Game Theory Evolving, Second Edition A Problem ...

Game Theory Evolving, Second Edition A Problem-Centered Introduction to Modeling Strategic Interaction

Herbert Gintis

Princeton University Press Princeton and Oxford

Contents Preface Probability Theory 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31

Basic Set Theory and Mathematical Notation Probability Spaces De Morgan's Laws Interocitors The Direct Evaluation of Probabilities Probability as Frequency Craps A Marksman Contest Sampling Aces Up Permutations Combinations and Sampling Mechanical Defects Mass Defection House Rules The Addition Rule for Probabilities A Guessing Game North Island, South Island Conditional Probability Bayes' Rule Extrasensory Perception Les Cinq Tiroirs Drug Testing Color Blindness Urns The Monty Hall Game The Logic of Murder and Abuse The Principle of Insufficient Reason The Greens and the Blacks The Brain and Kidney Problem The Value of Eyewitness Testimony

JCV

1 1 2 3 3 3 4 5 5 5 6 6 7 7 7 7 8 8 8 9 9 10 10 10 11 11 11 11 12 12 12 13

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Contents 1.32 1.33 1.34 1.35

When Weakness Is Strength The Uniform Distribution Laplace's Law of Succession From Uniform to Exponential

Bayesian Decision Theory 2.1 2.2 2.3 2.4 2.5

The Rational Actor Model Time Consistency and Exponential Discounting The Expected Utility Principle Risk and the Shape of the Utility Function The Scientific Status of the Rational Actor Model

Game Theory: Basic Concepts 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Big John and Little John The Extensive Form The Normal Form Mixed Strategies Nash Equilibrium The Fundamental Theorem of Game Theory Solving for Mixed-Strategy Nash Equilibria Throwing Fingers Battle of the Sexes The Hawk-Dove Game The Prisoner's Dilemma

Eliminating Dominated Strategies 4.1 Dominated Strategies 4.2 Backward Induction Exercises in Eliminating Dominated Strategies 4.3 4.4 Subgame Perfection 4.5 Stackelberg Leadership 4.6 The Second-Price Auction 4.7 The Mystery of Kidnapping 4.8 The Eviction Notice 4.9 Hagar's Battles 4.10 Military Strategy 4.11 The Dr. Strangelove Game

13 16 17 17 18 18 20 22 26 30 32 32 38 41 42 43 44 45 46 46 48 50 52 52 54 55 57 59 59 60 62 62 63 64

Contents 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 5

Strategic Voting Nuisance Suits An Armaments Game Football Strategy Poker with Bluffing The Little Miss Muffet Game Cooperation with Overlapping Generations Dominance-Solvable Games Agent-based Modeling Why Play a Nash Equilibrium? Modeling the Finitely-Repeated Prisoner's Dilemma Review of Basic Concepts

Pure-Strategy Nash Equilibria 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23

Price Matching as Tacit Collusion Competition on Main Street Markets as Disciplining Devices: Allied Widgets The Tobacco Market The Klingons and the Snarks Chess: The Trivial Pastime No-Draw, High-Low Poker An Agent-based Model of No-Draw, High-Low Poker The Truth Game The Rubinstein Bargaining Model Bargaining with Heterogeneous Impatience Bargaining with One Outside Option Bargaining with Dual Outside Options Huey, Dewey, and Louie Split a Dollar Twin Sisters The Samaritan's Dilemma The Rotten Kid Theorem The Shopper and the Fish Merchant Pure Coordination Games Pick Any Number Pure Coordination Games: Experimental Evidence Introductory Offers Web Sites (for Spiders)

64 65 67 67 68 69 70 71 72 75 77 79 80 80 81 81 87 87 88 89 91 92 94 96 97 98 102 104 104 106 107 109 109 110 111 112

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Contents Mixed-Strategy Nash Equilibria 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 6.35 6.36

The Algebra of Mixed Strategies Lions and Antelope A Patent Race Tennis Strategy Preservation of Ecology Game Hard Love Advertising Game Robin Hood and Little John The Motorist's Dilemma Family Politics Frankie and Johnny A Card Game Cheater-Inspector The Vindication of the Hawk Characterizing 2 x 2 Normal Form Games I Big John and Little John Revisited Dominance Revisited Competition on Main Street Revisited Twin Sisters Revisited Twin Sisters: An Agent-Based Model One-Card, Two-Round Poker with Bluffing An Agent-Based Model of Poker with Bluffing Trust in Networks ElFarol Decorated Lizards Sex Ratios as Nash Equilibria A Mating Game Coordination Failure Colonel Blotto Game Number Guessing Game Target Selection A Reconnaissance Game Attack on Hidden Object Two-Person, Zero-Sum Games Mutual Monitoring in a Partnership Mutual Monitoring in Teams

116 116 117 118 119 119 120 120 122 122 123 123 124 126 126 127 128 128 128 129 129 131 132 133 134 135 137 140 141 141 142 142 142 143 143 145 145

Contents 631 6.38 6.39 6.40 6.41 6.42 6.43 6.44 6.45 6.46 7

Principal-Agent Models 7.1 7.2 7.3 7.4 7.5 7.6 7.7

8

Gift Exchange Contract Monitoring Profit Signaling Properties of the Employment Relationship Peasant and Landlord Bob's Car Insurance A Generic Principal-Agent Model

Signaling Games 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

9

Altruism(?) in Bird Flocks The Groucho Marx Game Games of Perfect Information Correlated Equilibria Territoriality as a Correlated Equilibrium Haggling at the Bazaar Poker with Bluffing Revisited Algorithms for Finding Nash Equilibria Why Play Mixed Strategies? Reviewing of Basic Concepts

Signaling as a Coevolutionary Process A Generic Signaling Game Sex and Piety: The Darwin-Fisher Model Biological Signals as Handicaps The Shepherds Who Never Cry Wolf My Brother's Keeper Honest Signaling among Partial Altruists Educational Signaling Education as a Screening Device Capital as a Signaling Device

Repeated Games 9.1 9.2 9.3 9.4

Death and Discount Rates in Repeated Games Big Fish and Little Fish Alice and Bob Cooperate The Strategy of an Oil Cartel

146 147 151 151 153 154 156 157 160 161 162 162 163 164 168 169 173 174 179 179 180 182 187 189 190 193 195 197 199 201 202 202 204 205

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Contents 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17

Reputational Equilibrium Tacit Collusion The One-Stage Deviation Principle Tit for Tat I'd Rather Switch Than Fight The Folk Theorem The Folk Theorem and the Nature of Signaling The Folk Theorem Fails in Large Groups Contingent Renewal Markets Do Not Clear Short-Side Power in Contingent Renewal Markets Money Confers Power in Contingent Renewal Markets The Economy Is Controlled by the Wealthy Contingent Renewal Labor Markets

10 Evolutionarily Stable Strategies 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16

Evolutionarily Stable Strategies: Definition Properties of Evolutionarily Stable Strategies Characterizing Evolutionarily Stable Strategies A Symmetric Coordination Game A Dynamic Battle of the Sexes Symmetrical Throwing Fingers Hawks, Doves, and Bourgeois Trust in Networks II Cooperative Fishing Evolutionarily Stable Strategies Are Not Unbeatable A Nash Equilibrium That Is Not an EES Rock, Paper, and Scissors Has No ESS Invasion of the Pure-Strategy Mutants Multiple Evolutionarily Stable Strategies Evolutionarily Stable Strategies in Finite Populations Evolutionarily Stable Strategies in Asymmetric Games

11 Dynamical Systems 11.1 11.2 11.3 11.4

Dynamical Systems: Definition Population Growth Population Growth with Limited Carrying Capacity The Lotka-Volterra Predator-Prey Model

205 206 208 209 210 213 216 217 219 222 223 223 224 229 230 232 233 236 236 237 238 238 238 240 240 241 241 242 242 244 247 247 248 249 251

Contents 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14

Dynamical Systems Theory Existence and Uniqueness The Linearization Theorem Dynamical Systems in One Dimension Dynamical Systems in Two Dimensions Exercises in Two-Dimensional Linear Systems Lotka-Volterra with Limited Carrying Capacity Take No Prisoners The Hartman-Grobman Theorem Features of Two-Dimensional Dynamical Systems

12 Evolutionary Dynamics 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15 12.16 12.17 12.18 12.19

The Origins of Evolutionary Dynamics Strategies as Replicators A Dynamic Hawk-Dove Game Sexual Reproduction and the Replicator Dynamic Properties of the Replicator System The Replicator Dynamic in Two Dimensions Dominated Strategies and the Replicator Dynamic Equilibrium and Stability with a Replicator Dynamic Evolutionary Stability and Asymptotically Stability Trust in Networks III Characterizing 2 x 2 Normal Form Games II Invasion of the Pure-Strategy Nash Mutants II A Generalization of Rock, Paper, and Scissors Uta stansburiana in Motion The Dynamics of Rock, Paper, and Scissors The Lotka-Volterra Model and Biodiversity Asymmetric Evolutionary Games Asymmetric Evolutionary Games II The Evolution of Trust and Honesty

13 Markov Economies and Stochastic Dynamical Systems 13.1 13.2 13.3 13.4

Markov Chains The Ergodic Theorem for Markov Chains The Infinite Random Walk The Sisyphean Markov Chain

255 256 257 258 260 264 266 266 267 268 270 271 272 274 276 278 279 280 282 284 284 285 286 287 287 288 288 290 295 295 297 297 305 307 308

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Contents 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12

Andrei Andreyevich's Two-Urn Problem Solving Linear Recursion Equations Good Vibrations Adaptive Learning The Steady State of a Markov Chain Adaptive Learning II Adaptive Learning with Errors Stochastic Stability

309 310 311 312 314 315 316 317

14 Table of Symbols

319

15 Answers

321

Sources for Problems

373

References

375

Index

385