHANISM
Dean:to positive evidence gives salary = 90
to no evidence gives salary = 50
to negative evidence gives salary ≤ 50
SERGIU HART c© 2015 – p. 17
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Example 1: Mechanism t+ : 25% 90
t0 : 50% 60
t− : 25% 30
MECHANISM: (M1) Dean commits to salary policy(M2) then Professor provides evidence
OPTIMAL MECHANISM
Dean:to positive evidence gives salary = 90
to no evidence gives salary = 50
to negative evidence gives salary ≤ 50
OPTIMAL MECHANISM = EQUILIBRIUMSERGIU HART c© 2015 – p. 17
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
t− says t0
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
t− says t0
the value of t0 is higher than the value of t−
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
t− says t0
the value of t0 is higher than the value of t−
in MECHANISM:
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
t− says t0
the value of t0 is higher than the value of t−
in MECHANISM:
the only way to separate t− from t0is to pay t− strictly more than to t0
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
t− says t0
the value of t0 is higher than the value of t−
in MECHANISM:
the only way to separate t− from t0is to pay t− strictly more than to t0
this is not optimal
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
in EQUILIBRIUM :
t− says t0
the value of t0 is higher than the value of t−
in MECHANISM:
the only way to separate t− from t0is to pay t− strictly more than to t0
this is not optimal
OPTIMAL MECHANISM does notseparate more than EQUILIBRIUM
SERGIU HART c© 2015 – p. 18
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
OPTIMAL MECHANISM does notseparate more than EQUILIBRIUM
SERGIU HART c© 2015 – p. 19
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Example 1: Explanation t+ : 25% 90
t0 : 50% 60
t− : 25% 30
30 60
t− t0
partial truth:
OPTIMAL MECHANISM does notseparate more than EQUILIBRIUM
SERGIU HART c© 2015 – p. 19
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Example 2
y
SERGIU HART c© 2015 – p. 20
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Example 2
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):
SERGIU HART c© 2015 – p. 20
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Example 2
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 25%: negative evidence → value = 30
SERGIU HART c© 2015 – p. 20
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Example 2
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 25%: negative evidence → value = 30
[t+] 20%: positive evidence → value = 102
[t±] 5%: both evidences → value = 42
SERGIU HART c© 2015 – p. 20
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Example 2 t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
Professor wants salary as high as possible
Dean wants salary to be as close as possibleto the Professor’s value
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 25%: negative evidence → value = 30
[t+] 20%: positive evidence → value = 102
[t±] 5%: both evidences → value = 42
SERGIU HART c© 2015 – p. 20
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
SERGIU HART c© 2015 – p. 21
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30EQUILIBRIUM
SERGIU HART c© 2015 – p. 21
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30EQUILIBRIUM
Professor:t+, t± provide positive evidencet0, t− provide no evidence
SERGIU HART c© 2015 – p. 21
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30EQUILIBRIUM
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 negative evidence gives salary = 30
to both evidences gives salary = 42SERGIU HART c© 2015 – p. 21
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
SERGIU HART c© 2015 – p. 22
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
30 42 60 102value:
SERGIU HART c© 2015 – p. 22
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
30 42 60 102value:
t− t0 t+t±
SERGIU HART c© 2015 – p. 22
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
30 42 60 102value:
t− t0 t+t±
partial truth:
SERGIU HART c© 2015 – p. 22
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
30 42 60 102value:
t− t0 t+t±
partial truth:
prof says:
SERGIU HART c© 2015 – p. 22
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Example 2: Equilibrium t+ : 20% 102
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
SERGIU HART c© 2015 – p. 22
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Example 2: Mechanism t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
SERGIU HART c© 2015 – p. 23
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Example 2: Mechanism t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30OPTIMAL MECHANISM
SERGIU HART c© 2015 – p. 23
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Example 2: Mechanism t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30OPTIMAL MECHANISM
Dean:to positive evidence gives salary = 90
to no evidence gives salary = 50
to negative evidence gives salary ≤ 50
to both evidences gives salary ≤ 90
SERGIU HART c© 2015 – p. 23
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Example 2: Mechanism t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30OPTIMAL MECHANISM
Dean:to positive evidence gives salary = 90
to no evidence gives salary = 50
to negative evidence gives salary ≤ 50
to both evidences gives salary ≤ 90
OPTIMAL MECHANISM = EQUILIBRIUM
SERGIU HART c© 2015 – p. 23
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium
Professor:always provides no evidence
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium
Professor:always provides no evidence
Dean:ignores all evidence and gives salary = 60= 50%·60+25%·30+20%·102+5%·42
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium
Professor:always provides no evidence
Dean:ignores all evidence and gives salary = 60= 50%·60+25%·30+20%·102+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+
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium ("babbling" )
Professor:always provides no evidence
Dean:ignores all evidence and gives salary = 60= 50%·60+25%·30+20%·102+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+
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS
Professor:always provides no evidence
Dean:ignores all evidence and gives salary = 60= 50%·60+25%·30+20%·102+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+
SERGIU HART c© 2015 – p. 24
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS
SERGIU HART c© 2015 – p. 25
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS
30 42 60 102value:
t− t0 t+t±
partial truth:
SERGIU HART c© 2015 – p. 25
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS
30 42 60 102value:
t− t0 t+t±
partial truth:
prof says:
SERGIU HART c© 2015 – p. 25
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Example 2: Equilibrium t+ : 20% 102
t0 : 50% 60
t± : 5% 42
t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS
30 42 60 102value:
t− t0 t+t±
partial truth:
prof says:
dean pays: 30 6042 < 60
SERGIU HART c© 2015 – p. 25
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SERGIU HART c© 2015 – p. 26
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Example 3
y
SERGIU HART c© 2015 – p. 27
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Example 3
Professor’s evidence (verifiable):
SERGIU HART c© 2015 – p. 27
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Example 3
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
SERGIU HART c© 2015 – p. 27
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Example 3
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
Dean wants salary close to Professor’s value
SERGIU HART c© 2015 – p. 27
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Example 3
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
Dean wants salary close to Professor’s value
Professor t0 wants salary high
SERGIU HART c© 2015 – p. 27
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Example 3
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
Dean wants salary close to Professor’s value
Professor t0 wants salary high
Professor t− wants salary close to 50
SERGIU HART c© 2015 – p. 27
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Example 3
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM :
separation
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM :
separation ⇒ x0 = 60, x− = 30
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM :
separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM :
separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)⇒ contradiction
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM :
separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)⇒ contradiction
⇒ no separation (babbling), x = 45
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM : no separation , x = 45
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM : no separation , x = 45
MECHANISM:
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM : no separation , x = 45
MECHANISM:
to negative evidence gives salary = 30
to no evidence gives salary = 71
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM : no separation , x = 45
MECHANISM:
to negative evidence gives salary = 30
to no evidence gives salary = 71(separating: t− prefers 30 to 71)
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM : no separation , x = 45
MECHANISM:
to negative evidence gives salary = 30
to no evidence gives salary = 71(separating: t− prefers 30 to 71)
REQUIRES COMMITMENT :
after no evidence Dean prefers salary = 60
SERGIU HART c© 2015 – p. 28
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Example 3
EQUILIBRIUM : no separation , x = 45
MECHANISM:
to negative evidence gives salary = 30
to no evidence gives salary = 71(separating: t− prefers 30 to 71)
REQUIRES COMMITMENT :
after no evidence Dean prefers salary = 60
MECHANISM is strictly better for the Dean thanEQUILIBRIUM
SERGIU HART c© 2015 – p. 28
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Example 3: Commitment Helps
EQUILIBRIUM : no separation , x = 45
MECHANISM:
to negative evidence gives salary = 30
to no evidence gives salary = 71(separating: t− prefers 30 to 71)
REQUIRES COMMITMENT :
after no evidence Dean prefers salary = 60
MECHANISM is strictly better for the Dean thanEQUILIBRIUM
SERGIU HART c© 2015 – p. 28
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Example 3: Commitment Helps
SERGIU HART c© 2015 – p. 29
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Example 3: Commitment Helps
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
Dean wants salary close to Professor’s value
Professor t0 wants salary high
Professor t− wants salary close to 50
SERGIU HART c© 2015 – p. 29
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Example 3: Commitment Helps
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
Dean wants salary close to Professor’s value
Professor t0 wants salary high
Professor t− wants salary close to 50
SERGIU HART c© 2015 – p. 29
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Example 3: Commitment Helps
Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60
[t−] 50%: negative evidence → value = 30
Dean wants salary close to Professor’s value
Professor t0 wants salary high
Professor t− wants salary close to 50
↑NOT "evidence game"
SERGIU HART c© 2015 – p. 29
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SERGIU HART c© 2015 – p. 30
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Examples: Summary
SERGIU HART c© 2015 – p. 31
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Examples: Summary
Example 1 :equivalence (our result)
SERGIU HART c© 2015 – p. 31
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Examples: Summary
Example 1 :equivalence (our result)
Example 2 :EQUILIBRIUM different fromOPTIMAL MECHANISM
⇒ "truth-leaning"
SERGIU HART c© 2015 – p. 31
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Examples: Summary
Example 1 :equivalence (our result)
Example 2 :EQUILIBRIUM different fromOPTIMAL MECHANISM
⇒ "truth-leaning"
Example 3 :assumptions do not holdresult fails: commitment helps
SERGIU HART c© 2015 – p. 31
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Model
SERGIU HART c© 2015 – p. 32
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Model
AGENT (A)
PRINCIPAL (P ) (= "market")
SERGIU HART c© 2015 – p. 32
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Model
AGENT (A)
PRINCIPAL (P ) (= "market")
(finite) set of TYPES: T
the type t ∈ T is chosen according to aprobability distribution p ∈ ∆(T )
SERGIU HART c© 2015 – p. 32
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Model
AGENT (A)
PRINCIPAL (P ) (= "market")
(finite) set of TYPES: T
the type t ∈ T is chosen according to aprobability distribution p ∈ ∆(T )
the type t ∈ T is revealed to Agent and not toPrincipal
SERGIU HART c© 2015 – p. 32
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Model
AGENT (A)
PRINCIPAL (P ) (= "market")
(finite) set of TYPES: T
the type t ∈ T is chosen according to aprobability distribution p ∈ ∆(T )
the type t ∈ T is revealed to Agent and not toPrincipal
Agent’s MESSAGE: s ∈ T
SERGIU HART c© 2015 – p. 32
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Model
AGENT (A)
PRINCIPAL (P ) (= "market")
(finite) set of TYPES: T
the type t ∈ T is chosen according to aprobability distribution p ∈ ∆(T )
the type t ∈ T is revealed to Agent and not toPrincipal
Agent’s MESSAGE: s ∈ T
Principal’s DECISION: REWARD x ∈ R
SERGIU HART c© 2015 – p. 32
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Payoffs / Utilities
SERGIU HART c© 2015 – p. 33
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Payoffs / Utilities
UA and UP do not depend on the message s
SERGIU HART c© 2015 – p. 33
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Payoffs / Utilities
UA and UP do not depend on the message s
UA does not depend on the type t
UA(s, x; t) = x
SERGIU HART c© 2015 – p. 33
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Payoffs / Utilities
UA and UP do not depend on the message s
UA does not depend on the type t
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 )
SERGIU HART c© 2015 – p. 33
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Payoffs / Utilities
UA and UP do not depend on the message s
UA does not depend on the type t
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
SERGIU HART c© 2015 – p. 33
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Single Peakedness (SP)
SERGIU HART c© 2015 – p. 34
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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
SERGIU HART c© 2015 – p. 34
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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
⇔ 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)
SERGIU HART c© 2015 – p. 34
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Single Peakedness (SP)
SERGIU HART c© 2015 – p. 35
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Single Peakedness (SP)
Canonical example:ht(x) = −(x − v(t))2
SERGIU HART c© 2015 – p. 35
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Single Peakedness (SP)
Canonical example:ht(x) = −(x − v(t))2
v(q) = Eq[v(t)] =∑
t qt v(t)
SERGIU HART c© 2015 – p. 35
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Single Peakedness (SP)
Canonical example:ht(x) = −(x − v(t))2
v(q) = Eq[v(t)] =∑
t qt v(t)
More general:ht(x) is a differentiable strictly concavefunction of x, for each t
SERGIU HART c© 2015 – p. 35
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Single Peakedness (SP)
Canonical example:ht(x) = −(x − v(t))2
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
SERGIU HART c© 2015 – p. 35
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Evidence and Truth
SERGIU HART c© 2015 – p. 36
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Evidence and Truth
Agent reveals:
"the truth, nothing but the truth"
SERGIU HART c© 2015 – p. 36
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Evidence and Truth
Agent reveals:
"the truth, nothing but the truth"
NOT necessarily "the whole truth"
SERGIU HART c© 2015 – p. 36
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Evidence and Truth
Agent reveals:
"the truth, nothing but the truth"
all the evidence that the agent reveals mustbe true (it is verifiable)
NOT necessarily "the whole truth"
SERGIU HART c© 2015 – p. 36
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Evidence and Truth
Agent reveals:
"the truth, nothing but the truth"
all the evidence that the agent reveals mustbe true (it is verifiable)
NOT necessarily "the whole truth"
the agent does not have to reveal all theevidence that he has
SERGIU HART c© 2015 – p. 36
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Evidence and Truth
Agent reveals:
"the truth, nothing but the truth"
all the evidence that the agent reveals mustbe true (it is verifiable)
NOT necessarily "the whole truth"
the agent does not have to reveal all theevidence that he has
⇒ Agent can pretend to be a type thathas less evidence
SERGIU HART c© 2015 – p. 36
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Evidence and Truth
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
E = set of (verifiable) pieces of evidence
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
E = set of (verifiable) pieces of evidence
Et ⊆ E = set of pieces of evidence that t has(and can provide)
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
E = set of (verifiable) pieces of evidence
Et ⊆ E = set of pieces of evidence that t has(and can provide)
L(t) = {s ∈ T : Es ⊆ Et}
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
E = set of (verifiable) pieces of evidence
Et ⊆ E = set of pieces of evidence that t has(and can provide)
L(t) = {s ∈ T : Es ⊆ Et}
(L1) Reflexivity : t ∈ L(t)
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
E = set of (verifiable) pieces of evidence
Et ⊆ E = set of pieces of evidence that t has(and can provide)
L(t) = {s ∈ T : Es ⊆ Et}
(L1) Reflexivity : t ∈ L(t)
(L2) Transitivity : If s ∈ L(t) and r ∈ L(s)then r ∈ L(t)
SERGIU HART c© 2015 – p. 37
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Evidence and Truth
L(t) = the set of types that t can pretend to be= the set of possible messages of type t
E = set of (verifiable) pieces of evidence
Et ⊆ E = set of pieces of evidence that t has(and can provide)
L(t) = {s ∈ T : Es ⊆ Et}
(L1) Reflexivity : t ∈ L(t)
(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
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Type (summary)
SERGIU HART c© 2015 – p. 38
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Type (summary)
A type t has two characteristics:
SERGIU HART c© 2015 – p. 38
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Type (summary)
A type t has two characteristics:
Value to the Principal(expressed by h(t) and its peak v(t))
SERGIU HART c© 2015 – p. 38
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Type (summary)
A type t has two characteristics:
Value to the Principal(expressed by h(t) and its peak v(t))
Evidence that he can provide(expressed by L(t))
SERGIU HART c© 2015 – p. 38
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Type (summary)
A type t has two characteristics:
Value to the Principal(expressed by h(t) and its peak v(t))
Evidence that he can provide(expressed by L(t))
No relation is assumedbetween Value and Evidence
SERGIU HART c© 2015 – p. 38
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Game
SERGIU HART c© 2015 – p. 39
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Game
(G1) Agent sends message s ∈ L(t) to Principal
SERGIU HART c© 2015 – p. 39
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Game
(G1) Agent sends message s ∈ L(t) to Principal
(G2) Then Principal sets reward x ∈ R
SERGIU HART c© 2015 – p. 39
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Game
(G1) Agent sends message s ∈ L(t) to Principal
(G2) Then Principal sets reward x ∈ R
STRATEGIES
SERGIU HART c© 2015 – p. 39
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Game
(G1) Agent sends message s ∈ L(t) to Principal
(G2) Then Principal sets reward x ∈ R
STRATEGIES
(Agent) σ(s|t) = probability that type t sendsmessage s in L(t)
SERGIU HART c© 2015 – p. 39
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Game
(G1) Agent sends message s ∈ L(t) to Principal
(G2) Then Principal sets reward x ∈ R
STRATEGIES
(Agent) σ(s|t) = probability that type t sendsmessage s in L(t)
(Principal) ρ(s) ∈ R = reward to message s
SERGIU HART c© 2015 – p. 39
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Equilibrium
SERGIU HART c© 2015 – p. 40
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Equilibrium
(σ, ρ) is a NASH EQUILIBRIUM if
SERGIU HART c© 2015 – p. 40
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Equilibrium
(σ, ρ) is a NASH EQUILIBRIUM if
(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)
SERGIU HART c© 2015 – p. 40
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Equilibrium
(σ, ρ) is a NASH EQUILIBRIUM if
(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
SERGIU HART c© 2015 – p. 40
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Equilibrium
(σ, ρ) is a NASH EQUILIBRIUM if
(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
OUTCOME: πt = maxs∈L(t) ρ(s) = ρ(σ(·|t))
SERGIU HART c© 2015 – p. 40
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Equilibrium
(σ, ρ) is a NASH EQUILIBRIUM if
(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
OUTCOME: πt = maxs∈L(t) ρ(s) = ρ(σ(·|t))
π = (πt)t∈T ∈ RT
SERGIU HART c© 2015 – p. 40
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Truth-Leaning
SERGIU HART c© 2015 – p. 41
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Truth-Leaning
Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability
SERGIU HART c© 2015 – p. 41
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Truth-Leaning
Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability
(T1) Revealing the whole truth is preferablewhen the reward is the same
SERGIU HART c© 2015 – p. 41
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Truth-Leaning
Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability
(T1) Revealing the whole truth is preferablewhen the reward is the same(lexicographic preference)
SERGIU HART c© 2015 – p. 41
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Truth-Leaning
Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability
(T1) Revealing the whole truth is preferablewhen the reward is the same(lexicographic preference)
(T2) The whole truth is revealed with infinitesimalpositive probability
SERGIU HART c© 2015 – p. 41
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Truth-Leaning
Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability
(T1) Revealing the whole truth is preferablewhen the reward is the same(lexicographic preference)
(T2) The whole truth is revealed with infinitesimalpositive probability(by mistake, or because the agent may benon-strategic, or ... [UK])
SERGIU HART c© 2015 – p. 41
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Truth-Leaning
SERGIU HART c© 2015 – p. 42
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Truth-Leaning
A Nash equilibrium is TRUTH-LEANING if itsatisfies:
SERGIU HART c© 2015 – p. 42
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Truth-Leaning
A Nash equilibrium is TRUTH-LEANING if itsatisfies:
(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1
SERGIU HART c© 2015 – p. 42
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Truth-Leaning
A Nash equilibrium is TRUTH-LEANING if itsatisfies:
(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)
SERGIU HART c© 2015 – p. 42
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Truth-Leaning
A Nash equilibrium is TRUTH-LEANING if itsatisfies:
(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)
(T2) σ̄(t) = 0 ⇒ ρ(t) = v(t)
SERGIU HART c© 2015 – p. 42
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Truth-Leaning
A Nash equilibrium is TRUTH-LEANING if itsatisfies:
(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)
(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)
SERGIU HART c© 2015 – p. 42
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Truth-Leaning
SERGIU HART c© 2015 – p. 43
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Truth-Leaning
Truth-Leaning equilibria:
SERGIU HART c© 2015 – p. 43
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Truth-Leaning
Truth-Leaning equilibria:
coincide with the equilibria selected in the"voluntary disclosure" literature
SERGIU HART c© 2015 – p. 43
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Truth-Leaning
Truth-Leaning equilibria:
coincide with the equilibria selected in the"voluntary disclosure" literature
satisfy all the refinement conditions in theliterature
SERGIU HART c© 2015 – p. 43
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Truth-Leaning
Truth-Leaning equilibria:
coincide with the equilibria selected in the"voluntary disclosure" literature
satisfy all the refinement conditions in theliterature
eliminate "unreasonable" equilibria (such as"babbling" in Example 2)
SERGIU HART c© 2015 – p. 43
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Truth-Leaning
Truth-Leaning equilibria:
coincide with the equilibria selected in the"voluntary disclosure" literature
satisfy all the refinement conditions in theliterature
eliminate "unreasonable" equilibria (such as"babbling" in Example 2)
...
SERGIU HART c© 2015 – p. 43
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Mechanism
SERGIU HART c© 2015 – p. 44
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Mechanism
MECHANISM:
SERGIU HART c© 2015 – p. 44
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Mechanism
MECHANISM: Reward scheme ρ : T → R
(ρ(s) = reward to message s)
SERGIU HART c© 2015 – p. 44
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Mechanism
MECHANISM: Reward scheme ρ : T → R
(ρ(s) = reward to message s)
Agent’s payoff when type is t:
πt = maxs∈L(t)
ρ(s)
SERGIU HART c© 2015 – p. 44
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Mechanism
MECHANISM: Reward scheme ρ : T → R
(ρ(s) = reward to message s)
Agent’s payoff when type is t:
πt = maxs∈L(t)
ρ(s)
Outcome : π = (πt)t∈T ∈ RT
SERGIU HART c© 2015 – p. 44
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Mechanism
MECHANISM: Reward scheme ρ : T → R
(ρ(s) = reward to message s)
Agent’s payoff when type is t:
πt = maxs∈L(t)
ρ(s)
Outcome : π = (πt)t∈T ∈ RT
Principal’s payoff :
H(π) =∑
t∈T
pt ht(πt)
SERGIU HART c© 2015 – p. 44
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Incentive Compatibility
SERGIU HART c© 2015 – p. 45
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Incentive Compatibility
Outcome π = (πt)t∈T ∈ RT is generated by a
mechanism ρ
SERGIU HART c© 2015 – p. 45
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Incentive Compatibility
Outcome π = (πt)t∈T ∈ RT is generated by a
mechanism ρ
if and only if
SERGIU HART c© 2015 – p. 45
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Incentive Compatibility
Outcome π = (πt)t∈T ∈ RT is generated by a
mechanism ρ
if and only if
πt ≥ πs
for all s, t ∈ T with s ∈ L(t)
SERGIU HART c© 2015 – p. 45
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Incentive Compatibility
Outcome π = (πt)t∈T ∈ RT is generated by a
mechanism ρ
if and only if
πt ≥ πs
for all s, t ∈ T with s ∈ L(t)
Immediate because L satisfies reflexivity (L1)and transitivity (L2)
SERGIU HART c© 2015 – p. 45
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Incentive Compatibility
Outcome π = (πt)t∈T ∈ RT is generated by a
mechanism ρ
if and only if
πt ≥ πs
for all s, t ∈ T with s ∈ L(t)
Immediate because L satisfies reflexivity (L1)and transitivity (L2)
"direct" mechanism: ρ(t) = πt
SERGIU HART c© 2015 – p. 45
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Incentive Compatibility
Outcome π = (πt)t∈T ∈ RT is generated by a
mechanism ρ
if and only if
πt ≥ πs
for all s, t ∈ T with s ∈ L(t)
Immediate because L satisfies reflexivity (L1)and transitivity (L2)
"direct" mechanism: ρ(t) = πt
Green–Laffont 86SERGIU HART c© 2015 – p. 45
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Optimal Mechanism
SERGIU HART c© 2015 – p. 46
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Optimal Mechanism
OPTIMAL MECHANISM :
SERGIU HART c© 2015 – p. 46
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Optimal Mechanism
OPTIMAL MECHANISM :
Maximize H(π) =∑
t∈T pt ht(πt)
SERGIU HART c© 2015 – p. 46
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Optimal Mechanism
OPTIMAL MECHANISM :
Maximize H(π) =∑
t∈T pt ht(πt)
subject to (IC):
πt ≥ πs
for all s, t ∈ T with s ∈ L(t)
SERGIU HART c© 2015 – p. 46
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Optimal Mechanism
OPTIMAL MECHANISM :
Maximize H(π) =∑
t∈T pt ht(πt)
subject to (IC):
πt ≥ πs
for all s, t ∈ T with s ∈ L(t)
OPTIMAL MECHANISM = Maximum underIncentive Constraints
SERGIU HART c© 2015 – p. 46
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Main Result
SERGIU HART c© 2015 – p. 47
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Main Result
(i) There is a uniqueTRUTH-LEANING EQUILIBRIUM outcome.
SERGIU HART c© 2015 – p. 47
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Main Result
(i) There is a uniqueTRUTH-LEANING EQUILIBRIUM outcome.
(ii) There is a uniqueOPTIMAL MECHANISM outcome.
SERGIU HART c© 2015 – p. 47
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Main Result
(i) There is a uniqueTRUTH-LEANING EQUILIBRIUM outcome.
(ii) There is a uniqueOPTIMAL MECHANISM outcome.
(iii) These two outcomes COINCIDE.
SERGIU HART c© 2015 – p. 47
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Main Result: Equivalence
(i) There is a uniqueTRUTH-LEANING EQUILIBRIUM outcome.
(ii) There is a uniqueOPTIMAL MECHANISM outcome.
(iii) These two outcomes COINCIDE.
SERGIU HART c© 2015 – p. 47
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Main Result: Equivalence
SERGIU HART c© 2015 – p. 48
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Main Result: Equivalence
The equilibrium strategies need not be unique(happens only when the Agent is indifferent—andthen the Principal is also indifferent)
SERGIU HART c© 2015 – p. 48
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Main Result: Equivalence
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
All the conditions are indispensable :
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
All the conditions are indispensable :
Truth structure: reflexivity
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
All the conditions are indispensable :
Truth structure: reflexivity
Truth structure: transitivity
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
All the conditions are indispensable :
Truth structure: reflexivity
Truth structure: transitivity
Truth Leaning: whole truth slightly better
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
All the conditions are indispensable :
Truth structure: reflexivity
Truth structure: transitivity
Truth Leaning: whole truth slightly better
Truth Leaning: whole truth slightly possible
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
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
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
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
Principal’s utility: single-peaked with respectto Agent’s utility
SERGIU HART c© 2015 – p. 49
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Main Result: Equivalence
SERGIU HART c© 2015 – p. 50
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Main Result: Equivalence
EQUILIBRIUM (without commitment)yields the same as COMMITMENT
SERGIU HART c© 2015 – p. 50
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Main Result: Equivalence
EQUILIBRIUM (without commitment)yields the same as COMMITMENT
EQUILIBRIUM yields OPTIMAL SEPARATION(for the principal / "market")
SERGIU HART c© 2015 – p. 50
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Main Result: Equivalence
EQUILIBRIUM (without commitment)yields the same as COMMITMENT
EQUILIBRIUM yields OPTIMAL SEPARATION(for the principal / "market")
EQUILIBRIUM yields PARETO EFFICIENCY(in the canonical setup)
SERGIU HART c© 2015 – p. 50
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Main Result: Equivalence
UNDER INCENTIVE CONSTRAINTS
EQUILIBRIUM (without commitment)yields the same as COMMITMENT
EQUILIBRIUM yields OPTIMAL SEPARATION(for the principal / "market")
EQUILIBRIUM yields PARETO EFFICIENCY(in the canonical setup)
SERGIU HART c© 2015 – p. 50
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Applications
SERGIU HART c© 2015 – p. 51
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Applications: Finance
Disclosure by public firms
SERGIU HART c© 2015 – p. 51
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Applications: Finance
Disclosure by public firms
Disclosing false information is a criminal act
SERGIU HART c© 2015 – p. 51
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Applications: Finance
Disclosure by public firms
Disclosing false information is a criminal act
Withholding information is allowed in somecases
SERGIU HART c© 2015 – p. 51
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Applications: Finance
Disclosure by public firms
Disclosing false information is a criminal act
Withholding information is allowed in somecases, and is difficult (if not impossible) todetect
SERGIU HART c© 2015 – p. 51
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Applications: Finance
Disclosure by public firms
Disclosing false information is a criminal act
Withholding information is allowed in somecases, and is difficult (if not impossible) todetect
Impacts asset prices (e.g.: quarterly reports)
SERGIU HART c© 2015 – p. 51
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Applications: Finance
SERGIU HART c© 2015 – p. 52
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Applications: Finance
Our result:
SERGIU HART c© 2015 – p. 52
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Applications: Finance
Our result:Market’s behavior in EQUILIBRIUM yields theOPTIMAL SEPARATION between "good" and "bad"firms (even if mechanisms and commitmentswere possible)
SERGIU HART c© 2015 – p. 52
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Applications: Finance
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
SERGIU HART c© 2015 – p. 52
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Applications: Law
SERGIU HART c© 2015 – p. 53
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Applications: Law
SERGIU HART c© 2015 – p. 53
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Applications: Law
Commitments: constitutions, laws, legaldoctrine, precedents, ...
SERGIU HART c© 2015 – p. 53
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Applications: Law
Commitments: constitutions, laws, legaldoctrine, precedents, ...
Affect evidence provided in court
SERGIU HART c© 2015 – p. 53
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Applications: Law
Commitments: constitutions, laws, legaldoctrine, precedents, ...
Affect evidence provided in court
Our result:The power of these commitments may not gobeyond selecting the truth-leaning equilibria
SERGIU HART c© 2015 – p. 53
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Applications: Law
Commitments: constitutions, laws, legaldoctrine, precedents, ...
Affect evidence provided in court
Our result:The power of these commitments may not gobeyond selecting the truth-leaning equilibria– which are most natural here
SERGIU HART c© 2015 – p. 53
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Law
SERGIU HART c© 2015 – p. 54
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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)
SERGIU HART c© 2015 – p. 54
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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)
SERGIU HART c© 2015 – p. 54
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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)
SERGIU HART c© 2015 – p. 54
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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)
SERGIU HART c© 2015 – p. 54
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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 )
SERGIU HART c© 2015 – p. 54
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Law: Right to Remain Silent
SERGIU HART c© 2015 – p. 55
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Law: Right to Remain Silent
EQUILIBRIA entail (Bayesian) inferences
SERGIU HART c© 2015 – p. 55
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Law: Right to Remain Silent
EQUILIBRIA entail (Bayesian) inferences
Our equivalence result implies that the sameinferences apply to OPTIMAL MECHANISMS
SERGIU HART c© 2015 – p. 55
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Law: Right to Remain Silent
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
SERGIU HART c© 2015 – p. 55
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Law: Right to Remain Silent
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
SERGIU HART c© 2015 – p. 55
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Law: Right to Remain Silent
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(reinforce the advantages of truth-telling)
SERGIU HART c© 2015 – p. 55
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Medical Over-Treatment
SERGIU HART c© 2015 – p. 56
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Medical Over-Treatment
Doctors and hospitals prefer to over-treat asthey are paid more when doing so
SERGIU HART c© 2015 – p. 56
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Medical Over-Treatment
Doctors and hospitals prefer to over-treat asthey are paid more when doing so
Give doctors incentives to provide evidence
SERGIU HART c© 2015 – p. 56
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Equivalence Theorem: Intuition
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
SERGIU HART c© 2015 – p. 57
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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)
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
No one pretends to be worth less than they are
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
v(t) < πt = πs ≤ v(s)
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
NOTE: These conclusions need not hold forequilibria that are not truth-leaning
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
⇒ t and s cannot be separated in any OPTIMALMECHANISM
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
⇒ t and s cannot be separated in any OPTIMALMECHANISM
- To separate: ρ(t) > ρ(s) (else t says s)
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
⇒ t and s cannot be separated in any OPTIMALMECHANISM
- To separate: ρ(t) > ρ(s) (else t says s)- Not optimal: decreasing ρ(t) or increasing ρ(s)brings rewards closer to values
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
⇒ t and s cannot be separated in any OPTIMALMECHANISM
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Intuition
In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:
s reveals his type (i.e., says s)
v(t) < v(s)
⇒ t and s cannot be separated in any OPTIMALMECHANISM
CONCLUSION:OPTIMAL MECHANISM cannot separate
more than TRUTH-LEANING EQUILIBRIUM
SERGIU HART c© 2015 – p. 57
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Equivalence Theorem: Proof
SERGIU HART c© 2015 – p. 58
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Equivalence Theorem: Proof
0. Preliminaries
SERGIU HART c© 2015 – p. 58
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Equivalence Theorem: Proof
0. Preliminaries
1. Every TL-EQUILIBRIUM outcome equalsthe unique OPTIMAL MECHANISM outcome
SERGIU HART c© 2015 – p. 58
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Equivalence Theorem: Proof
0. Preliminaries
1. Every TL-EQUILIBRIUM outcome equalsthe unique OPTIMAL MECHANISM outcome
2. A TL-EQUILIBRIUM exists
SERGIU HART c© 2015 – p. 58
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Proof: 0. Preliminaries
SERGIU HART c© 2015 – p. 59
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Proof: 0. Preliminaries
"In betweenness" : v(t1) ≦ v(t2) implies
v(t1) ≦ v({t1, t2}) ≦ v(t2)
SERGIU HART c© 2015 – p. 59
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Proof: 0. Preliminaries
"In betweenness" : v(t1) ≦ v(t2) implies
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)
SERGIU HART c© 2015 – p. 59
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Proof: 0. Preliminaries
"In betweenness" : v(t1) ≦ v(t2) implies
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
SERGIU HART c© 2015 – p. 59
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Proof: 0. Preliminaries
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
Let (σ, ρ) be a TL-EQUILIBRIUM with outcome π.Then:
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).
Pf. v(s) ≥ ρ(s)
(s is used)
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).
Pf. v(s) ≥ ρ(s) = πt
(s is optimal for t)
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).
Pf. v(s) ≥ ρ(s) = πt > v(t)
(t is not used)
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).
Pf. v(s) ≥ ρ(s) = πt > v(t) 2
SERGIU HART c© 2015 – p. 60
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Proof: 0. Preliminaries
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)
Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).
Pf. v(s) ≥ ρ(s) = πt > v(t) 2
Note. May not hold for NON-TL-equilibria.SERGIU HART c© 2015 – p. 60
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Proof: 1. equilibrium → mechanism
SERGIU HART c© 2015 – p. 61
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Proof: 1. equilibrium → mechanism
Let (σ, ρ) be TL-EQUILIBRIUM , with outcome π.
SERGIU HART c© 2015 – p. 61
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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).
SERGIU HART c© 2015 – p. 61
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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
SERGIU HART c© 2015 – p. 61
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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
SERGIU HART c© 2015 – p. 61
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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
⇒ 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,
SERGIU HART c© 2015 – p. 61
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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
⇒ 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,because v(t) < v(T ) ≤ v(s) for all t 6= s
SERGIU HART c© 2015 – p. 61
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Proof: 1. equilibrium → mechanism
Special Case :
SERGIU HART c© 2015 – p. 62
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Proof: 1. equilibrium → mechanism
Special Case :
v(t) v(t′) v(s)
t t′ s
L:
SERGIU HART c© 2015 – p. 62
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Proof: 1. equilibrium → mechanism
Special Case :
v(t) v(t′) v(s)
t t′ s
L:v(T )
SERGIU HART c© 2015 – p. 62
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Proof: 1. equilibrium → mechanism
Special Case :
v(t) v(t′) v(s)
t t′ s
L:v(T )
IC: πs ≤ πt
πs ≤ πt′
SERGIU HART c© 2015 – p. 62
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Proof: 1. equilibrium → mechanism
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
General Case.
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
General Case. For each message s that isused (i.e., σ̄(s) > 0)
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
General Case. For each message s that isused (i.e., σ̄(s) > 0) apply the Special Casewith:
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
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)
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
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)probability distribution = q(s)(the posterior given message s)
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
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)probability distribution = q(s)(the posterior given message s)
⇒ π restricted to Ts is the unique OPTIMALMECHANISM, for each s
SERGIU HART c© 2015 – p. 63
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Proof: 1. equilibrium → mechanism
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)probability distribution = q(s)(the posterior given message s)
⇒ π restricted to Ts is the unique OPTIMALMECHANISM, for each s
⇒ π is the unique OPTIMAL MECHANISM
SERGIU HART c© 2015 – p. 63
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Proof: 2. existence of TL-equilibrium
SERGIU HART c© 2015 – p. 64
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Proof: 2. existence of TL-equilibrium
For every ε > 0, let Γε be the perturbation of theGAME Γ:
SERGIU HART c© 2015 – p. 64
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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)
SERGIU HART c© 2015 – p. 64
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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)
σ(t|t) ≥ ε(probability of revealing the whole truth is atleast ε)
SERGIU HART c© 2015 – p. 64
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Proof: 2. existence of TL-equilibrium
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
Proposition. Γε has a Nash equilibrium.
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
Proposition. Γε has a Nash equilibrium.
Proof. Standard use of Kakutani’s Fixed PointTheorem.
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.σ(t|t) < 1 ⇒
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.σ(t|t) < 1 ⇒ σ(t|r) = 0 for all r
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.σ(t|t) < 1 ⇒ σ(t|r) = 0 for all r
⇒ q(t) = 1t
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.σ(t|t) < 1 ⇒ σ(t|r) = 0 for all r
⇒ q(t) = 1t
⇒ ρ(t) = v(t)
SERGIU HART c© 2015 – p. 65
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Proof: 2. existence of TL-equilibrium
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.σ(t|t) < 1 ⇒ σ(t|r) = 0 for all r
⇒ q(t) = 1t
⇒ ρ(t) = v(t)
If t ∈ BRA(t) then put σ′(t|t) = 1SERGIU HART c© 2015 – p. 65
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Proof: 2’. mechanism → equilibrium
y
SERGIU HART c© 2015 – p. 66
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Proof: 2’. mechanism → equilibrium
Let π be an OPTIMAL MECHANISM outcome.
SERGIU HART c© 2015 – p. 66
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Proof: 2’. mechanism → equilibrium
Let π be an OPTIMAL MECHANISM outcome.
We will construct a TL-EQUILIBRIUM (σ, ρ) withoutcome π.
SERGIU HART c© 2015 – p. 66
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Proof: 2’. mechanism → equilibrium
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
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)
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
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)
Principal’s strategy: put
ρ(t) = min{πt, v(t)} for each t
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
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)
Agent’s strategy:
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
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)
Agent’s strategy:
Let S := {t : v(t) ≥ πt} – these are themessages that will be used in equilibrium
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
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)
Agent’s strategy:
Let S := {t : v(t) ≥ πt} – these are themessages that will be used in equilibrium
If t ∈ S then put σ(t|t) = 1
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
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)
Agent’s strategy:
Let S := {t : v(t) ≥ πt} – these are themessages that will be used in equilibrium
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
SERGIU HART c© 2015 – p. 67
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
SERGIU HART c© 2015 – p. 68
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)
SERGIU HART c© 2015 – p. 68
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)
A simple case: S is a singleton
SERGIU HART c© 2015 – p. 68
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)
A simple case: S is a singleton
The general case: Partition T into disjointsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S
SERGIU HART c© 2015 – p. 68
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Hall’s Marriage Theorem
SERGIU HART c© 2015 – p. 69
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Hall’s Marriage Theorem
A set B of n boys, and a set G of n girls
SERGIU HART c© 2015 – p. 69
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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
SERGIU HART c© 2015 – p. 69
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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
SERGIU HART c© 2015 – p. 69
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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:
SERGIU HART c© 2015 – p. 69
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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
SERGIU HART c© 2015 – p. 69
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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
Theorem (Hall 1935). The condition is alsosufficient .
SERGIU HART c© 2015 – p. 69
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The Hull of Hall’s Theorem
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such that
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B
Theorem . There exists a partitionof ∪b∈BGb into disjointsets (Fb)b∈B such that for every b ∈ B
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B
Theorem . There exists a partitionof ∪b∈BGb into disjointsets (Fb)b∈B such that for every b ∈ B
Fb ⊆ Gb
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B
Theorem . There exists a partitionof ∪b∈BGb into disjointsets (Fb)b∈B such that for every b ∈ B
Fb ⊆ Gb
γ(Fb) = β({b})
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B
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})
SERGIU HART c© 2015 – p. 70
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The Hull of Hall’s Theorem
Finite sets B and G
For each b ∈ B a subset Gb ⊆ G
Measures β on B and γ on G such thatγ(∪b∈CGb) ≥ β(C) for every C ⊆ Bwith equality for C = B
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
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)
Partition T into disjointsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S
SERGIU HART c© 2015 – p. 71
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)
Partition T into disjoint fractionalsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S
SERGIU HART c© 2015 – p. 71
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Proof: 2’. mechanism → equilibrium
Agent’s strategy:
S := {t : v(t) ≥ πt} (messages used)
We need to determine which messages in Stypes t /∈ S will use
For each s ∈ S putRs := {t /∈ S : s ∈ L(t), πt = πs} ∪ {s}(set of types that may use message s)
Partition T into disjoint fractionalsets Qs ⊆ Rs such that v(Qs) = πs for everys ∈ S↔ the strategy σ
SERGIU HART c© 2015 – p. 71
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SERGIU HART c© 2015 – p. 72
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Glazer and Rubinstein (2004, 2006)
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
Result :{optimal mechanisms} ⊆ {equilibria}
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
Result :{optimal mechanisms} ⊆ {equilibria}
Rewards : mixtures of two outcomes
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
Result :{optimal mechanisms} ⊆ {equilibria}
Rewards : mixtures of two outcomes⇒ linear ht
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
Result :{optimal mechanisms} ⊆ {equilibria}
Rewards : mixtures of two outcomes⇒ linear ht
Extended to concave ht: Sher (2011)
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
Result :{optimal mechanisms} ⊆ {equilibria}
Rewards : mixtures of two outcomes⇒ linear ht
Extended to concave ht: Sher (2011)General condition:"G ENERALIZED SINGLE-PEAKEDNESS "
SERGIU HART c© 2015 – p. 73
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Glazer and Rubinstein (2004, 2006)
Messages : arbitrary ⇒ no "truth" structure
Result :{optimal mechanisms} ⊆ {equilibria}
Rewards : mixtures of two outcomes⇒ linear ht
Extended to concave ht: Sher (2011)General condition:"G ENERALIZED SINGLE-PEAKEDNESS "⇔ "P RINCIPAL’S UNIFORM BEST"(includes convex ht, ...)
SERGIU HART c© 2015 – p. 73
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Summary
SERGIU HART c© 2015 – p. 74
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Summary
EVIDENCE GAMES model very commonsetups
SERGIU HART c© 2015 – p. 74
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Summary
EVIDENCE GAMES model very commonsetups
In EVIDENCE GAMES there is EQUIVALENCEbetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)
SERGIU HART c© 2015 – p. 74
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Summary
EVIDENCE GAMES model very commonsetups
In EVIDENCE GAMES there is EQUIVALENCEbetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)
EQUILIBRIUM is CONSTRAINED EFFICIENT(in the canonical case)
SERGIU HART c© 2015 – p. 74
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Summary
EVIDENCE GAMES model very commonsetups
In EVIDENCE GAMES there is EQUIVALENCEbetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)
EQUILIBRIUM is CONSTRAINED EFFICIENT(in the canonical case)
The conditions of EVIDENCE GAMES areindispensable for this EQUIVALENCE
SERGIU HART c© 2015 – p. 74
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And That Is The Whole Truth ...
SERGIU HART c© 2015 – p. 75
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And That Is The Whole Truth ...
c© Mick Stevens SERGIU HART c© 2015 – p. 75