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Repeated Games and Reputations: Long-Run Relationships
by George J. Mailath (Author), Larry Samuelson (Author)
"Repeated Games is comprehensive, self-contained, and extremely clear, with proofs that not infrequently improve on the originals. The book is an ideal text for part or all of a second graduate class in game theory, and will be a valuable aid for any student of the field."--Drew Fudenberg, Professor of Economics, Harvard University
"George Mailath and Larry Samuelson have written a landmark book in game theory, which takes stock of decades of research on repeated games and dynamic games more generally. The book not only provides an insightful synthesis of the extensive literatures relating to folk theorems, reputation, and play under a variety of information and monitoring structures; but perhaps more importantly it provides some original proofs that shed new light on some of the central results in these areas. This book will be an invaluable resource for researchers in the area, and should also quickly become a standard reading for advanced graduate students."--Matthew O. Jackson, Edie and Lew Wasserman Professor of Economics, California Institute of Technology
"The study of repeated games has been one of the most fruitful and important developments in economic theory in the last thirty years. In this beautifully lucid book, George Mailath and Larry Samuelson--two leading researchers in the field--lay out the classic results in detail and also bring the reader up to date with the latest findings."--Eric S. Maskin, A.O. Hirschman Professor of Social Science, Institute for Advanced Study
"The theory of reputations in repeated games has become one of the most important areas of research in economic theory, because it offers essential insights into the foundations of economic and political institutions. The past decade has seen great progress in this area, especially in the study of games with imperfect private monitoring. George Mailath and Larry Samuelson have been active leaders in this research, and here they systematically lay out the state of the art. This book will be an important text and reference for years to come."--Roger Myerson, University of Chicago
"Theorists use repeated games to understand self-enforcing contracts, and to explore the power of reputation formation in strategic settings. The centrality of these ideas explains why, despite the technical challenges involved, the literature on repeated games has grown rapidly in recent years. With their masterful treatment of many of the most important parts of this vast territory, Mailath and Samuelson have done a great service to both students and researchers."--David G. Pearce, Department of Economics, New York University
Book Description
Personalized and continuing relationships play a central role in any society. Economists have built upon the theories of repeated games and reputations to make important advances in understanding such relationships. Repeated Games and Reputations begins with a careful development of the fundamental concepts in these theories, including the notions of a repeated game, strategy, and equilibrium. Mailath and Samuelson then present the classic folk theorem and reputation results for games of perfect and imperfect public monitoring, with the benefit of the modern analytical tools of decomposability and self-generation. They also present more recent developments, including results beyond folk theorems and recent work in games of private monitoring and alternative approaches to reputations. Repeated Games and Reputations synthesizes and unifies the vast body of work in this area, bringing the reader to the research frontier. Detailed arguments and proofs are given throughout, interwoven with examples, discussions of how the theory is to be used in the study of relationships, and economic applications. The book will be useful to those doing basic research in the theory of repeated games and reputations as well as those using these tools in more applied research.
1 Introduction 1
1.1 Intertemporal Incentives 1
1.2 The Prisoners’ Dilemma 3
1.3 Oligopoly 4
1.4 The Prisoner’s Dilemma under Imperfect Monitoring 5
1.5 The Product-Choice Game 7
1.6 Discussion 8
1.7 A Reader’s Guide 10
1.8 The Scope of the Book 10
Part I Games with Perfect Monitoring
2 The Basic Structure of Repeated Games with Perfect Monitoring 15
2.1 The Canonical Repeated Game 15
2.1.1 The Stage Game 15
2.1.2 Public Correlation 17
2.1.3 The Repeated Game 19
2.1.4 Subgame-Perfect Equilibrium of the Repeated Game 22
2.2 The One-Shot Deviation Principle 24
2.3 Automaton Representations of Strategy Profiles 29
2.4 Credible Continuation Promises 32
2.5 Generating Equilibria 37
2.5.1 Constructing Equilibria: Self-Generation 37
2.5.2 Example: Mutual Effort 40
2.5.3 Example: The Folk Theorem 41
2.5.4 Example: Constructing Equilibria for Low δ 44
2.5.5 Example: Failure of Monotonicity 46
2.5.6 Example: Public Correlation 49
2.6 Constructing Equilibria: Simple Strategies and Penal Codes 51
2.6.1 Simple Strategies and Penal Codes 51
2.6.2 Example: Oligopoly 54
2.7 Long-Lived and Short-Lived Players 61
2.7.1 Minmax Payoffs 63
2.7.2 Constraints on Payoffs 66
x Contents
3 The Folk Theorem with Perfect Monitoring 69
3.1 Examples 70
3.2 Interpreting the Folk Theorem 72
3.2.1 Implications 72
3.2.2 Patient Players 73
3.2.3 Patience and Incentives 75
3.2.4 Observable Mixtures 76
3.3 The Pure-Action Folk Theorem for Two Players 76
3.4 The Folk Theorem with More than Two Players 80
3.4.1 A Counterexample 80
3.4.2 Player-Specific Punishments 82
3.5 Non-Equivalent Utilities 87
3.6 Long-Lived and Short-Lived Players 91
3.7 Convexifying the Equilibrium Payoff SetWithout
Public Correlation 96
3.8 Mixed-Action Individual Rationality 101
4 How Long Is Forever? 105
4.1 Is the Horizon Ever Infinite? 105
4.2 Uncertain Horizons 106
4.3 Declining Discount Factors 107
4.4 Finitely Repeated Games 112
4.5 Approximate Equilibria 118
4.6 Renegotiation 120
4.6.1 Finitely Repeated Games 122
4.6.2 Infinitely Repeated Games 134
5 Variations on the Game 145
5.1 Random Matching 145
5.1.1 Public Histories 146
5.1.2 Personal Histories 147
5.2 Relationships in Context 152
5.2.1 A Frictionless Market 153
5.2.2 Future Benefits 154
5.2.3 Adverse Selection 155
5.2.4 Starting Small 158
5.3 Multimarket Interactions 161
5.4 Repeated Extensive Forms 162
5.4.1 Repeated Extensive-Form Games Have More Subgames 163
5.4.2 Player-Specific Punishments in Repeated Extensive-Form
Games 165
5.4.3 Extensive-Form Games and Imperfect Monitoring 167
5.4.4 Extensive-Form Games and Weak Individual Rationality 168
5.4.5 Asynchronous Moves 169
5.4.6 Simple Strategies 172
Contents xi
5.5 Dynamic Games: Introduction 174
5.5.1 The Game 175
5.5.2 Markov Equilibrium 177
5.5.3 Examples 178
5.6 Dynamic Games: Foundations 186
5.6.1 Consistent Partitions 187
5.6.2 Coherent Consistency 188
5.6.3 Markov Equilibrium 190
5.7 Dynamic Games: Equilibrium 192
5.7.1 The Structure of Equilibria 192
5.7.2 A Folk Theorem 195
6 Applications 201
6.1 PriceWars 201
6.1.1 Independent Price Shocks 201
6.1.2 Correlated Price Shocks 203
6.2 Time Consistency 204
6.2.1 The Stage Game 204
6.2.2 Equilibrium, Commitment, and Time Consistency 206
6.2.3 The Infinitely Repeated Game 207
6.3 Risk Sharing 208
6.3.1 The Economy 209
6.3.2 Full Insurance Allocations 210
6.3.3 Partial Insurance 212
6.3.4 Consumption Dynamics 213
6.3.5 Intertemporal Consumption Sensitivity 219
Part II Games with (Imperfect) Public Monitoring
7 The Basic Structure of Repeated Games with Imperfect Public
Monitoring 225
7.1 The Canonical Repeated Game 225
7.1.1 The Stage Game 225
7.1.2 The Repeated Game 226
7.1.3 Recovering a Recursive Structure: Public Strategies and
Perfect Public Equilibria 228
7.2 A Repeated Prisoners’ Dilemma Example 232
7.2.1 Punishments Happen 233
7.2.2 Forgiving Strategies 235
7.2.3 Strongly Symmetric Behavior Implies Inefficiency 239
7.3 Decomposability and Self-Generation 241
7.4 The Impact of Increased Precision 249
7.5 The Bang-Bang Result 251
xii Contents
7.6 An Example with Short-Lived Players 255
7.6.1 Perfect Monitoring 256
7.6.2 Imperfect Public Monitoring of the Long-Lived Player 260
7.7 The Repeated Prisoners’ Dilemma Redux 264
7.7.1 Symmetric Inefficiency Revisited 264
7.7.2 Enforcing a Mixed-Action Profile 267
7.8 Anonymous Players 269
8 Bounding Perfect Public Equilibrium Payoffs 273
8.1 Decomposing on Half-Spaces 273
8.2 The Inefficiency of Strongly Symmetric Equilibria 278
8.3 Short-Lived Players 280
8.3.1 The Upper Bound on Payoffs 280
8.3.2 Binding Moral Hazard 281
8.4 The Prisoners’ Dilemma 282
8.4.1 Bounds on Efficiency: Pure Actions 282
8.4.2 Bounds on Efficiency: Mixed Actions 284
8.4.3 A Characterization with Two Signals 287
8.4.4 Efficiency with Three Signals 289
8.4.5 Efficient Asymmetry 291
9 The Folk Theorem with Imperfect Public Monitoring 293
9.1 Characterizing the Limit Set of PPE Payoffs 293
9.2 The Rank Conditions and a Public Monitoring Folk Theorem 298
9.3 Perfect Monitoring Characterizations 303
9.3.1 The Folk Theorem with Long-Lived Players 303
9.3.2 Long-Lived and Short-Lived Players 303
9.4 Enforceability and Identifiability 305
9.5 Games with a Product Structure 309
9.6 Repeated Extensive-Form Games 311
9.7 Games of Symmetric Incomplete Information 316
9.7.1 Equilibrium 318
9.7.2 A Folk Theorem 320
9.8 Short Period Length 326
10 Private Strategies in Games with Imperfect Public Monitoring 329
10.1 Sequential Equilibrium 329
10.2 A Reduced-Form Example 331
10.2.1 Pure Strategies 331
10.2.2 Public Correlation 332
10.2.3 Mixed Public Strategies 332
10.2.4 Private Strategies 333
10.3 Two-Period Examples 334
10.3.1 Equilibrium Punishments Need Not Be Equilibria 334
10.3.2 Payoffs by Correlation 337
10.3.3 Inconsistent Beliefs 338
Contents xiii
10.4 An Infinitely Repeated Prisoner’s Dilemma 340
10.4.1 Public Transitions 340
10.4.2 An Infinitely Repeated Prisoners’ Dilemma: Indifference 343
11 Applications 347
11.1 Oligopoly with Imperfect Monitoring 347
11.1.1 The Game 347
11.1.2 Optimal Collusion 348
11.1.3 Which News Is Bad News? 350
11.1.4 Imperfect Collusion 352
11.2 Repeated Adverse Selection 354
11.2.1 General Structure 354
11.2.2 An Oligopoly with Private Costs: The Game 355
11.2.3 A Uniform-Price Equilibrium 356
11.2.4 A Stationary-Outcome Separating Equilibrium 357
11.2.5 Efficiency 359
11.2.6 Nonstationary-Outcome Equilibria 360
11.3 Risk Sharing 365
11.4 Principal-Agent Problems 370
11.4.1 Hidden Actions 370
11.4.2 Incomplete Contracts: The Stage Game 371
11.4.3 Incomplete Contracts: The Repeated Game 372
11.4.4 Risk Aversion: The Stage Game 374
11.4.5 Risk Aversion: Review Strategies in the Repeated Game 375
Part III Games with Private Monitoring
12 Private Monitoring 385
12.1 ATwo-Period Example 385
12.1.1 Almost Public Monitoring 387
12.1.2 Conditionally Independent Monitoring 389
12.1.3 Intertemporal Incentives from Second-Period
Randomization 392
12.2 Private Monitoring Games: Basic Structure 394
12.3 Almost Public Monitoring: Robustness in the Infinitely Repeated
Prisoner’s Dilemma 397
12.3.1 The Forgiving Profile 398
12.3.2 Grim Trigger 400
12.4 Independent Monitoring: A Belief-Based Equilibrium for the
Infinitely Repeated Prisoner’s Dilemma 404
12.5 A Belief-Free Example 410
13 Almost Public Monitoring Games 415
13.1 When Is Monitoring Almost Public? 415
13.2 Nearby Games with Almost Public Monitoring 418
xiv Contents
13.2.1 Payoffs 418
13.2.2 Continuation Values 419
13.2.3 Best Responses 421
13.2.4 Equilibrium 421
13.3 Public Profiles with Bounded Recall 423
13.4 Failure of Coordination under Unbounded Recall 425
13.4.1 Examples 425
13.4.2 Incentives to Deviate 427
13.4.3 Separating Profiles 428
13.4.4 Rich Monitoring 432
13.4.5 Coordination Failure 434
13.5 Patient Players 434
13.5.1 Patient Strictness 435
13.5.2 Equilibria in Nearby Games 437
13.6 A Folk Theorem 441
14 Belief-Free Equilibria in Private Monitoring Games 445
14.1 Definition and Examples 445
14.1.1 Repeated Prisoners’ Dilemma with Perfect Monitoring 447
14.1.2 Repeated Prisoners’ Dilemma with Private Monitoring 451
14.2 Strong Self-Generation 453
Part IV Reputations
15 Reputations with Short-Lived Players 459
15.1 The Adverse Selection Approach to Reputations 459
15.2 Commitment Types 463
15.3 Perfect Monitoring Games 466
15.3.1 Building a Reputation 470
15.3.2 The Reputation Bound 474
15.3.3 An Example: Time Consistency 477
15.4 Imperfect Monitoring Games 478
15.4.1 Stackelberg Payoffs 480
15.4.2 The Reputation Bound 484
15.4.3 Small Players with Idiosyncratic Signals 492
15.5 Temporary Reputations 493
15.5.1 Asymptotic Beliefs 494
15.5.2 Uniformly Disappearing Reputations 496
15.5.3 Asymptotic Equilibrium Play 497
15.6 Temporary Reputations: The Proof of Proposition 15.5.1 500
15.6.1 Player 2’s Posterior Beliefs 500
15.6.2 Player 2’s Beliefs about Her Future Behavior 502
Contents xv
15.6.3 Player 1’s Beliefs about Player 2’s Future Behavior 503
15.6.4 Proof of Proposition 15.5.1 509
16 Reputations with Long-Lived Players 511
16.1 The Basic Issue 511
16.2 Perfect Monitoring and Minmax-Action Reputations 515
16.2.1 Minmax-Action Types and Conflicting Interests 515
16.2.2 Examples 518
16.2.3 Two-Sided Incomplete Information 520
16.3 Weaker Reputations for Any Action 521
16.4 Imperfect Public Monitoring 524
16.5 Commitment Types Who Punish 531
16.6 Equal Discount Factors 533
16.6.1 Example 1: Common Interests 534
16.6.2 Example 2: Conflicting Interests 537
16.6.3 Example 3: Strictly Dominant Action Games 540
16.6.4 Example 4: Strictly Conflicting Interests 541
16.6.5 Bounded Recall 544
16.6.6 Reputations and Bargaining 546
16.7 Temporary Reputations 547
17 Finitely Repeated Games 549
17.1 The Chain Store Game 550
17.2 The Prisoners’ Dilemma 554
17.3 The Product-Choice Game 560
17.3.1 The Last Period 562
17.3.2 The First Period, Player 1 562
17.3.3 The First Period, Player 2 565
18 Modeling Reputations 567
18.1 An Alternative Model of Reputations 568
18.1.1 Modeling Reputations 568
18.1.2 The Market 570
18.1.3 Reputation with Replacements 573
18.1.4 How Different Is It? 576
18.2 The Role of Replacements 576
18.3 Good Types and Bad Types 580
18.3.1 Bad Types 580
18.3.2 Good Types 581
18.4 Reputations with Common Consumers 584
18.4.1 Belief-Free Equilibria with Idiosyncratic Consumers 585
18.4.2 Common Consumers 586
18.4.3 Reputations 587
18.4.4 Replacements 588
18.4.5 Continuity at the Boundary and Markov Equilibria 590
18.4.6 Competitive Markets 594
xvi Contents
18.5 Discrete Choices 596
18.6 Lost Consumers 599
18.6.1 The Purchase Game 599
18.6.2 Bad Reputations: The Stage Game 600
18.6.3 The Repeated Game 601
18.6.4 Incomplete Information 603
18.6.5 Good Firms 607
18.6.6 Captive Consumers 608
18.7 Markets for Reputations 610
18.7.1 Reputations Have Value 610
18.7.2 Buying Reputations 613
Bibliography 619
Symbols 629
Index 631
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