Game Theory: An Introduction

Cover......Page 1
GAME THEORY......Page 4
Title......Page 6
Copyright......Page 7
Contents......Page 8
Preface......Page 14
PART I Rational Decision Making......Page 20
Chapter 1 The Single-Person Decision Problem......Page 22
1.1 Actions, Outcomes, and Preferences......Page 23
1.1.1 Preference Relations......Page 24
1.1.2 Payoff Functions......Page 26
1.2 The Rational Choice Paradigm......Page 28
1.3 Summary......Page 30
2.1 Risk, Nature, and Random Outcomes......Page 33
2.1.1 Finite Outcomes and Simple Lotteries......Page 34
2.1.2 Simple versus Compound Lotteries......Page 35
2.1.3 Lotteries over Continuous Outcomes......Page 36
2.2 Evaluating Random Outcomes......Page 37
2.2.1 Expected Payoff: The Finite Case......Page 38
2.2.2 Expected Payoff: The Continuous Case......Page 39
2.2.3 Caveat: It’s Not Just the Order Anymore......Page 40
2.2.4 Risk Attitudes......Page 41
2.2.5 The St. Petersburg Paradox......Page 42
2.3.2 Maximizing Expected Payoffs......Page 43
2.4.1 Backward Induction......Page 45
2.4.2 Discounting Future Payoffs......Page 47
2.5.1 The Value of Information......Page 48
2.5.2 Discounted Future Consumption......Page 50
2.6 Theory versus Practice......Page 51
2.8 Exercises......Page 52
PART II Static Games of Complete Information......Page 60
Chapter 3 Preliminaries......Page 62
3.1 Normal-Form Games with Pure Strategies......Page 65
3.1.1 Example: The Prisoner’s Dilemma......Page 67
3.1.3 Example: Voting on a New Agenda......Page 68
3.2 Matrix Representation: Two-Player Finite Game......Page 69
3.2.1 Example: The Prisoner’s Dilemma......Page 70
3.3 Solution Concepts......Page 71
3.3.1 Assumptions and Setup......Page 73
3.3.2 Evaluating Solution Concepts......Page 74
3.3.3 Evaluating Outcomes......Page 75
3.4 Summary......Page 76
3.5 Exercises......Page 77
4.1.1 Dominated Strategies......Page 78
4.1.2 Dominant Strategy Equilibrium......Page 80
4.1.3 Evaluating Dominant Strategy Equilibrium......Page 81
4.2.1 Iterated Elimination and Common Knowledge of Rationality......Page 82
4.2.2 Example: Cournot Duopoly......Page 84
4.2.3 Evaluating IESDS......Page 86
4.3.1 The Best Response......Page 88
4.3.2 Beliefs and Best-Response Correspondences......Page 90
4.3.4 The Cournot Duopoly Revisited......Page 92
4.3.5 The “p-Beauty Contest”......Page 93
4.5 Exercises......Page 95
Chapter 5 Pinning Down Beliefs: Nash Equilibrium......Page 98
5.1 Nash Equilibrium in Pure Strategies......Page 99
5.1.1 Pure-Strategy Nash Equilibrium in a Matrix......Page 100
5.2.1 Two Kinds of Societies......Page 102
5.2.2 The Tragedy of the Commons......Page 103
5.2.3 Cournot Duopoly......Page 106
5.2.4 Bertrand Duopoly......Page 107
5.2.5 Political Ideology and Electoral Competition......Page 112
5.4 Exercises......Page 114
Chapter 6 Mixed Strategies......Page 120
6.1.1 Finite Strategy Sets......Page 121
6.1.2 Continuous Strategy Sets......Page 123
6.1.4 Expected Payoffs......Page 124
6.2 Mixed-Strategy Nash Equilibrium......Page 126
6.2.1 Example: Matching Pennies......Page 127
6.2.2 Example: Rock-Paper-Scissors......Page 130
6.2.3 Multiple Equilibria: Pure and Mixed......Page 132
6.3 IESDS and Rationalizability Revisited......Page 133
6.4 Nash’s Existence Theorem......Page 136
6.6 Exercises......Page 142
PART III Dynamic Games of Complete Information......Page 146
Chapter 7 Preliminaries......Page 148
7.1 The Extensive-Form Game......Page 149
7.1.1 Game Trees......Page 151
7.1.2 Imperfect versus Perfect Information......Page 155
7.2.1 Pure Strategies......Page 156
7.2.2 Mixed versus Behavioral Strategies......Page 158
7.2.3 Normal-Form Representation of Extensive-Form Games......Page 162
7.3 Nash Equilibrium and Paths of Play......Page 164
7.5 Exercises......Page 166
Chapter 8 Credibility and Sequential Rationality......Page 170
8.1 Sequential Rationality and Backward Induction......Page 171
8.2 Subgame-Perfect Nash Equilibrium: Concept......Page 172
8.3.1 The Centipede Game......Page 178
8.3.2 Stackelberg Competition......Page 179
8.3.3 Mutually Assured Destruction......Page 182
8.3.4 Time-Inconsistent Preferences......Page 185
8.4 Summary......Page 188
8.5 Exercises......Page 189
Chapter 9 Multistage Games......Page 194
9.1 Preliminaries......Page 195
9.2 Payoffs......Page 196
9.3 Strategies and Conditional Play......Page 197
9.4 Subgame-Perfect Equilibria......Page 199
9.5 The One-Stage Deviation Principle......Page 203
9.7 Exercises......Page 205
10.1 Finitely Repeated Games......Page 209
10.2 Infinitely Repeated Games......Page 211
10.2.1 Payoffs......Page 212
10.2.2 Strategies......Page 214
10.3 Subgame-Perfect Equilibria......Page 215
10.4 Application: Tacit Collusion......Page 220
10.5.1 Cooperation as Reputation......Page 223
10.5.2 Third-Party Institutions as Reputation Mechanisms......Page 224
10.5.3 Reputation Transfers without Third Parties......Page 226
10.6 The Folk Theorem: Almost Anything Goes......Page 228
10.7 Summary......Page 233
10.8 Exercises......Page 234
Chapter 11 Strategic Bargaining......Page 239
11.1 One Round of Bargaining: The Ultimatum Game......Page 241
11.2 Finitely Many Rounds of Bargaining......Page 243
11.3 The Infinite-Horizon Game......Page 247
11.4 Application: Legislative Bargaining......Page 248
11.4.1 Closed-Rule Bargaining......Page 249
11.4.2 Open-Rule Bargaining......Page 251
11.5 Summary......Page 254
11.6 Exercises......Page 255
PART IV Static Games of Incomplete Information......Page 258
Chapter 12 Bayesian Games......Page 260
12.1.1 Players, Actions, Information, and Preferences......Page 265
12.1.2 Deriving Posteriors from a Common Prior: A Player’s Beliefs......Page 266
12.1.3 Strategies and Bayesian Nash Equilibrium......Page 268
12.2.1 Teenagers and the Game of Chicken......Page 271
12.2.2 Study Groups......Page 274
12.3 Inefficient Trade and Adverse Selection......Page 277
12.4 Committee Voting......Page 280
12.5 Mixed Strategies Revisited: Harsanyi’s Interpretation......Page 283
12.7 Exercises......Page 285
Chapter 13 Auctions and Competitive Bidding......Page 289
13.1.1 Second-Price Sealed-Bid Auctions......Page 291
13.1.2 English Auctions......Page 294
13.1.3 First-Price Sealed-Bid and Dutch Auctions......Page 295
13.1.4 Revenue Equivalence......Page 298
13.2 Common Values and the Winner’s Curse......Page 301
13.4 Exercises......Page 304
14.1.1 The Players......Page 307
14.1.2 The Mechanism Designer......Page 308
14.1.3 The Mechanism Game......Page 309
14.2 The Revelation Principle......Page 311
14.3.2 Vickrey-Clarke-Groves Mechanisms......Page 314
14.5 Exercises......Page 318
PART V Dynamic Games of Incomplete Information......Page 320
15.1 The Problem with Subgame Perfection......Page 322
15.2 Perfect Bayesian Equilibrium......Page 326
15.3 Sequential Equilibrium......Page 331
15.5 Exercises......Page 333
Chapter 16 Signaling Games......Page 337
16.1 Education Signaling: The MBA Game......Page 338
16.2 Limit Pricing and Entry Deterrence......Page 342
16.2.1 Separating Equilibria......Page 343
16.2.2 Pooling Equilibria......Page 349
16.3 Refinements of Perfect Bayesian Equilibrium in Signaling Games......Page 351
16.5 Exercises......Page 354
17.1 Cooperation in a Finitely Repeated Prisoner’s Dilemma......Page 358
17.2 Driving a Tough Bargain......Page 361
17.3 A Reputation for Being “Nice”......Page 368
17.5 Exercises......Page 373
Chapter 18 Information Transmission and Cheap Talk......Page 376
18.1 Information Transmission: A Finite Example......Page 377
18.2 Information Transmission: The Continuous Case......Page 380
18.3 Application: Information and Legislative Organization......Page 384
18.5 Exercises......Page 386
19.1.1 Basic Definitions......Page 388
19.1.2 Basic Set Operations......Page 389
19.2.1 Basic Definitions......Page 390
19.2.2 Continuity......Page 391
19.3.1 Basic Definitions......Page 392
19.3.2 Differentiation and Optimization......Page 393
19.3.3 Integration......Page 396
19.4.1 Basic Definitions......Page 397
19.4.2 Cumulative Distribution and Density Functions......Page 398
19.4.3 Independence, Conditional Probability, and Bayes’ Rule......Page 399
19.4.4 Expected Values......Page 401
References......Page 404
Index......Page 408