Game-Theoretical Models in Biology

Front Cover ... 1
Dedication ... 8
Contents ... 10
Preface ... 22
Authors ... 26
Chapter 1: Introduction ... 28
1.1 The history of evolutionary games ... 28
1.1.1 Early game playing and strategic decisions ... 30
1.1.2 The birth of modern game theory ... 31
1.1.3 The beginnings of evolutionary games ... 32
1.2 The key mathematical developments ... 34
1.2.1 Static games ... 34
1.2.2 Dynamic games ... 35
1.3 The range of applications ... 37
1.4 Reading this book ... 38
Chapter 2: What is a game? ... 40
2.1 Key game elements ... 41
2.1.1 Players ... 41
2.1.2 Strategies ... 42
2.1.2.1 Pure strategies ... 42
2.1.2.2 Mixed strategies ... 43
2.1.2.3 Pure or mixed strategies? ... 45
2.1.3 Payo?s ... 45
2.1.3.1 Representation of payo?s by matrices ... 46
2.1.3.2 Payo?s from contests between mixed strategists ... 47
2.1.3.3 Generic payo?s ... 48
2.1.4 Games in normal form ... 50
2.2 Games in biological settings ... 51
2.2.1 Representing the population ... 52
2.2.2 Payo?s in matrix games ... 53
2.3 Further reading ... 54
2.4 Exercises ... 54
Chapter 3: Two approaches to game analysis ... 56
3.1 The dynamical approach ... 56
3.1.1 Replicator dynamics ... 56
3.1.1.1 Discrete replicator dynamics ... 56
3.1.1.2 Continuous replicator dynamics ... 57
3.1.2 Adaptive dynamics ... 58
3.1.3 Other dynamics ... 59
3.1.4 Timescales in evolution ... 60
3.2 The static approach—Evolutionarily Stable Strategy (ESS) ... 61
3.2.1 Nash equilibria ... 61
3.2.2 Evolutionarily Stable Strategies ... 64
3.2.2.1 ESSs for matrix games ... 65
3.2.3 Some di?erences between polymorphic and monomorphic populations ... 66
3.2.4 Stability of Nash equilibria and of ESSs ... 68
3.3 Dynamics versus statics ... 69
3.3.1 ESS and replicator dynamics in matrix games ... 70
3.3.2 Replicator dynamics and ?nite populations ... 71
3.4 MATLAB program ... 72
3.5 Further reading ... 73
3.6 Exercises ... 73
Chapter 4: Some classical games ... 76
4.1 The Hawk-Dove game ... 76
4.1.1 The underlying con?ict situation ... 76
4.1.2 The mathematical model ... 77
4.1.3 Mathematical analysis ... 77
4.1.4 An adjusted Hawk-Dove game ... 78
4.1.5 Replicator dynamics in the Hawk-Dove game ... 78
4.1.6 Polymorphic mixture versus mixed strategy ... 78
4.2 The Prisoner’s Dilemma ... 80
4.2.1 The underlying con?ict situation ... 81
4.2.2 The mathematical model ... 81
4.2.3 Mathematical analysis ... 82
4.2.4 Interpretation of the results ... 82
4.2.5 The Iterated Prisoner’s Dilemma, computer tournaments and Tit for Tat ... 83
4.3 The war of attrition ... 85
4.3.1 The underlying con?ict situation ... 85
4.3.2 The mathematical model ... 85
4.3.3 Mathematical analysis ... 86
4.3.4 Some remarks on the above analysis and results ... 88
4.3.5 A war of attrition game with limited contest duration ... 88
4.3.6 A war of attrition with ?nite strategies ... 89
4.3.7 The asymmetric war of attrition ... 90
4.4 The sex ratio game ... 90
4.4.1 The underlying con?ict situation ... 91
4.4.2 The mathematical model ... 91
4.4.3 Mathematical analysis ... 92
4.5 MATLAB program ... 92
4.6 Further reading ... 94
4.7 Exercises ... 95
Chapter 5: The underlying biology ... 98
5.1 Darwin and natural selection ... 98
5.2 Genetics ... 100
5.2.1 Hardy-Weinberg equilibrium ... 102
5.2.2 Genotypes with di?erent ?tnesses ... 104
5.3 Games involving genetics ... 107
5.3.1 Genetic version of the Hawk-Dove game ... 107
5.3.2 A rationale for symmetric games ... 108
5.3.3 Restricted repertoire and the streetcar theory ... 109
5.4 Fitness, strategies and players ... 109
5.4.1 Fitness 1 ... 110
5.4.2 Fitness 2 ... 110
5.4.3 Fitness 3 ... 110
5.4.4 Fitness 4 ... 111
5.4.5 Fitness 5 ... 111
5.4.6 Further considerations ... 111
5.5 Sel?sh genes: How can non-bene?cal genes propagate? ... 112
5.5.1 Genetic hitchhiking ... 112
5.5.2 Sel?sh genes ... 114
5.5.3 Memes and cultural evolution ... 115
5.5.4 Selection at the level of the cell ... 115
5.6 The role of simple mathematical models ... 116
5.7 MATLAB program ... 117
5.8 Further reading ... 118
5.9 Exercises ... 118
Chapter 6: Matrix games ... 120
6.1 Properties of ESSs ... 120
6.1.1 An equivalent de?nition of an ESS ... 120
6.1.2 A uniform invasion barrier ... 121
6.1.3 Local superiority of an ESS ... 123
6.1.4 ESS supports and the Bishop-Cannings theorem ... 124
6.2 ESSs in a 2 × 2 matrix game ... 126
6.3 Haigh’s procedure to locate all ESSs ... 128
6.4 ESSs in a 3 × 3 matrix game ... 130
6.4.1 Pure strategies ... 130
6.4.2 A mixture of two strategies ... 131
6.4.3 Internal ESSs ... 131
6.4.4 No ESS ... 132
6.5 Patterns of ESSs ... 133
6.5.1 Attainable patterns ... 134
6.5.2 Exclusion results ... 135
6.5.3 Construction methods ... 136
6.5.4 How many ESSs can there be? ... 137
6.6 Extensions to the Hawk-Dove game ... 138
6.6.1 The extended Hawk-Dove game with generic payo?s ... 139
6.6.2 ESSs on restricted strategy sets ... 140
6.6.3 Sequential introduction of strategies ... 140
6.7 MATLAB program ... 141
6.8 Further reading ... 144
6.9 Exercises ... 145
Chapter 7: Nonlinear games ... 148
7.1 Overview and general theory ... 148
7.2 Linearity in the focal player strategy and playing the ?eld ... 151
7.2.1 A generalisation of results for linear games ... 151
7.2.2 Playing the ?eld ... 154
7.2.2.1 Parker’s matching principle ... 154
7.3 Nonlinearity due to non-constant interaction rates ... 156
7.3.1 Nonlinearity in pairwise games ... 156
7.3.2 Other games with nonlinear interaction rates ... 158
7.4 Nonlinearity in the strategy of the focal player ... 158
7.4.1 A sperm allocation game ... 159
7.4.2 A tree height competition game ... 160
7.5 Some di?erences between linear and nonlinear theory ... 161
7.6 MATLAB program ... 162
7.7 Further reading ... 164
7.8 Exercises ... 164
Chapter 8: Asymmetric games ... 168
8.1 Selten’s theorem for games with two roles ... 169
8.2 Bimatrix games ... 171
8.2.1 Dynamics in bimatrix games ... 173
8.3 Uncorrelated asymmetry—The Owner-Intruder game ... 175
8.4 Correlated asymmetry ... 177
8.4.1 Asymmetry in the probability of victory ... 178
8.4.2 A game of brood care and desertion ... 179
8.4.2.1 Linear version ... 179
8.4.2.2 Nonlinear version ... 180
8.4.3 Asymmetries in rewards and costs: the asymmetric war of attrition ... 182
8.5 MATLAB program ... 184
8.6 Further reading ... 185
8.7 Exercises ... 185
Chapter 9: Multi-player games ... 188
9.1 Multi-player matrix games ... 189
9.1.1 Two-strategy games ... 190
9.1.2 ESSs for multi-player games ... 192
9.1.3 Patterns of ESSs ... 194
9.1.4 More on two-strategy, m-player matrix games ... 194
9.1.5 Dynamics of multi-player matrix games ... 197
9.2 The multi-player war of attrition ... 199
9.2.1 The multi-player war of attrition without strategy adjustments ... 199
9.2.2 The multi-player war of attrition with strategy adjustments ... 201
9.2.3 Multi-player war of attrition with several rewards ... 202
9.3 Structures of dependent pairwise games ... 203
9.3.1 Knockout contests ... 203
9.4 MATLAB program ... 206
9.5 Further reading ... 208
9.6 Exercises ... 208
Chapter 10: Extensive form games and other concepts in game theory ... 212
10.1 Games in extensive form ... 212
10.1.1 Key components ... 213
10.1.1.1 The game tree ... 213
10.1.1.2 The player partition ... 213
10.1.1.3 Choices ... 213
10.1.1.4 Strategy ... 214
10.1.1.5 The payo? function ... 214
10.1.2 Backwards induction and sequential equilibria ... 214
10.1.3 Games in extensive form and games in normal form ... 218
10.2 Perfect, imperfect and incomplete information ... 220
10.2.1 Disturbed games ... 221
10.2.2 Games in extensive form with imperfect information—The information partition ... 223
10.3 Repeated games ... 226
10.4 MATLAB program ... 228
10.5 Further reading ... 229
10.6 Exercises ... 230
Chapter 11: State-based games ... 234
11.1 State-based games ... 235
11.1.1 Optimal foraging ... 235
11.1.2 The general theory of state-based games ... 237
11.1.3 A simple foraging game ... 238
11.1.4 Evolutionary games based upon state ... 239
11.2 A question of size ... 242
11.2.1 Setting up the model ... 243
11.2.2 ESS analysis ... 244
11.2.3 A numerical example ... 244
11.3 Life history theory ... 245
11.4 MATLAB program ... 247
11.5 Further reading ... 248
11.6 Exercises ... 249
Chapter 12: Games in finite and structured populations ... 252
12.1 Finite populations and stochastic games ... 252
12.1.1 The Moran process ... 252
12.1.2 The ?xation probability ... 254
12.1.3 General Birth-Death processes ... 256
12.1.4 The Moran process and discrete replicator dynamics ... 257
12.1.5 Fixation and absorption times ... 258
12.1.5.1 Exact formulae ... 258
12.1.5.2 The di?usion approximation ... 259
12.1.6 Games in ?nite populations ... 260
12.2 Evolution on graphs ... 263
12.2.1 The ?xed ?tness case ... 266
12.2.1.1 Regular graphs ... 267
12.2.1.2 Selection suppressors and ampli?ers ... 268
12.2.2 Games on graphs ... 269
12.2.3 Dynamics and ?tness ... 270
12.3 Spatial games and cellular automata ... 272
12.4 MATLAB program ... 274
12.5 Further reading ... 275
12.6 Exercises ... 276
Chapter 13: Adaptive dynamics ... 278
13.1 Introduction and philosophy ... 278
13.2 Fitness functions and the ?tness landscape ... 279
13.2.1 Taylor expansion of s(y, x) ... 281
13.2.2 Adaptive dynamics for matrix games ... 282
13.3 Pairwise invasibility and Evolutionarily Singular Strategies ... 283
13.3.1 Four key properties of Evolutionarily Singular Strategies ... 283
13.3.1.1 Non-invasible strategies ... 283
13.3.1.2 When an ess can invade nearby strategies ... 284
13.3.1.3 Convergence stability ... 284
13.3.1.4 Protected polymorphism ... 284
13.3.2 Classi?cation of Evolutionarily Singular Strategies ... 284
13.3.2.1 Case 5 ... 285
13.3.2.2 Case 7 ... 287
13.3.2.3 Case 3—Branching points ... 287
13.4 Adaptive dynamics with multiple traits ... 289
13.5 The assumptions of adaptive dynamics ... 291
13.6 MATLAB program ... 292
13.7 Further reading ... 293
13.8 Exercises ... 294
Chapter 14: The evolution of cooperation ... 298
14.1 Kin selection and inclusive ?tness ... 299
14.2 Greenbeard genes ... 301
14.3 Direct reciprocity: developments of the Prisoner’s Dilemma ... 304
14.3.1 An error-free environment ... 304
14.3.2 An error-prone environment ... 306
14.3.3 ESSs in the IPD game ... 307
14.3.4 A simple rule for the evolution of cooperation by direct reciprocity ... 308
14.4 Punishment ... 308
14.5 Indirect reciprocity and reputation dynamics ... 310
14.6 The evolution of cooperation on graphs ... 313
14.7 Multi-level selection ... 315
14.8 MATLAB program ... 316
14.9 Further reading ... 317
14.10 Exercises ... 318
Chapter 15: Group living ... 320
15.1 The costs and bene?ts of group living ... 320
15.2 Dominance hierarchies: formation and maintenance ... 321
15.2.1 Stability and maintenance of dominance hierarchies ... 321
15.2.2 Dominance hierarchy formation ... 324
15.2.2.1 Winner and loser models ... 325
15.2.3 Swiss tournaments ... 326
15.3 The enemy without: responses to predators ... 328
15.3.1 Setting up the game ... 329
15.3.1.1 Modelling scanning for predators ... 329
15.3.1.2 Payo?s ... 330
15.3.2 Analysis of the game ... 331
15.4 The enemy within: infanticide and other anti-social behaviour ... 332
15.4.1 Infanticide ... 332
15.4.2 Other behaviour which negatively a?ects groups ... 334
15.5 MATLAB program ... 335
15.6 Further reading ... 336
15.7 Exercises ... 337
Chapter 16: Mating games ... 340
16.1 Introduction and overview ... 340
16.2 Direct con?ict ... 341
16.2.1 Setting up the model ... 341
16.2.1.1 Analysis of a single contest ... 342
16.2.1.2 The case of a limited number of contests per season ... 342
16.2.2 An unlimited number of contests ... 345
16.2.3 Determining rewards and costs ... 346
16.3 Indirect con?ict and sperm competition ... 347
16.3.1 Setting up the model ... 347
16.3.1.1 Modelling sperm production ... 347
16.3.1.2 Model parameters ... 348
16.3.1.3 Modelling fertilization and payo?s ... 348
16.3.2 The ESS if males have no knowledge ... 349
16.3.3 The ESS if males have partial knowledge ... 350
16.3.4 Summary ... 351
16.4 The Battle of the Sexes ... 351
16.4.1 Analysis as a bimatrix game ... 352
16.4.2 The coyness game ... 352
16.4.2.1 The model ... 353
16.4.2.2 Fitness ... 354
16.4.2.3 Determining the ESS ... 356
16.5 Selecting mates: signalling and the handicap principle ... 357
16.5.1 Setting up the model ... 359
16.5.2 Assumptions about the game parameters ... 359
16.5.3 ESSs ... 361
16.5.4 A numerical example ... 362
16.5.5 Properties of the ESS—honest signalling ... 363
16.6 Other signalling scenarios ... 364
16.6.1 Limited options ... 364
16.6.2 Signalling without cost ... 365
16.7 MATLAB program ... 367
16.8 Further Reading ... 368
16.9 Exercises ... 369
Chapter 17: Food competition ... 372
17.1 Introduction ... 372
17.2 Ideal Free Distribution for a single species ... 372
17.2.1 The model ... 372
17.3 Ideal Free Distribution for multiple species ... 376
17.3.1 The model ... 376
17.3.2 Both patches occupied by both species ... 377
17.3.3 One patch occupied by one species, another by both ... 377
17.3.4 Species on di?erent patches ... 378
17.3.5 Species on the same patch ... 378
17.4 Distributions at and deviations from the Ideal Free Distribution ... 378
17.5 Compartmental models of kleptoparasitism ... 380
17.5.1 The model ... 381
17.5.2 Analysis ... 382
17.5.3 Extensions of the model ... 386
17.6 Compartmental models of interference ... 389
17.7 Producer-scrounger models ... 390
17.7.1 The Finder-Joiner game—the sequential version with complete information ... 391
17.7.1.1 The model ... 391
17.7.1.2 Analysis ... 391
17.7.1.3 Discussion ... 392
17.7.2 The Finder-Joiner game—the sequential version with partial information ... 394
17.8 MATLAB program ... 395
17.9 Further reading ... 397
17.10 Exercises ... 397
Chapter 18: Predator-prey and host-parasite interactions ... 400
18.1 Game-theoretical predator-prey models ... 400
18.1.1 The model ... 401
18.1.2 Analysis ... 402
18.1.3 Results ... 403
18.2 The evolution of defence and signalling ... 403
18.2.1 The model ... 404
18.2.1.1 Interaction of prey with a predator ... 404
18.2.1.2 Payo? to an individual prey ... 405
18.2.2 Analysis and results ... 406
18.2.3 An alternative model ... 406
18.2.4 Cheating ... 408
18.3 Brood parasitism ... 409
18.3.1 The model ... 409
18.3.2 Results ... 411
18.4 Parasitic wasps and the asymmetric war of attrition ... 412
18.4.1 The model ... 413
18.4.2 Analysis—evaluating the payo?s ... 415
18.4.3 Discussion ... 416
18.5 Complex parasite lifecycles ... 417
18.5.1 A model of upwards incorporation ... 417
18.5.2 Analysis and results ... 419
18.6 MATLAB program ... 419
18.7 Further reading ... 422
18.8 Exercises ... 422
Chapter 19: Epidemic models ... 426
19.1 SIS and SIR models ... 426
19.1.1 The SIS epidemic ... 427
19.1.1.1 The model ... 427
19.1.1.2 Analysis ... 428
19.1.1.3 Summary of results ... 429
19.1.2 The SIR epidemic ... 429
19.1.2.1 The model ... 430
19.1.2.2 Analysis and results ... 431
19.1.2.3 Some other models ... 431
19.1.3 Epidemics on graphs ... 433
19.2 The evolution of virulence ... 434
19.2.1 An SI model for single epidemics with immigration and death ... 434
19.2.1.1 Model and results ... 435
19.2.2 An SI model for two epidemics with immigration and death and no superinfection ... 435
19.2.2.1 Model and results ... 436
19.2.3 Superinfection ... 436
19.2.3.1 Model and results ... 437
19.3 Viruses and the Prisoner’s Dilemma ... 438
19.3.1 The model ... 438
19.3.2 Results ... 438
19.3.3 A real example ... 439
19.4 MATLAB program ... 440
19.5 Further reading ... 441
19.6 Exercises ... 441
Chapter 20: Conclusions ... 444
20.1 Types of evolutionary games used in biology ... 444
20.1.1 Classical games, linearity on the left and replicator dynamics ... 444
20.1.2 Strategies as a continuous trait and nonlinearity on the left ... 446
20.1.3 Departures from in?nite, well-mixed populations of identical individuals ... 446
20.1.4 More complex interactions and other mathematical complications ... 448
20.1.5 Some biological issues ... 449
20.1.6 Models of speci?c behaviours ... 450
20.2 What makes a good mathematical model? ... 451
20.3 Future developments ... 453
20.3.1 Agent-based modelling ... 453
20.3.2 Multi-level selection ... 453
20.3.3 Unifying timescales ... 454
20.3.4 Games in structured populations ... 454
20.3.5 Nonlinear games ... 454
20.3.6 Asymmetries in populations ... 455
20.3.7 What is a payo?? ... 455
20.3.8 A more uni?ed approach to model applications ... 455
Appendix A: Intro to MATLAB ... 456
Bibliography ... 472