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