Game Theory Evolving

Herbert Gintis holds faculty positions at the Santa Fe Institute, Central European University, and University of Siena. He has coedited numerous books, including "Moral Sentiments and Material Interests, Unequal Chances" (Princeton), and "Foundations of Human Sociality".

• Preface xv Chapter 1: Probability Theory 1 1.1 Basic Set Theory and Mathematical Notation 1 1.2 Probability Spaces 2 1.3 De Morgan's Laws 3 1.4 Interocitors 3 1.5 The Direct Evaluation of Probabilities 3 1.6 Probability as Frequency 4 1.7 Craps 5 1.8 A Marksman Contest 5 1.9 Sampling 5 1.10 Aces Up 6 1.11 Permutations 6 1.12 Combinations and Sampling 7 1.13 Mechanical Defects 7 1.14 Mass Defection 7 1.15 House Rules 7 1.16 The Addition Rule for Probabilities 8 1.17 A Guessing Game 8 1.18 North Island, South Island 8 1.19 Conditional Probability 9 1.20 Bayes' Rule 9 1.21 Extrasensory Perception 10 1.22 Les Cinq Tiroirs 10 1.23 Drug Testing 10 1.24 Color Blindness 11 1.25 Urns 11 1.26 The Monty Hall Game 11 1.27 The Logic of Murder and Abuse 11 1.28 The Principle of Insufficient Reason 12 1.29 The Greens and the Blacks 12 1.30 The Brain and Kidney Problem 12 1.31 The Value of Eyewitness Testimony 13 1.32 When Weakness Is Strength 13 1.33 The Uniform Distribution 16 1.34 Laplace's Law of Succession 17 1.35 From Uniform to Exponential 17 Chapter 2: Bayesian Decision Theory 18 2.1 The Rational Actor Model 18 2.2 Time Consistency and Exponential Discounting 20 2.3 The Expected Utility Principle 22 2.4 Risk and the Shape of the Utility Function 26 2.5 The Scientific Status of the Rational Actor Model 30 Chapter 3: Game Theory: Basic Concepts 32 3.1 Big John and Little John 32 3.2 The Extensive Form 38 3.3 The Normal Form 41 3.4 Mixed Strategies 42 3.5 Nash Equilibrium 43 3.6 The Fundamental Theorem of Game Theory 44 3.7 Solving for Mixed-Strategy Nash Equilibria 44 3.8 Throwing Fingers 46 3.9 Battle of the Sexes 46 3.10 The Hawk-Dove Game 48 3.11 The Prisoner's Dilemma 50 Chapter 4: Eliminating Dominated Strategies 52 4.1 Dominated Strategies 52 4.2 Backward Induction 54 4.3 Exercises in Eliminating Dominated Strategies 55 4.4 Subgame Perfection 57 4.5 Stackelberg Leadership 59 4.6 The Second-Price Auction 59 4.7 The Mystery of Kidnapping 60 4.8 The Eviction Notice 62 4.9 Hagar's Battles 62 4.10 Military Strategy 63 4.11 The Dr. Strangelove Game 64 4.12 Strategic Voting 64 4.13 Nuisance Suits 65 4.14 An Armaments Game 67 4.15 Football Strategy 67 4.16 Poker with Bluffing 68 4.17 The Little Miss Muffet Game 69 4.18 Cooperation with Overlapping Generations 70 4.19 Dominance-Solvable Games 71 4.20 Agent-based Modeling 72 4.21 Why Play a Nash Equilibrium? 75 4.22 Modeling the Finitely-Repeated Prisoner's Dilemma 77 4.23 Review of Basic Concepts 79 Chapter 5: Pure-Strategy Nash Equilibria 80 5.1 Price Matching as Tacit Collusion 80 5.2 Competition on Main Street 81 5.3 Markets as Disciplining Devices: Allied Widgets 81 5.4 The Tobacco Market 87 5.5 The Klingons and the Snarks 87 5.6 Chess: The Trivial Pastime 88 5.7 No-Draw, High-Low Poker 89 5.8 An Agent-based Model of No-Draw, High-Low Poker 91 5.9 The Truth Game 92 5.10 The Rubinstein Bargaining Model 94 5.11 Bargaining with Heterogeneous Impatience 96 5.12 Bargaining with One Outside Option 97 5.13 Bargaining with Dual Outside Options 98 5.14 Huey, Dewey, and Louie Split a Dollar 102 5.15 Twin Sisters 104 5.16 The Samaritan's Dilemma 104 5.17 The Rotten Kid Theorem 106 5.18 The Shopper and the Fish Merchant 107 5.19 Pure Coordination Games 109 5.20 Pick Any Number 109 5.21 Pure Coordination Games: Experimental Evidence 110 5.22 Introductory Offers 111 5.23 Web Sites (for Spiders) 112 Chapter 6: Mixed-Strategy Nash Equilibria 116 6.1 The Algebra of Mixed Strategies 116 6.2 Lions and Antelope 117 6.3 A Patent Race 118 6.4 Tennis Strategy 119 6.5 Preservation of Ecology Game 119 6.6 Hard Love 120 6.7 Advertising Game 120 6.8 Robin Hood and Little John 122 6.9 The Motorist's Dilemma 122 6.10 Family Politics 123 6.11 Frankie and Johnny 123 6.12 A Card Game 124 6.13 Cheater-Inspector 126 6.14 The Vindication of the Hawk 126 6.15 Characterizing 2 x 2 Normal Form Games I 127 6.16 Big John and Little John Revisited 128 6.17 Dominance Revisited 128 6.18 Competition on Main Street Revisited 128 6.19 Twin Sisters Revisited 129 6.20 Twin Sisters: An Agent-Based Model 129 6.21 One-Card, Two-Round Poker with Bluffing 131 6.22 An Agent-Based Model of Poker with Bluffing 132 6.23 Trust in Networks 133 6.24 El Farol 134 6.25 Decorated Lizards 135 6.26 Sex Ratios as Nash Equilibria 137 6.27 A Mating Game 140 6.28 Coordination Failure 141 6.29 Colonel Blotto Game 141 6.30 Number Guessing Game 142 6.31 Target Selection 142 6.32 A Reconnaissance Game 142 6.33 Attack on Hidden Object 143 6.34 Two-Person, Zero-Sum Games 143 6.35 Mutual Monitoring in a Partnership 145 6.36 Mutual Monitoring in Teams 145 6.37 Altruism(?) in Bird Flocks 146 6.38 The Groucho Marx Game 147 6.39 Games of Perfect Information 151 6.40 Correlated Equilibria 151 6.41 Territoriality as a Correlated Equilibrium 153 6.42 Haggling at the Bazaar 154 6.43 Poker with Bluffing Revisited 156 6.44 Algorithms for Finding Nash Equilibria 157 6.45 Why Play Mixed Strategies? 160 6.46 Reviewing of Basic Concepts 161 Chapter 7: Principal-AgentModels 162 7.1 Gift Exchange 162 7.2 Contract Monitoring 163 7.3 Profit Signaling 164 7.4 Properties of the Employment Relationship 168 7.5 Peasant and Landlord 169 7.6 Bob's Car Insurance 173 7.7 A Generic Principal-Agent Model 174 Chapter 8: Signaling Games 179 8.1 Signaling as a Coevolutionary Process 179 8.2 A Generic Signaling Game 180 8.3 Sex and Piety: The Darwin-Fisher Model 182 8.4 Biological Signals as Handicaps 187 8.5 The ShepherdsWho Never Cry Wolf 189 8.6 My Brother's Keeper 190 8.7 Honest Signaling among Partial Altruists 193 8.8 Educational Signaling 195 8.9 Education as a Screening Device 197 8.10 Capital as a Signaling Device 199 Chapter 9: Repeated Games 201 9.1 Death and Discount Rates in Repeated Games 202 9.2 Big Fish and Little Fish 202 9.3 Alice and Bob Cooperate 204 9.4 The Strategy of an Oil Cartel 205 9.5 Reputational Equilibrium 205 9.6 Tacit Collusion 206 9.7 The One-Stage Deviation Principle 208 9.8 Tit for Tat 209 9.9 I'd Rather Switch Than Fight 210 9.10 The Folk Theorem 213 9.11 The Folk Theorem and the Nature of Signaling 216 9.12 The Folk Theorem Fails in Large Groups 217 9.13 Contingent Renewal Markets Do Not Clear 219 9.14 Short-Side Power in Contingent Renewal Markets 222 9.15 Money Confers Power in Contingent Renewal Markets 223 9.16 The Economy Is Controlled by the Wealthy 223 9.17 Contingent Renewal Labor Markets 224 Chapter 10: Evolutionarily Stable Strategies 229 10.1 Evolutionarily Stable Strategies: Definition 230 10.2 Properties of Evolutionarily Stable Strategies 232 10.3 Characterizing Evolutionarily Stable Strategies 233 10.4 A Symmetric Coordination Game 236 10.5 A Dynamic Battle of the Sexes 236 10.6 Symmetrical Throwing Fingers 237 10.7 Hawks, Doves, and Bourgeois 238 10.8 Trust in Networks II 238 10.9 Cooperative Fishing 238 10.10 Evolutionarily Stable Strategies Are Not Unbeatable 240 10.11 A Nash Equilibrium That Is Not an EES 240 10.12 Rock, Paper, and Scissors Has No ESS 241 10.13 Invasion of the Pure-Strategy Mutants 241 10.14 Multiple Evolutionarily Stable Strategies 242 10.15 Evolutionarily Stable Strategies in Finite Populations 242 10.16 Evolutionarily Stable Strategies in Asymmetric Games 244 Chapter 11: Dynamical Systems 247 11.1 Dynamical Systems: Definition 247 11.2 Population Growth 248 11.3 Population Growth with Limited Carrying Capacity 249 11.4 The Lotka-Volterra Predator-Prey Model 251 11.5 Dynamical Systems Theory 255 11.6 Existence and Uniqueness 256 11.7 The Linearization Theorem 257 11.8 Dynamical Systems in One Dimension 258 11.9 Dynamical Systems in Two Dimensions 260 11.10 Exercises in Two-Dimensional Linear Systems 264 11.11 Lotka-Volterra with Limited Carrying Capacity 266 11.12 Take No Prisoners 266 11.13 The Hartman-Grobman Theorem 267 11.14 Features of Two-Dimensional Dynamical Systems 268 Chapter 12: Evolutionary Dynamics 270 12.1 The Origins of Evolutionary Dynamics 271 12.2 Strategies as Replicators 272 12.3 A Dynamic Hawk-Dove Game 274 12.4 Sexual Reproduction and the Replicator Dynamic 276 12.5 Properties of the Replicator System 278 12.6 The Replicator Dynamic in Two Dimensions 279 12.7 Dominated Strategies and the Replicator Dynamic 280 12.8 Equilibrium and Stability with a Replicator Dynamic 282 12.9 Evolutionary Stability and Asymptotically Stability 284 12.10 Trust in Networks III 284 12.11 Characterizing 2 x 2 Normal Form Games II 285 12.12 Invasion of the Pure-Strategy Nash Mutants II 286 12.13 A Generalization of Rock, Paper, and Scissors 287 12.14 Uta stansburiana in Motion 287 12.15 The Dynamics of Rock, Paper, and Scissors 288 12.16 The Lotka-VolterraModel and Biodiversity 288 12.17 Asymmetric Evolutionary Games 290 12.18 Asymmetric Evolutionary Games II 295 12.19 The Evolution of Trust and Honesty 295 Chapter 13: Markov Economies and Stochastic Dynamical Systems 297 13.1 Markov Chains 297 13.2 The Ergodic Theorem for Markov Chains 305 13.3 The Infinite Random Walk 307 13.4 The Sisyphean Markov Chain 308 13.5 Andrei Andreyevich's Two-Urn Problem 309 13.6 Solving Linear Recursion Equations 310 13.7 Good Vibrations 311 13.8 Adaptive Learning 312 13.9 The Steady State of a Markov Chain 314 13.10 Adaptive Learning II 315 13.11 Adaptive Learning with Errors 316 13.12 Stochastic Stability 317 Chapter 14: Table of Symbols 319 Chapter 15: Answers 321 Sources for Problems 373 References 375

"Gintis has wholeheartedly embraced the evolutionary approach to games... The author is an accomplished economist raised in the classical mold, and his background shows in many aspects of the book ... [He] has important things to say."--Karl Sigmund, Science "Game Theory Evolving is an exceptionally well-written and constructed introduction to the field. And with Gintis' outline of agent-based modeling and his tips for programming, many readers may be motivated to take up his invitation and experiment with a problem in evolutionary dynamics of their own."--Jennifer M. Wilson, Mathematical Reviews