Implementation without commitment in moral hazard environments - - PowerPoint PPT Presentation

Implementation without commitment in moral hazard environments

Bruno Salcedo

Pennsylvania State University LAMES – November 2013

1 Introduction 2 Interdependent-choice equilibrium 3 Nash implementation without commitment 4 Equilibrium refinements

Coordination

• Coordination or interdependent choices – the choices of some agents

depending on the choices of others

• e.g. others will be good to me if and only if I am good to them
• Standard settings: repeated games or games with contracts
• A player might be willing to choose some alternative today (or sign a

contract) only because of the way his opponents will react in the future to his current choice

• This paper investigates coordination in single-shot interactions without

commitment

Coordination

• Coordination or interdependent choices – the choices of some agents

depending on the choices of others

• e.g. others will be good to me if and only if I am good to them
• Standard settings: repeated games or games with contracts
• A player might be willing to choose some alternative today (or sign a

contract) only because of the way his opponents will react in the future to his current choice

• This paper investigates coordination in single-shot interactions without

commitment

Coordination

• Coordination or interdependent choices – the choices of some agents

depending on the choices of others

• e.g. others will be good to me if and only if I am good to them
• Standard settings: repeated games or games with contracts
• A player might be willing to choose some alternative today (or sign a

contract) only because of the way his opponents will react in the future to his current choice

• This paper investigates coordination in single-shot interactions without

commitment

Moral hazard

• Moral hazard – individual and social incentives are misaligned,

e.g. Nash equilibria are Pareto inefficient

• Moral hazard disappears with complete contracts (Coase’s theorem) or with

repetition and patience (folk theorems)

Question Which outcomes can be sustained as equilibria with the help of a non-strategic mediator but without repetition, monetary transfers or binding contracts?

• Different literatures ask related questions, e.g. Aumann (1974, 1987);

Forges (1986); Myerson (1986)

Moral hazard

• Moral hazard – individual and social incentives are misaligned,

e.g. Nash equilibria are Pareto inefficient

• Moral hazard disappears with complete contracts (Coase’s theorem) or with

repetition and patience (folk theorems)

Question Which outcomes can be sustained as equilibria with the help of a non-strategic mediator but without repetition, monetary transfers or binding contracts?

• Different literatures ask related questions, e.g. Aumann (1974, 1987);

Forges (1986); Myerson (1986)

Moral hazard

• Moral hazard – individual and social incentives are misaligned,

e.g. Nash equilibria are Pareto inefficient

• Moral hazard disappears with complete contracts (Coase’s theorem) or with

repetition and patience (folk theorems)

Question Which outcomes can be sustained as equilibria with the help of a non-strategic mediator but without repetition, monetary transfers or binding contracts?

• Different literatures ask related questions, e.g. Aumann (1974, 1987);

Forges (1986); Myerson (1986)

Moral hazard

• Moral hazard – individual and social incentives are misaligned,

e.g. Nash equilibria are Pareto inefficient

• Moral hazard disappears with complete contracts (Coase’s theorem) or with

repetition and patience (folk theorems)

Question Which outcomes can be sustained as equilibria with the help of a non-strategic mediator but without repetition, monetary transfers or binding contracts?

• Different literatures ask related questions, e.g. Aumann (1974, 1987);

Forges (1986); Myerson (1986)

A prisoner’s dilemma – Nishihara (1997, 1999)

• Two suspects of a crime are arrested
• The DA has enough information to convict them for a misdemeanor but

needs a signed confession to charge them for the alleged crime

• Each prisoner is offered a sentence reduction in exchange for such

confession C D C 1 , 1 −k , 1 + g D 1 + g , −k 0 , 0 k, g > 0, g < 1

. . . managed by a trusted lawyer

• The prisoners have a constitutional right to hire the services of a lawyer

(she) who will schedule and be present in all their interactions with the DA

• They could hire the same lawyer and instruct her as follows:
• You must uniformly randomize the order of our appointments, so that each
• f us will be the first one to receive the offer with probability 1/2
• You must always recommend that we not confess, unless our accomplice has

already confessed

• In that case you should instruct us to also confess
• Other than that, you must not reveal any additional information

. . . managed by a trusted lawyer

• The prisoners have a constitutional right to hire the services of a lawyer

(she) who will schedule and be present in all their interactions with the DA

• They could hire the same lawyer and instruct her as follows:
• You must uniformly randomize the order of our appointments, so that each
• f us will be the first one to receive the offer with probability 1/2
• You must always recommend that we not confess, unless our accomplice has

already confessed

• In that case you should instruct us to also confess
• Other than that, you must not reveal any additional information

. . . managed by a trusted lawyer

• The prisoners have a constitutional right to hire the services of a lawyer

(she) who will schedule and be present in all their interactions with the DA

• They could hire the same lawyer and instruct her as follows:
• You must uniformly randomize the order of our appointments, so that each
• f us will be the first one to receive the offer with probability 1/2
• You must always recommend that we not confess, unless our accomplice has

already confessed

• In that case you should instruct us to also confess
• Other than that, you must not reveal any additional information

Extensive form mechanism

bc b b b b b b b b b b b b b b

1

2

• 1

2

• 1

2 2 2 1 1 D D D D D D C C C C C C 1 + g −k −k 1 + g 1 1 1 1 1 + g −k −k 1 + g

Outline of the paper

1 Interdependent-choice equilibrium

• Mediated games in which a trusted mediator manages the play
• ICE are Nash outcomes of mediated games

2 Revelation principle

• Large class of mechanisms consistent with no commitment, no transfers, and

no repetition

• Corresponding equilibrium outcomes are ICE
• These restrictions rule out a folk theorem

3 Subgame perfection [not in this talk]

Outline of the paper

1 Interdependent-choice equilibrium

• Mediated games in which a trusted mediator manages the play
• ICE are Nash outcomes of mediated games

2 Revelation principle

• Large class of mechanisms consistent with no commitment, no transfers, and

no repetition

• Corresponding equilibrium outcomes are ICE
• These restrictions rule out a folk theorem

3 Subgame perfection [not in this talk]

Outline of the paper

1 Interdependent-choice equilibrium

• Mediated games in which a trusted mediator manages the play
• ICE are Nash outcomes of mediated games

2 Revelation principle

• Large class of mechanisms consistent with no commitment, no transfers, and

no repetition

• Corresponding equilibrium outcomes are ICE
• These restrictions rule out a folk theorem

3 Subgame perfection [not in this talk]

1 Introduction 2 Interdependent-choice equilibrium 3 Nash implementation without commitment 4 Equilibrium refinements

Environment

• Strategic environment partially characterized by:
• Players I = {1, 2}
• Finite sets of actions Ai,

A = ×i∈I Ai

• Utility functions ui : A → R
• NOT a simultaneous move game!!
• Remain agnostic about the sequential and informational structures
• Do not assume that choices are “simultaneous” nor independent

Environment

• Strategic environment partially characterized by:
• Players I = {1, 2}
• Finite sets of actions Ai,

A = ×i∈I Ai

• Utility functions ui : A → R
• NOT a simultaneous move game!!
• Remain agnostic about the sequential and informational structures
• Do not assume that choices are “simultaneous” nor independent

Mediated games

• A non-strategic mediator “manages” the play through sequential private

recommendations

• A mediated game is characterized by a tuple
• α, θ, B = ×i Bi
• α(a) is the probability that the mediator chooses to implement a
• θ(i|a) is the probability that the mediator chooses i to move first, conditional
• n choosing to implement a
• Bi is the set of additional punishments that i can use as credible threats, the

set of actual punishments is B∗

i = Bi ∪ supp(αi)

Mediated games

• A non-strategic mediator “manages” the play through sequential private

recommendations

• A mediated game is characterized by a tuple
• α, θ, B = ×i Bi
• α(a) is the probability that the mediator chooses to implement a
• θ(i|a) is the probability that the mediator chooses i to move first, conditional
• n choosing to implement a
• Bi is the set of additional punishments that i can use as credible threats, the

set of actual punishments is B∗

i = Bi ∪ supp(αi)

Mediated games

• A non-strategic mediator “manages” the play through sequential private

recommendations

• A mediated game is characterized by a tuple
• α, θ, B = ×i Bi
• α(a) is the probability that the mediator chooses to implement a
• θ(i|a) is the probability that the mediator chooses i to move first, conditional
• n choosing to implement a
• Bi is the set of additional punishments that i can use as credible threats, the

set of actual punishments is B∗

i = Bi ∪ supp(αi)

Mediated games

• A non-strategic mediator “manages” the play through sequential private

recommendations

• A mediated game is characterized by a tuple
• α, θ, B = ×i Bi
• α(a) is the probability that the mediator chooses to implement a
• θ(i|a) is the probability that the mediator chooses i to move first, conditional
• n choosing to implement a
• Bi is the set of additional punishments that i can use as credible threats, the

set of actual punishments is B∗

i = Bi ∪ supp(αi)

Mediated games

• α, θ, B
• Mediator choices

1 The game begins with the mediator privately choosing the action profile

a∗ ∈ A to implement and the player i∗ ∈ I to move first

• a∗ is randomly chosen according to α ∈ ∆(A)
• i∗ is randomly chosen according to θ( · |a∗) ∈ ∆(I)

Mediated games

• α, θ, B
• Sequential visits

1 The game begins with the mediator choosing a∗ ∼ α and i∗ ∼ θ( · |a∗) 2 She then “visits” players one by one, i∗ first and −i∗ second

• In each visit, she recommends an action ar

j and observes the action ap j

actually played

• Players are only informed about the recommendations they receive

Mediated games

• α, θ, B
• Sequential visits

1 The game begins with the mediator choosing a∗ ∼ α and i∗ ∼ θ( · |a∗) 2 She then “visits” players one by one, i∗ first and −i∗ second

• In each visit, she recommends an action ar

j and observes the action ap j

actually played

• Players are only informed about the recommendations they receive

Mediated games

• α, θ, B
• Recommendations

1 The game begins with the mediator choosing a∗ ∼ α and i∗ ∼ θ( · |a∗) 2 She then “visits” players in the chose order, recommends an action ar j and

• bserves the action ap

j actually played 3 The recommendations are chosen as follows:

• i∗ is always recommended to play the intended action a∗

i

• She recommends the intended action to −i∗ if i∗ complied, and the worst

available punishment otherwise: ar

−i∗ = a∗ −i∗

if ap

i∗ = ar i∗

ar

−i∗ ∈ arg min a−i∗ ∈B∗

−i∗

ui∗(ap

i∗, a−i∗)

if ap

i∗ = ar i∗

Mediated games

• α, θ, B
• Recommendations

1 The game begins with the mediator choosing a∗ ∼ α and i∗ ∼ θ( · |a∗) 2 She then “visits” players in the chose order, recommends an action ar j and

• bserves the action ap

j actually played 3 The recommendations are chosen as follows:

• i∗ is always recommended to play the intended action a∗

i

• She recommends the intended action to −i∗ if i∗ complied, and the worst

available punishment otherwise: ar

−i∗ = a∗ −i∗

if ap

i∗ = ar i∗

ar

−i∗ ∈ arg min a−i∗ ∈B∗

−i∗

ui∗(ap

i∗, a−i∗)

if ap

i∗ = ar i∗

Interdependent-choice equilibrium

• A mediated game is incentive compatible if following the mediator’s

recommendations constitutes a Nash equilibrium

• That is, if for every player i and actions ai, a′

i:

• a−i ∈A−i

α

• a
• ·
• ui(a) − θ(−i|a) ui(a′

i, a−i) − θ(i|a) wi(a′ i|B∗)

• ≥ 0

Definition An interdependent-choice equilibrium with respect to B is a distribution α ∈ ∆(A) such that (α, θ, B) is IC for some θ

Interdependent-choice equilibrium

• A mediated game is incentive compatible if following the mediator’s

recommendations constitutes a Nash equilibrium

• That is, if for every player i and actions ai, a′

i:

• a−i ∈A−i

α

• a
• ·
• ui(a) − θ(−i|a) ui(a′

i, a−i) − θ(i|a) wi(a′ i|B∗)

• ≥ 0

Definition An interdependent-choice equilibrium with respect to B is a distribution α ∈ ∆(A) such that (α, θ, B) is IC for some θ

Observations

• The set of ICE (with respect to A) is a closed and convex polytope
• It contains the set of correlated equilibria and is contained in the set of

individually rational distributions

• The containments can be strict
• It is ⊆-monotone with respect to B
• Stronger incentive conditions can be accommodated adjusting B

Observations

• The set of ICE (with respect to A) is a closed and convex polytope
• It contains the set of correlated equilibria and is contained in the set of

individually rational distributions

• The containments can be strict
• It is ⊆-monotone with respect to B
• Stronger incentive conditions can be accommodated adjusting B

Observations

• The set of ICE (with respect to A) is a closed and convex polytope
• It contains the set of correlated equilibria and is contained in the set of

individually rational distributions

• The containments can be strict
• It is ⊆-monotone with respect to B
• Stronger incentive conditions can be accommodated adjusting B

Example: a game of chicken

b b b b b

S W S 0 , 0 5 , 1 W 1 , 5 4 , 4 u1 u2 (S, S) (S, W) (W, S) (W, W) 1 4 5 1 4 5

Example: a game of chicken

Individually rational Nash hull u1 u2 (S, S) (S, W) (W, S) (W, W) 1 4 5 1 4 5

Example: a game of chicken

Individually rational Correlated Nash hull u1 u2 (S, S) (S, W) (W, S) (W, W) 1 4 5 1 4 5

Example: a game of chicken

Individually rational Correlated Nash hull Interdependent-choice u1 u2 (S, S) (S, W) (W, S) (W, W) 1 4 5 1 4 5

1 Introduction 2 Interdependent-choice equilibrium 3 Nash implementation without commitment 4 Equilibrium refinements

Nash implementation

• An extensive form mechanism is any EFG consistent with:
• The information that we have about the environment, i.e. payoff-relevant

actions and preferences

• No-transfers, no-commitment and no-repetition
• Say that α is Nash implementable iff it can result as a NE of an EFM
• Revelation principle – the set of ICE with B = A characterizes the

Nash-implementable outcomes

Nash implementation

• An extensive form mechanism is any EFG consistent with:
• The information that we have about the environment, i.e. payoff-relevant

actions and preferences

• No-transfers, no-commitment and no-repetition
• Say that α is Nash implementable iff it can result as a NE of an EFM
• Revelation principle – the set of ICE with B = A characterizes the

Nash-implementable outcomes

Nash implementation

• An extensive form mechanism is any EFG consistent with:
• The information that we have about the environment, i.e. payoff-relevant

actions and preferences

• No-transfers, no-commitment and no-repetition
• Say that α is Nash implementable iff it can result as a NE of an EFM
• Revelation principle – the set of ICE with B = A characterizes the

Nash-implementable outcomes

Extensive form mechanisms

• Impose only three requirements:

1 Outcome equivalence

Each terminal node (outcome of the game) can be identified with an action profile (outcome of the environment)

2 Strategic equivalence

Each player i chooses an action ai along every path

3 No partial commitment

At the moment of choosing his action, each player i could have choosen any

• ther action in Ai

Extensive form mechanisms

• Impose only three requirements:

1 Outcome equivalence

Each terminal node (outcome of the game) can be identified with an action profile (outcome of the environment)

2 Strategic equivalence

Each player i chooses an action ai along every path

3 No partial commitment

At the moment of choosing his action, each player i could have choosen any

• ther action in Ai

Extensive form mechanisms

• Impose only three requirements:

1 Outcome equivalence

Each terminal node (outcome of the game) can be identified with an action profile (outcome of the environment)

2 Strategic equivalence

Each player i chooses an action ai along every path

3 No partial commitment

At the moment of choosing his action, each player i could have choosen any

• ther action in Ai

Examples of mechanisms

b b b b b b b b b b b b b bc

1 2 1 1 2 2 pass a1 a′

1

a2 a′

2

a1 a′

1

a1 a′

1

a2 a′

2

a2 a′

2

b b b b b b b b b bc

1 2 2 2 a1, show a′

1

a1, hide a2 a′

2

a2 a′

2

a2 a′

2

Revelation principle

Theorem A distribution over outcome profiles is Nash implementable if and only if it is an interdependent-choice equilibrium

• The sufficiency is straightforward because mediated games are extensive

form mechanisms

• Necessity follows from standard arguments:
• Given any EFM and equilibrium, one can recover a mediated game in which

all the non-relevant choices are delegated to the mediator

• Players have less information and deviations are less profitable

Revelation principle

Theorem A distribution over outcome profiles is Nash implementable if and only if it is an interdependent-choice equilibrium

• The sufficiency is straightforward because mediated games are extensive

form mechanisms

• Necessity follows from standard arguments:
• Given any EFM and equilibrium, one can recover a mediated game in which

all the non-relevant choices are delegated to the mediator

• Players have less information and deviations are less profitable

A prisoner’s dilemma

C D C 1 , 1 −k , 1 + g D 1 + g , −k 0 , 0

Proposition In a prisoner’s dilemma game, a distribution α is Nash implementable if and only if: (1 − g) · α(C, C) ≥ k

• α(D, C) + α(C, D)

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 1.00

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.95

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.90

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.85

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.80

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.75

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.70

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.65

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.60

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.55

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.50

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.45

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.40

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.35

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.30

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.25

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.20

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.15

A prisoner’s dilemma – ICE payoffs

b b b b

u1 u2 g = 0.10

1 Introduction 2 Interdependent-choice equilibrium 3 Nash implementation without commitment 4 Equilibrium refinements

Sequential rationality

• ICE is defined as if players could commit to punish off the equilibrium path
• Different forms of sequential rationality can be guaranteed by restricting credible

threats B to be:

• Actions which can be played with positive probability in some ICE with

respect to ∅

• Actions with are not absolutely dominated (Halpern and Pass, 2012)
• Combinations of the previous two

Sequential rationality

• ICE is defined as if players could commit to punish off the equilibrium path
• Different forms of sequential rationality can be guaranteed by restricting credible

threats B to be:

• Actions which can be played with positive probability in some ICE with

respect to ∅

• Actions with are not absolutely dominated (Halpern and Pass, 2012)
• Combinations of the previous two

. . . has little bite

Theorem For generic 2 × 2 games, and for games with a completely mixed ICE or without strict dominance, ‘almost’ every ICE is sequentially implementable

u1 u2

Thank you!

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Related literature

• Mediators
• Aumann (1974, 1987); Forges (1986); Myerson (1986);

Bergemann and Morris (2011) . . .

• Ashlagi et al. (2009); Monderer and Tennenholtz (2009);

Forgó (2010); Moulin et al. (2013); Moulin and Vial (1978) . . .

• Preplay negotiation

Kalai et al. (2010); Jackson and Wilkie (2005); Renou (2009); Bade et al. (2009); Kalai (1981, 2004) . . .

• Asynchronous moves

Kamada and Kandori (2009, 2012); Solan and Yariv (2004); Van Damme and Hurkens (1999) . . .

• Implicit interdependence

Halpern and Rong (2010); Halpern and Pass (2012); Eisert et al. (1999); Figuiéres et al. (2004); Rapoport (1965) . . . return

Related literature

• Mediators
• Aumann (1974, 1987); Forges (1986); Myerson (1986);

Bergemann and Morris (2011) . . .

• Ashlagi et al. (2009); Monderer and Tennenholtz (2009);

Forgó (2010); Moulin et al. (2013); Moulin and Vial (1978) . . .

• Preplay negotiation

Kalai et al. (2010); Jackson and Wilkie (2005); Renou (2009); Bade et al. (2009); Kalai (1981, 2004) . . .

• Asynchronous moves

Kamada and Kandori (2009, 2012); Solan and Yariv (2004); Van Damme and Hurkens (1999) . . .

• Implicit interdependence

Halpern and Rong (2010); Halpern and Pass (2012); Eisert et al. (1999); Figuiéres et al. (2004); Rapoport (1965) . . . return

Related literature

• Mediators
• Aumann (1974, 1987); Forges (1986); Myerson (1986);

Bergemann and Morris (2011) . . .

• Ashlagi et al. (2009); Monderer and Tennenholtz (2009);

Forgó (2010); Moulin et al. (2013); Moulin and Vial (1978) . . .

• Preplay negotiation

Kalai et al. (2010); Jackson and Wilkie (2005); Renou (2009); Bade et al. (2009); Kalai (1981, 2004) . . .

• Asynchronous moves

Kamada and Kandori (2009, 2012); Solan and Yariv (2004); Van Damme and Hurkens (1999) . . .

• Implicit interdependence

Halpern and Rong (2010); Halpern and Pass (2012); Eisert et al. (1999); Figuiéres et al. (2004); Rapoport (1965) . . . return

Related literature

• Mediators
• Aumann (1974, 1987); Forges (1986); Myerson (1986);

Bergemann and Morris (2011) . . .

• Ashlagi et al. (2009); Monderer and Tennenholtz (2009);

Forgó (2010); Moulin et al. (2013); Moulin and Vial (1978) . . .

• Preplay negotiation

Kalai et al. (2010); Jackson and Wilkie (2005); Renou (2009); Bade et al. (2009); Kalai (1981, 2004) . . .

• Asynchronous moves

Kamada and Kandori (2009, 2012); Solan and Yariv (2004); Van Damme and Hurkens (1999) . . .

• Implicit interdependence

Halpern and Rong (2010); Halpern and Pass (2012); Eisert et al. (1999); Figuiéres et al. (2004); Rapoport (1965) . . . return