Evolutionary Dynamics and Extensive Form Games

✍ Scribed by Ross Cressman

• Publisher: The MIT Press
• Year: 2003
• Tongue: English
• Leaves: 331
• Series: Economic Learning and Social Evolution 5
• Category: Library

✦ Synopsis

Evolutionary game theory attempts to predict individual behavior (whether of humans or other species) when interactions between individuals are modeled as a noncooperative game. Most dynamic analyses of evolutionary games are based on their normal forms, despite the fact that many interesting games are specified more naturally through their extensive forms. Because every extensive form game has a normal form representation, some theorists hold that the best way to analyze an extensive form game is simply to ignore the extensive form structure and study the game in its normal form representation. This book rejects that suggestion, arguing that a game's normal form representation often omits essential information from the perspective of dynamic evolutionary game theory.The book offers a synthesis of current knowledge about extensive form games from an evolutionary perspective, emphasizing connections between the extensive form representation and dynamic models that traditionally have been applied to biological and economic phenomena. It develops a general theory to analyze dynamically arbitrary extensive form games and applies this theory to a range of examples. It lays the foundation for the analysis of specific extensive form models of behavior and for the further theoretical study of extensive form evolutionary games.

✦ Table of Contents

Contents......Page 6
Series Foreword......Page 10
Preface......Page 12
1 Introduction......Page 16
1.1 Extensive Form versus Normal Form......Page 17
1.2 Biology versus Economics......Page 23
1.3 Imitation......Page 27
1.4 Organizational Matters......Page 30
1.5 Notes......Page 31
2.1 The Replicator Dynamic......Page 34
2.2 Dynamics for Two-Strategy Games......Page 38
2.3 Monotone Selection Dynamics......Page 42
2.4 Fictitious Play and Best Response Dynamic......Page 46
2.5 Convergence and Stability: NE and ESS......Page 49
2.6.1 Rock–Scissors–Paper Games......Page 52
2.6.2 ESSets and NE Components......Page 57
2.6.3 More Three-Strategy Games......Page 58
2.7 Dynamic Stability for General Games......Page 61
2.8.1 Discrete-Time Viability Selection......Page 68
2.8.2 Continuous-Time Natural Selection......Page 69
2.9 One-Stage Simultaneity Games......Page 71
2.10 Multi-armed Bandits......Page 73
2.11 Appendix......Page 79
2.12 Notes......Page 81
3 Bimatrix Games......Page 84
3.1 Nash Equilibria and Strict Equilibrium Sets......Page 85
3.2 Bimatrix Replicator and Best Response Dynamics......Page 86
3.2.1 The Owner-Intruder Game......Page 89
3.3 Dynamics for Two-Strategy Bimatrix Games......Page 90
3.3.1 Nondegenerate Bimatrix Games......Page 91
3.3.2 Degenerate Bimatrix Games......Page 94
3.4 Symmetrized Bimatrix Games......Page 100
3.4.1 The Symmetrized Bimatrix Replicator Dynamic......Page 103
3.4.2 The Symmetrized Best Response Dynamic......Page 108
3.5 Bimatrix Monotone Selection Dynamics......Page 111
3.6 Notes......Page 116
4 Asymmetric Games......Page 118
4.1 The Normal Form......Page 119
4.2 The Extensive Form: NE and ESSets......Page 121
4.2.1 An Age-Structured Owner-Intruder Game......Page 123
4.3 SESets and Agent Normal Forms......Page 125
4.4 Dynamics and the Wright Manifold......Page 128
4.4.1 The Replicator Dynamic and Subgames......Page 129
4.4.2 Best Response Dynamics......Page 131
4.5 Truly Asymmetric Two-Player Games......Page 132
4.5.1 The Age-Structured Owner-Intruder Game Dynamic......Page 136
4.6 Truly Symmetric Two-Player Games......Page 138
4.6.1 A Truly Symmetric Game Dynamic Counterexample......Page 140
4.6.2 Parallel Bandits......Page 143
4.7.1 A Family of Asymmetric Games......Page 152
4.7.2 Two-Species Evolutionarily Stable Strategies......Page 155
4.8 A Hierarchical Hawk-Dove Game......Page 161
4.9 Appendix A......Page 165
4.10 Appendix B......Page 167
4.11 Notes......Page 168
5.1 Continuous-Time Selection-Recombination......Page 170
5.2 Symmetric Extensive Form with Additive Fitness......Page 172
5.3 Recombination......Page 175
5.4 Selection and Recombination......Page 177
5.5 Notes......Page 178
6 Extensive Form Games......Page 180
6.1 N-Player Extensive Form Games......Page 181
6.1.1 Strategies and Payoffs......Page 183
6.1.2 Nash Equilibria, Subgames, and Backward Induction......Page 186
6.2 Normal Forms and the Replicator Dynamic......Page 188
6.3 The Wright Manifold and Replicator Dynamic......Page 190
6.4 Symmetric Extensive Form Games......Page 195
6.5 Appendix......Page 199
6.6 Notes......Page 200
7 Simultaneity Games......Page 202
7.1 Elementary Two-Stage Simultaneity Games......Page 203
7.2 Two-Stage Two-Strategy Games......Page 207
7.2.1 Two-Stage Two-Strategy Repeated Games......Page 210
7.2.2 Symmetric Signaling Games......Page 212
7.2.3 Cheap Talk Games......Page 215
7.3.1 Simultaneity Games with No Asymmetric Subgames......Page 216
7.3.2 Simultaneity Games with Asymmetric Subgames......Page 219
7.3.3 Simultaneity Games with Moves by Nature......Page 221
7.4 The War of Attrition......Page 222
7.4.1 The Discrete War of Attrition......Page 223
7.4.2 The Continuous War of Attrition......Page 228
7.4.3 The Discrete War of Aggression......Page 230
7.5 The Finitely Repeated Prisoner's Dilemma Game......Page 232
7.5.1 The Replicator and Monotone Selection Dynamics......Page 235
7.5.2 The Best Response Dynamic and Fictitious Play......Page 237
7.6 Appendix A: Proof of Theorem 7.5.3......Page 243
7.7 Appendix B: Maximal Attractor......Page 247
7.8 Notes......Page 248
8 Perfect Information Games......Page 250
8.1 Elementary Perfect Information Games......Page 252
8.2 Equilibrium Selection: Dynamic Approach......Page 255
8.2.1 The Replicator and Monotone Selection Dynamics......Page 257
8.2.2 Fictitious Play and Best Response Dynamic......Page 263
8.2.3 Behavior Strategy Fictitious Play......Page 267
8.3 The Centipede Game......Page 270
8.3.1 Centipede Games of Lengths Two and Three......Page 271
8.3.2 Centipede Games of Length N ≥ 4......Page 273
8.4 Extensive Form Bandits......Page 275
8.4.1 The Centipede Bandit......Page 283
8.5 Appendix A......Page 291
8.6 Appendix B......Page 298
8.7 Notes......Page 302
9.1 Monotone Trajectories......Page 304
9.2 Subgame Monotone Trajectories......Page 307
9.3 An Imitation Example......Page 316
9.4 Discussion......Page 318
9.5 Notes......Page 319
Bibliography......Page 322
C......Page 328
L......Page 329
R......Page 330
Z......Page 331