Game Theory

Steven Tadelis is associate professor and Barbara and Gerson Bakar Faculty Fellow at the Haas School of Business at the University of California, Berkeley, and a Distinguished Economist at eBay Research Labs.

• Preface xi PART I Rational Decision Making Chapter 1 The Single-Person Decision Problem 3 *1.1 Actions, Outcomes, and Preferences 4 * 1.1.1 Preference Relations 5 * 1.1.2 Payoff Functions 7 *1.2 The Rational Choice Paradigm 9 *1.3 Summary 11 *1.4 Exercises 11 Chapter 2 Introducing Uncertainty and Time 14 *2.1 Risk, Nature, and Random Outcomes 14 2.1.1 Finite Outcomes and Simple Lotteries 15 2.1.2 Simple versus Compound Lotteries 16 2.1.3 Lotteries over Continuous Outcomes 17 *2.2 Evaluating Random Outcomes 18 2.2.1 Expected Payoff: The Finite Case 19 2.2.2 Expected Payoff: The Continuous Case 20 2.2.3 Caveat: It's Not Just the Order Anymore 21 2.2.4 Risk Attitudes 22 2.2.5 The St. Petersburg Paradox 23 *2.3 Rational Decision Making with Uncertainty 24 2.3.1 Rationality Revisited 24 2.3.2 Maximizing Expected Payoffs 24 *2.4 Decisions over Time 26 2.4.1 Backward Induction 26 2.4.2 Discounting Future Payoffs 28 *2.5 Applications 29 2.5.1 The Value of Information 29 2.5.2 Discounted Future Consumption 31 *2.6 Theory versus Practice 32 *2.7 Summary 33 *2.8 Exercises 33 PART II Static Games of Complete Information Chapter 3 Preliminaries 43 *3.1 Normal-Form Games with Pure Strategies 46 3.1.1 Example: The Prisoner's Dilemma 48 3.1.2 Example: Cournot Duopoly 49 3.1.3 Example: Voting on a New Agenda 49 *3.2 Matrix Representation: Two-Player Finite Game 50 3.2.1 Example: The Prisoner's Dilemma 51 3.2.2 Example: Rock-Paper-Scissors 52 *3.3 Solution Concepts 52 3.3.1 Assumptions and Setup 54 3.3.2 Evaluating Solution Concepts 55 3.3.3 Evaluating Outcomes 56 *3.4 Summary 57 *3.5 Exercises 58 Chapter 4 Rationality and Common Knowledge 59 *4.1 Dominance in Pure Strategies 59 4.1.1 Dominated Strategies 59 4.1.2 Dominant Strategy Equilibrium 61 4.1.3 Evaluating Dominant Strategy Equilibrium 62 *4.2 Iterated Elimination of Strictly Dominated Pure Strategies 63 4.2.1 Iterated Elimination and Common Knowledge of Rationality 63 4.2.2 Example: Cournot Duopoly 65 4.2.3 Evaluating IESDS 67 *4.3 Beliefs, Best Response, and Rationalizability 69 4.3.1 The Best Response 69 4.3.2 Beliefs and Best-Response Correspondences 71 4.3.3 Rationalizability 73 4.3.4 The Cournot Duopoly Revisited 73 4.3.5 The "p-Beauty Contest" 74 4.3.6 Evaluating Rationalizability 76 *4.4 Summary 76 *4.5 Exercises 76 Chapter 5 Pinning Down Beliefs: Nash Equilibrium 79 *5.1 Nash Equilibrium in Pure Strategies 80 5.1.1 Pure-Strategy Nash Equilibrium in a Matrix 81 5.1.2 Evaluating the Nash Equilibria Solution 83 *5.2 Nash Equilibrium: Some Classic Applications 83 5.2.1 Two Kinds of Societies 83 5.2.2 The Tragedy of the Commons 84 5.2.3 Cournot Duopoly 87 5.2.4 Bertrand Duopoly 88 5.2.5 Political Ideology and Electoral Competition 93 *5.3 Summary 95 *5.4 Exercises 95 Chapter 6 Mixed Strategies 101 *6.1 Strategies, Beliefs, and Expected Payoffs 102 6.1.1 Finite Strategy Sets 102 6.1.2 Continuous Strategy Sets 104 6.1.3 Beliefs and Mixed Strategies 105 6.1.4 Expected Payoffs 105 *6.2 Mixed-Strategy Nash Equilibrium 107 6.2.1 Example: Matching Pennies 108 6.2.2 Example: Rock-Paper-Scissors 111 6.2.3 Multiple Equilibria: Pure and Mixed 113 *6.3 IESDS and Rationalizability Revisited 114 *6.4 Nash's Existence Theorem 117 *6.5 Summary 123 *6.6 Exercises 123 PART III Dynamic Games of Complete Information Chapter 7 Preliminaries 129 *7.1 The Extensive-Form Game 130 7.1.1 Game Trees 132 7.1.2 Imperfect versus Perfect Information 136 *7.2 Strategies and Nash Equilibrium 137 7.2.1 Pure Strategies 137 7.2.2 Mixed versus Behavioral Strategies 139 7.2.3 Normal-Form Representation of Extensive-Form Games 143 *7.3 Nash Equilibrium and Paths of Play 145 *7.4 Summary 147 *7.5 Exercises 147 Chapter 8 Credibility and Sequential Rationality 151 *8.1 Sequential Rationality and Backward Induction 152 *8.2 Subgame-Perfect Nash Equilibrium: Concept 153 *8.3 Subgame-Perfect Nash Equilibrium: Examples 159 8.3.1 The Centipede Game 159 8.3.2 Stackelberg Competition 160 8.3.3 Mutually Assured Destruction 163 8.3.4 Time-Inconsistent Preferences 166 *8.4 Summary 169 *8.5 Exercises 170 Chapter 9 Multistage Games 175 *9.1 Preliminaries 176 *9.2 Payoffs 177 *9.3 Strategies and Conditional Play 178 *9.4 Subgame-Perfect Equilibria 180 *9.5 The One-Stage Deviation Principle 184 *9.6 Summary 186 *9.7 Exercises 186 Chapter 10 Repeated Games 190 *10.1 Finitely Repeated Games 190 *10.2 Infinitely Repeated Games 192 10.2.1 Payoffs 193 10.2.2 Strategies 195 *10.3 Subgame-Perfect Equilibria 196 *10.4 Application: Tacit Collusion 201 *10.5 Sequential Interaction and Reputation 204 10.5.1 Cooperation as Reputation 204 10.5.2 Third-Party Institutions as Reputation Mechanisms 205 10.5.3 Reputation Transfers without Third Parties 207 *10.6 The Folk Theorem: Almost Anything Goes 209 *10.7 Summary 214 *10.8 Exercises 215 Chapter 11 Strategic Bargaining 220 *11.1 One Round of Bargaining: The Ultimatum Game 222 *11.2 Finitely Many Rounds of Bargaining 224 *11.3 The Infinite-Horizon Game 228 *11.4 Application: Legislative Bargaining 229 11.4.1 Closed-Rule Bargaining 230 11.4.2 Open-Rule Bargaining 232 *11.5 Summary 235 *11.6 Exercises 236 PART IV Static Games of Incomplete Information Chapter 12 Bayesian Games 241 *12.1 Strategic Representation of Bayesian Games 246 12.1.1 Players, Actions, Information, and Preferences 246 12.1.2 Deriving Posteriors from a Common Prior: A Player's Beliefs 247 12.1.3 Strategies and Bayesian Nash Equilibrium 249 *12.2 Examples 252 12.2.1 Teenagers and the Game of Chicken 252 12.2.2 Study Groups 255 *12.3 Inefficient Trade and Adverse Selection 258 *12.4 Committee Voting 261 *12.5 Mixed Strategies Revisited: Harsanyi's Interpretation 264 *12.6 Summary 266 *12.7 Exercises 266 Chapter 13 Auctions and Competitive Bidding 270 *13.1 Independent Private Values 272 13.1.1 Second-Price Sealed-Bid Auctions 272 13.1.2 English Auctions 275 13.1.3 First-Price Sealed-Bid and Dutch Auctions 276 13.1.4 Revenue Equivalence 279 *13.2 Common Values and the Winner's Curse 282 *13.3 Summary 285 *13.4 Exercises 285 Chapter 14 Mechanism Design 288 *14.1 Setup: Mechanisms as Bayesian Games 288 14.1.1 The Players 288 14.1.2 The Mechanism Designer 289 14.1.3 The Mechanism Game 290 *14.2 The Revelation Principle 292 *14.3 Dominant Strategies and Vickrey-Clarke-Groves Mechanisms 295 14.3.1 Dominant Strategy Implementation 295 14.3.2 Vickrey-Clarke-Groves Mechanisms 295 *14.4 Summary 299 *14.5 Exercises 299 PART V Dynamic Games of Incomplete Information Chapter 15 Sequential Rationality with Incomplete Information 303 *15.1 The Problem with Subgame Perfection 303 *15.2 Perfect Bayesian Equilibrium 307 *15.3 Sequential Equilibrium 312 *15.4 Summary 314 *15.5 Exercises 314 Chapter 16 Signaling Games 318 *16.1 Education Signaling: The MBA Game 319 *16.2 Limit Pricing and Entry Deterrence 323 16.2.1 Separating Equilibria 324 16.2.2 Pooling Equilibria 330 *16.3 Refinements of Perfect Bayesian Equilibrium in Signaling Games 332 *16.4 Summary 335 *16.5 Exercises 335 Chapter 17 Building a Reputation 339 *17.1 Cooperation in a Finitely Repeated Prisoner's Dilemma 339 *17.2 Driving a Tough Bargain 342 *17.3 A Reputation for Being "Nice" 349 *17.4 Summary 354 *17.5 Exercises 354 Chapter 18 Information Transmission and Cheap Talk 357 *18.1 Information Transmission: A Finite Example 358 *18.2 Information Transmission: The Continuous Case 361 *18.3 Application: Information and Legislative Organization 365 *18.4 Summary 367 *18.5 Exercises 367 Chapter 19 Mathematical Appendix 369 *19.1 Sets and Sequences 369 19.1.1 Basic Definitions 369 19.1.2 Basic Set Operations 370 *19.2 Functions 371 19.2.1 Basic Definitions 371 19.2.2 Continuity 372 *19.3 Calculus and Optimization 373 19.3.1 Basic Definitions 373 19.3.2 Differentiation and Optimization 374 19.3.3 Integration 377 *19.4 Probability and Random Variables 378 19.4.1 Basic Definitions 378 19.4.2 Cumulative Distribution and Density Functions 379 19.4.3 Independence, Conditional Probability, and Bayes' Rule 380 19.4.4 Expected Values 382 References 385 Index 389

"The book is enjoyable to read and truly an enrichment in game theory. It is widely well-structured and well-written and mathematically correct. The purpose is given perfectly. I recommend the book for researchers and graduate students who wants to get some insight in the area of game theory."--Sirma Zeynep, Zentralblatt MATH "The book aims to be precise and rigorous, yet accessible and reader-friendly, and, to a great extent, it does hit these apparently conflicting targets... The depth of the book is intermediate, with a conventional, yet clear, style of writing. It will please mainstream economists... It can help advanced undergraduates and also students at honors or master's levels. It can also be used by PhD students seeking a fast, not so mathematized introduction to the field."--Jose Rodriques-Neto, Economic Record