Introduction to Game Theory Steve Schecter Department of Mathematics North Carolina State University Herbert Gintis Santa Fe Institute Contents Preface vii Chapter 1. Backward induction 3 1.1. Tony’s Accident 3 1.2. Games in extensive form with complete information 5 1.3. Strategies 7 1.4. Backward induction 8 1.5. Big Monkey and Little Monkey 1 11 1.6. Threats, promises, commitments 12 1.7. Ultimatum Game 14 1.8. Rosenthal’s Centipede Game 15 1.9. Continuous games 17 1.10. Stackelberg’s model of duopoly 18 1.11. Economics and calculus background 22 1.12. The Samaritan’s Dilemma 23 1.13. The Rotten Kid Theorem 28 1.14. Backward induction for finite horizon games 30 1.15. Critique of backward induction 31 1.16. Problems 33 Chapter 2. Eliminating dominated strategies 43 2.1. Prisoner’s Dilemma 43 2.2. Games in normal form 45 2.3. Dominated strategies 46 2.4. Israelis and Palestinians 47 2.5. Global Warming 49 2.6. Hagar’s Battles 50 2.7. Second-price auctions 52 2.8. Iterated elimination of dominated strategies 54 2.9. The Battle of the Bismarck Sea 54 2.10. Normal form of a game in extensive form with complete information 55 2.11. Big Monkey and Little Monkey 2 56 2.12. Backward induction and iterated elimination of dominated strategies 58 2.13. Critique of elimination of dominated strategies 60 2.14. Problems 60 iii iv CONTENTS Chapter 3. Nash equilibria 65 3.1. Big Monkey and Little Monkey 3 and the definition of Nash equilibria 65 3.2. Finding Nash equilibria by inspection: important examples 67 3.3. Water Pollution 1 69 3.4. Tobacco Market 71 3.5. Finding Nash equilibria by iterated elimination of dominated strategies 73 3.6. Big Monkey and Little Monkey 4: threats, promises, and commitments revisited 76 3.7. Finding Nash equilibria using best response 77 3.8. Big Monkey and Little Monkey 5 78 3.9. Water Pollution 2 78 3.10. Cournot’s model of duopoly 80 3.11. Problems 80 Chapter 4. Games in extensive form with incomplete information 89 4.1. Lotteries 89 4.2. Buying Fire Insurance 89 4.3. Games in extensive form with incomplete information 90 4.4. Buying a Used Car 91 4.5. The Travails of Boss Gorilla 95 4.6. Cuban Missile Crisis 97 4.7. Problems 104 Chapter 5. Mixed-strategy Nash equilibria 109 5.1. Mixed-strategy Nash equilibria 109 5.2. Tennis 114 5.3. Other ways to find mixed-strategy Nash equilibria 115 5.4. One-card Two-round Poker 117 5.5. Two-player zero-sum games 122 5.6. The Ultimatum Minigame 125 5.7. Colonel Blotto vs. the People’s Militia 127 5.8. Water Pollution 3 132 5.9. Equivalent games 133 5.10. Software for computing Nash equilibria 135 5.11. Critique of Nash Equilibrium 136 5.12. Problems 136 Chapter 6. Subgame perfect Nash equilibria and infinite-horizon games 141 6.1. Subgame perfect Nash equilibria 141 6.2. Big Monkey and Little Monkey 6 142 6.3. Subgame perfect equilibria and backward induction 143 6.4. Duels and Truels 145 6.5. The Rubinstein bargaining model 150 6.6. Repeated games 153 6.7. The Wine Merchant and the Connoisseur 153 CONTENTS v 6.8. The Folk Theorem 157 6.9. Problems 160 Chapter 7. Symmetries of games 167 7.1. Interchangeable players 167 7.2. Reporting a Crime 169 7.3. Sex Ratio 171 7.4. Other symmetries of games 173 7.5. Problems 178 Chapter 8. Alternatives to the Nash Equilibrium 181 8.1. Correlated equilibrium 181 8.2. Epistemic game theory 183 8.3. Evolutionary stability 184 8.4. Evolutionary stability with two pure strategies 187 8.5. Sex Ratio 190 8.6. Problems 191 Chapter 9. Differential equations 193 9.1. Differential equations and scientific laws 193 9.2. The phase line 195 9.3. Vector fields 196 9.4. Functions and differential equations 198 9.5. Linear differential equations 201 9.6. Linearization 204 Chapter 10. Evolutionary dynamics 209 10.1. Replicator system 209 10.2. Evolutionary dynamics with two pure strategies 212 10.3. Microsoft vs. Apple 213 10.4. Hawks and Doves revisited 215 10.5. Orange-throat, Blue-throat, and Yellow-striped Lizards 216 10.6. Equilibria of the replicator system 220 10.7. Cooperators, Defectors, and Tit-for-tatters 221 10.8. Dominated strategies and the replicator system 225 10.9. Asymmetric evolutionary games 227 10.10. Big Monkey and Little Monkey 7 229 10.11. Hawks and Doves with Unequal Value 231 10.12. The Ultimatum Minigame revisited 233 10.13. Problems 234 Chapter 11. Sources for examples and problems 241 Bibliography 245 Preface Game theory deals with situations in which your payoff depends not only on your own choices but on the choices of others. How are you supposed to decide what to do, since you cannot control what others will do? In calculus you learn to maximize and minimize functions, for example to find the cheapest way to build something. This field of mathematics is called optimiza- tion. Game theory differs from optimization in that in optimization problems, your payoff depends only on your own choices. Like the field of optimization, game theory is defined by the problems it deals with, not by the mathematical techniques that are used to deal with them. The techniques are whatever works best. Also, like the field of optimization, the problems of game theory come from many different areas of study. It is nevertheless helpful to treat game theory as a single mathematical field, since then techniques developed for problems in one area, for example evolutionary biology, become available to another, for example economics. Game theory has three uses: (1) Understand the world. For example, game theory helps explain why animals sometimes fight over territory and sometimes don’t. (2) Respond to the world. For example, game theory has been used to develop strategies to win money at poker. (3) Change the world. Often the world is the way it is because people are responding to the rules of a game. Changing the game can change how they act. For example, rules on using energy can be designed to encourage conservation and innovation. The idea behind the organization of this book is: learn an idea, then try to use it in as many interesting ways as possible. Because of this organization, the most important idea in game theory, the Nash equilibrium, does not make an appearance until Chapter 3. Two ideas that are more basic—backward induction for games in extensive form, and elimination of dominated strategies for games in normal form— are treated first. vii Traditionally, game theory has been viewed as a way to find rational answers to dilemmas. However, since the 1970s it has been applied to animal behavior, and animals presumably do not make rational analyses. A more reasonable view of animal behavior is that predominant strategies emerge over time as more successful ones replace less successful ones. This point of view on game theory is now called evolutionary game theory. Once one thinks of strategies as changing over time, the mathematical field of differential equations becomes relevant. Because students do not always have a good background in differential equations, we have included an introduction to the area in Chapter 9. This text grew out of Herb’s book [3], which is “a problem-centered intro- duction to modeling strategic interaction.” Steve began using Herb’s book in 2005 to teach a game theory course in the North Carolina State University Mathemat- ics Department. The course was aimed at upper division mathematics majors and other interested students with some mathematical background (calculus including some differential equations). Over the following years Steve produced a set of class notes to supplement [3], which was superseded in 2009 by [4]. This text combines material from those two books by Herb, and from his recent book [5], with Steve’s notes, and adds some new material. Examples and problems are the heart of the book. The text also includes proofs of general results, written in a fairly typical mathematical style. Steve usually covers just a few of these in his course, since the course is open to students with a limited mathematical background. However, mathematics students who have had previous proof-oriented courses should be able to handle them. June 29, 2013 1 CHAPTER 1 Backward induction This chapter deals with interactions in which two or more opponents take actions one after the other. If you are involved in such an interaction, you can try to think ahead to how your opponent might respond to each of your possible actions, bearing in mind that he is trying to achieve his own objectives, not yours. As we shall see in Sections 1.12 and 1.13, this simple idea underlies work of two Nobel Prize-winning economists. However, we shall also see that it may not be helpful to carry this idea too far. 1.1. Tony’s Accident When one of us (Steve) was a college student, his friend Tony caused a minor traffic accident. We’ll let Steve tell the story: The car of the victim, whom I’ll call Vic, was slightly scraped. Tony didn’t want to tell his insurance company. The next morning, Tony and I went with Vic to visit some body shops. The upshot was that the repair would cost $80. Tony and I had lunch with a bottle of wine, and thought over the situation.