Repeated Games
and Reputations
Long-Run Relationships
George J. Mailath and Larry Samuelson
1
2006
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Contents
1 Introduction
1.1 Intertemporal Incentives
1.2 The Prisoners’ Dilemma
1.3 Oligopoly
1.4 The Prisoner’s Dilemma under Imperfect Monitoring
1.5 The Product-Choice Game
1.6 Discussion
1.7 A Reader’s Guide
1.8 The Scope of the Book
Part I
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Games with Perfect Monitoring
2 The Basic Structure of Repeated Games with Perfect Monitoring
2.1
2.2
2.3
2.4
2.5
2.6
2.7
The Canonical Repeated Game
2.1.1 The Stage Game
2.1.2 Public Correlation
2.1.3 The Repeated Game
2.1.4 Subgame-Perfect Equilibrium of the Repeated Game
The One-Shot Deviation Principle
Automaton Representations of Strategy Profiles
Credible Continuation Promises
Generating Equilibria
2.5.1 Constructing Equilibria: Self-Generation
2.5.2 Example: Mutual Effort
2.5.3 Example: The Folk Theorem
2.5.4 Example: Constructing Equilibria for Low δ
2.5.5 Example: Failure of Monotonicity
2.5.6 Example: Public Correlation
Constructing Equilibria: Simple Strategies and Penal Codes
2.6.1 Simple Strategies and Penal Codes
2.6.2 Example: Oligopoly
Long-Lived and Short-Lived Players
2.7.1 Minmax Payoffs
2.7.2 Constraints on Payoffs
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3 The Folk Theorem with Perfect Monitoring
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Examples
Interpreting the Folk Theorem
3.2.1 Implications
3.2.2 Patient Players
3.2.3 Patience and Incentives
3.2.4 Observable Mixtures
The Pure-Action Folk Theorem for Two Players
The Folk Theorem with More than Two Players
3.4.1 A Counterexample
3.4.2 Player-Specific Punishments
Non-Equivalent Utilities
Long-Lived and Short-Lived Players
Convexifying the Equilibrium Payoff Set Without
Public Correlation
Mixed-Action Individual Rationality
4 How Long Is Forever?
4.1 Is the Horizon Ever Infinite?
4.2 Uncertain Horizons
4.3 Declining Discount Factors
4.4 Finitely Repeated Games
4.5 Approximate Equilibria
4.6 Renegotiation
4.6.1 Finitely Repeated Games
4.6.2 Infinitely Repeated Games
5 Variations on the Game
Random Matching
5.1.1 Public Histories
5.1.2 Personal Histories
5.2 Relationships in Context
5.2.1 A Frictionless Market
5.2.2 Future Benefits
5.2.3 Adverse Selection
5.2.4 Starting Small
5.3 Multimarket Interactions
5.4 Repeated Extensive Forms
5.4.1 Repeated Extensive-Form Games Have More Subgames
5.4.2 Player-Specific Punishments in Repeated Extensive-Form
Games
5.4.3 Extensive-Form Games and Imperfect Monitoring
5.4.4 Extensive-Form Games and Weak Individual Rationality
5.4.5 Asynchronous Moves
5.4.6 Simple Strategies
5.1
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Dynamic Games: Introduction
5.5.1 The Game
5.5.2 Markov Equilibrium
5.5.3 Examples
5.6 Dynamic Games: Foundations
5.6.1 Consistent Partitions
5.6.2 Coherent Consistency
5.6.3 Markov Equilibrium
5.7 Dynamic Games: Equilibrium
5.7.1 The Structure of Equilibria
5.7.2 A Folk Theorem
5.5
6 Applications
Price Wars
6.1.1 Independent Price Shocks
6.1.2 Correlated Price Shocks
6.2 Time Consistency
6.2.1 The Stage Game
6.2.2 Equilibrium, Commitment, and Time Consistency
6.2.3 The Infinitely Repeated Game
6.3 Risk Sharing
6.3.1 The Economy
6.3.2 Full Insurance Allocations
6.3.3 Partial Insurance
6.3.4 Consumption Dynamics
6.3.5 Intertemporal Consumption Sensitivity
6.1
Part II
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Games with (Imperfect) Public Monitoring
7 The Basic Structure of Repeated Games with Imperfect Public
Monitoring
7.1
7.2
7.3
7.4
7.5
The Canonical Repeated Game
7.1.1 The Stage Game
7.1.2 The Repeated Game
7.1.3 Recovering a Recursive Structure: Public Strategies and
Perfect Public Equilibria
A Repeated Prisoners’ Dilemma Example
7.2.1 Punishments Happen
7.2.2 Forgiving Strategies
7.2.3 Strongly Symmetric Behavior Implies Inefficiency
Decomposability and Self-Generation
The Impact of Increased Precision
The Bang-Bang Result
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7.6 An Example with Short-Lived Players
7.6.1 Perfect Monitoring
7.6.2 Imperfect Public Monitoring of the Long-Lived Player
7.7 The Repeated Prisoners’ Dilemma Redux
7.7.1 Symmetric Inefficiency Revisited
7.7.2 Enforcing a Mixed-Action Profile
7.8 Anonymous Players
8 Bounding Perfect Public Equilibrium Payoffs
Decomposing on Half-Spaces
The Inefficiency of Strongly Symmetric Equilibria
Short-Lived Players
8.3.1 The Upper Bound on Payoffs
8.3.2 Binding Moral Hazard
8.4 The Prisoners’ Dilemma
8.4.1 Bounds on Efficiency: Pure Actions
8.4.2 Bounds on Efficiency: Mixed Actions
8.4.3 A Characterization with Two Signals
8.4.4 Efficiency with Three Signals
8.4.5 Efficient Asymmetry
8.1
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8.3
9 The Folk Theorem with Imperfect Public Monitoring
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Characterizing the Limit Set of PPE Payoffs
The Rank Conditions and a Public Monitoring Folk Theorem
Perfect Monitoring Characterizations
9.3.1 The Folk Theorem with Long-Lived Players
9.3.2 Long-Lived and Short-Lived Players
Enforceability and Identifiability
Games with a Product Structure
Repeated Extensive-Form Games
Games of Symmetric Incomplete Information
9.7.1 Equilibrium
9.7.2 A Folk Theorem
Short Period Length
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10 Private Strategies in Games with Imperfect Public Monitoring
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10.1 Sequential Equilibrium
10.2 A Reduced-Form Example
10.2.1 Pure Strategies
10.2.2 Public Correlation
10.2.3 Mixed Public Strategies
10.2.4 Private Strategies
10.3 Two-Period Examples
10.3.1 Equilibrium Punishments Need Not Be Equilibria
10.3.2 Payoffs by Correlation
10.3.3 Inconsistent Beliefs
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10.4 An Infinitely Repeated Prisoner’s Dilemma
10.4.1 Public Transitions
10.4.2 An Infinitely Repeated Prisoners’ Dilemma: Indifference
11 Applications
11.1 Oligopoly with Imperfect Monitoring
11.1.1 The Game
11.1.2 Optimal Collusion
11.1.3 Which News Is Bad News?
11.1.4 Imperfect Collusion
11.2 Repeated Adverse Selection
11.2.1 General Structure
11.2.2 An Oligopoly with Private Costs: The Game
11.2.3 A Uniform-Price Equilibrium
11.2.4 A Stationary-Outcome Separating Equilibrium
11.2.5 Efficiency
11.2.6 Nonstationary-Outcome Equilibria
11.3 Risk Sharing
11.4 Principal-Agent Problems
11.4.1 Hidden Actions
11.4.2 Incomplete Contracts: The Stage Game
11.4.3 Incomplete Contracts: The Repeated Game
11.4.4 Risk Aversion: The Stage Game
11.4.5 Risk Aversion: Review Strategies in the Repeated Game
Part III
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Games with Private Monitoring
12 Private Monitoring
12.1 A Two-Period Example
12.1.1 Almost Public Monitoring
12.1.2 Conditionally Independent Monitoring
12.1.3 Intertemporal Incentives from Second-Period
12.2
12.3
12.4
12.5
Randomization
Private Monitoring Games: Basic Structure
Almost Public Monitoring: Robustness in the Infinitely Repeated
Prisoner’s Dilemma
12.3.1 The Forgiving Profile
12.3.2 Grim Trigger
Independent Monitoring: A Belief-Based Equilibrium for the
Infinitely Repeated Prisoner’s Dilemma
A Belief-Free Example
13 Almost Public Monitoring Games
13.1 When Is Monitoring Almost Public?
13.2 Nearby Games with Almost Public Monitoring
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13.2.1 Payoffs
13.2.2 Continuation Values
13.2.3 Best Responses
13.2.4 Equilibrium
13.3 Public Profiles with Bounded Recall
13.4 Failure of Coordination under Unbounded Recall
13.4.1 Examples
13.4.2 Incentives to Deviate
13.4.3 Separating Profiles
13.4.4 Rich Monitoring
13.4.5 Coordination Failure
13.5 Patient Players
13.5.1 Patient Strictness
13.5.2 Equilibria in Nearby Games
13.6 A Folk Theorem
14 Belief-Free Equilibria in Private Monitoring Games
14.1 Definition and Examples
14.1.1 Repeated Prisoners’ Dilemma with Perfect Monitoring
14.1.2 Repeated Prisoners’ Dilemma with Private Monitoring
14.2 Strong Self-Generation
Part IV
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15 Reputations with Short-Lived Players
15.1 The Adverse Selection Approach to Reputations
15.2 Commitment Types
15.3 Perfect Monitoring Games
15.3.1 Building a Reputation
15.3.2 The Reputation Bound
15.3.3 An Example: Time Consistency
15.4 Imperfect Monitoring Games
15.4.1 Stackelberg Payoffs
15.4.2 The Reputation Bound
15.4.3 Small Players with Idiosyncratic Signals
15.5 Temporary Reputations
15.5.1 Asymptotic Beliefs
15.5.2 Uniformly Disappearing Reputations
15.5.3 Asymptotic Equilibrium Play
15.6 Temporary Reputations: The Proof of Proposition 15.5.1
15.6.1 Player 2’s Posterior Beliefs
15.6.2 Player 2’s Beliefs about Her Future Behavior
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15.6.3 Player 1’s Beliefs about Player 2’s Future Behavior
15.6.4 Proof of Proposition 15.5.1
16 Reputations with Long-Lived Players
16.1 The Basic Issue
16.2 Perfect Monitoring and Minmax-Action Reputations
16.2.1 Minmax-Action Types and Conflicting Interests
16.2.2 Examples
16.2.3 Two-Sided Incomplete Information
16.3 Weaker Reputations for Any Action
16.4 Imperfect Public Monitoring
16.5 Commitment Types Who Punish
16.6 Equal Discount Factors
16.6.1 Example 1: Common Interests
16.6.2 Example 2: Conflicting Interests
16.6.3 Example 3: Strictly Dominant Action Games
16.6.4 Example 4: Strictly Conflicting Interests
16.6.5 Bounded Recall
16.6.6 Reputations and Bargaining
16.7 Temporary Reputations
17 Finitely Repeated Games
17.1 The Chain Store Game
17.2 The Prisoners’ Dilemma
17.3 The Product-Choice Game
17.3.1 The Last Period
17.3.2 The First Period, Player 1
17.3.3 The First Period, Player 2
18 Modeling Reputations
18.1 An Alternative Model of Reputations
18.1.1 Modeling Reputations
18.1.2 The Market
18.1.3 Reputation with Replacements
18.1.4 How Different Is It?
18.2 The Role of Replacements
18.3 Good Types and Bad Types
18.3.1 Bad Types
18.3.2 Good Types
18.4 Reputations with Common Consumers
18.4.1 Belief-Free Equilibria with Idiosyncratic Consumers
18.4.2 Common Consumers
18.4.3 Reputations
18.4.4 Replacements
18.4.5 Continuity at the Boundary and Markov Equilibria
18.4.6 Competitive Markets
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18.5 Discrete Choices
18.6 Lost Consumers
18.6.1 The Purchase Game
18.6.2 Bad Reputations: The Stage Game
18.6.3 The Repeated Game
18.6.4 Incomplete Information
18.6.5 Good Firms
18.6.6 Captive Consumers
18.7 Markets for Reputations
18.7.1 Reputations Have Value
18.7.2 Buying Reputations
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Bibliography
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Symbols
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Index
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