Game-Theoretical Models in Biology (Chapman & Hall/CRC Mathematical Biology Series)

"Covering the major topics of evolutionary game theory, Game-Theoretical Models in Biology, Second Edition presents both abstract and practical mathematical models of real biological situations. It discusses the static aspects of game theory in a mathematically rigorous way that is appealing to mathematicians. In addition, the authors explore many applications of game theory to biology, making the text useful to biologists as well. The book describes a wide range of topics in evolutionary games, including matrix games, replicator dynamics, the hawk-dove game, and the prisoner's dilemma. It covers the evolutionarily stable strategy, a key concept in biological games, and offers in-depth details of the mathematical models. Most chapters illustrate how to use Python to solve various games. Important biological phenomena, such as the sex ratio of so many species being close to a half, the evolution of cooperative behaviour, and the existence of adornments (for example, the peacock's tail), have been explained using ideas underpinned by game theoretical modelling. Suitable for readers studying and working at the interface of mathematics and the life sciences, this book shows how evolutionary game theory is used in the modelling of these diverse biological phenomena. In this thoroughly revised new edition, the authors have added three new chapters on the evolution of structured populations, biological signalling games, and a topical new chapter on evolutionary models of cancer. There are also new sections on games with time constraints that convert simple games to potentially complex nonlinear ones; new models on extortion strategies for the Iterated Prisoner's Dilemma and on social dilemmas; and on evolutionary models of vaccination, a timely section given the current Covid pandemic. Features Presents a wide range of biological applications of game theory Suitable for researchers and professionals in mathematical biology and the life sciences, and as a text for postgraduate courses in mathematical biology Provides numerous examples, exercises, and Python code"-- Provided by publisher Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Authors 1. Introduction 1.1. The history of evolutionary games 1.1.1. Early game playing and strategic decisions 1.1.2. The birth of modern game theory 1.1.3. The beginnings of evolutionary games 1.2. The key mathematical developments 1.2.1. Static games 1.2.2. Dynamic games 1.3. The range of applications 1.4. Reading this book 2. What is a game? 2.1. Key game elements 2.1.1. Players 2.1.2. Strategies 2.1.2.1. Pure strategies 2.1.2.2. Mixed strategies 2.1.2.3. Pure or mixed strategies? 2.1.3. Payoffs 2.1.3.1. Representation of payoffs by matrices 2.1.3.2. Contests between mixed strategists 2.1.3.3. Generic payoffs 2.1.4. Games in normal form 2.2. Games in biological settings 2.2.1. Representing the population 2.2.2. payoffs in matrix games 2.3. Further reading 2.4. Exercises 3. Two approaches to game analysis 3.1. The dynamical approach 3.1.1. Replicator dynamics 3.1.1.1. Discrete replicator dynamics 3.1.1.2. Continuous replicator dynamics 3.1.2. Adaptive dynamics 3.1.3. Other dynamics 3.1.4. Timescales in evolution 3.2. The static approach — ESS 3.2.1. Nash equilibria 3.2.2. Evolutionarily Stable Strategies 3.2.2.1. ESSs for matrix games 3.2.3. Polymorphic versus monomorphic populations 3.2.4. Stability of Nash equilibria and of ESSs 3.3. Dynamics versus statics 3.3.1. ESS and replicator dynamics in matrix games 3.3.2. Replicator dynamics and finite populations 3.4. Python code 3.5. Further reading 3.6. Exercises 4. Some classical games 4.1. The Hawk-Dove game 4.1.1. The underlying conflict situation 4.1.2. The mathematical model 4.1.3. Mathematical analysis 4.1.4. An adjusted Hawk-Dove game 4.1.5. Replicator dynamics in the Hawk-Dove game 4.1.6. Polymorphic mixture versus mixed strategy 4.2. The Prisoner's Dilemma 4.2.1. The underlying conflict situation 4.2.2. The mathematical model 4.2.3. Mathematical analysis 4.2.4. Interpretation of the results 4.2.5. The IPD, computer tournaments and Tit for Tat 4.3. The war of attrition 4.3.1. The underlying conflict situation 4.3.2. The mathematical model 4.3.3. Mathematical analysis 4.3.4. Some remarks on the above analysis and results 4.3.5. A war of attrition game with limited contest duration 4.3.6. A war of attrition with finite strategies 4.3.7. The asymmetric war of attrition 4.4. The sex ratio game 4.4.1. The underlying conflict situation 4.4.2. The mathematical model 4.4.3. Mathematical analysis 4.5. Python code 4.6. Further reading 4.7. Exercises 5. The underlying biology 5.1. Darwin and natural selection 5.2. Genetics 5.2.1. Hardy-Weinberg equilibrium 5.2.2. Genotypes with different fitnesses 5.3. Games involving genetics 5.3.1. Genetic version of the Hawk-Dove game 5.3.2. A rationale for symmetric games 5.3.3. Restricted repertoire and the streetcar theory 5.4. Fitness, strategies and players 5.4.1. Fitness 1 5.4.2. Fitness 2 5.4.3. Fitness 3 5.4.4. Fitness 4 5.4.5. Fitness 5 5.4.6. Further considerations 5.5. Selfish genes: How can non-beneficial genes propagate? 5.5.1. Genetic hitchhiking 5.5.2. Selfish genes 5.5.3. Memes and cultural evolution 5.5.4. Selection at the level of the cell 5.6. The role of simple mathematical models 5.7. Python code 5.8. Further reading 5.9. Exercises 6. Matrix games 6.1. Properties of ESSs 6.1.1. An equivalent definition of an ESS 6.1.2. A uniform invasion barrier 6.1.3. Local superiority of an ESS 6.1.4. ESS supports and the Bishop-Cannings theorem 6.2. ESSs in a 2 x 2 matrix game 6.3. Haigh's procedure to locate all ESSs 6.4. ESSs in a 3 x 3 matrix game 6.4.1. Pure strategies 6.4.2. A mixture of two strategies 6.4.3. Internal ESSs 6.4.4. No ESS 6.5. Patterns of ESSs 6.5.1. Attainable patterns 6.5.2. Exclusion results 6.5.3. Construction methods 6.5.4. How many ESSs can there be? 6.6. Extensions to the Hawk-Dove game 6.6.1. The extended Hawk-Dove game with generic payoffs 6.6.2. ESSs on restricted strategy sets 6.6.3. Sequential introduction of strategies 6.7. Python code 6.8. Further reading 6.9. Exercises 7. Nonlinear games 7.1. Overview and general theory 7.2. Linearity in the focal player strategy and playing the field 7.2.1. A generalisation of results for linear games 7.2.2. Playing the field 7.2.2.1. Parker's matching principle 7.3. Nonlinearity due to non-constant interaction rates 7.3.1. Nonlinearity in pairwise games 7.3.2. Other games with nonlinear interaction rates 7.4. Nonlinearity due to games with time constraints 7.4.1. The model 7.5. Nonlinearity in the strategy of the focal player 7.5.1. A sperm allocation game 7.5.2. A tree height competition game 7.6. Linear versus nonlinear theory 7.7. Python code 7.8. Further reading 7.9. Exercises 8. Asymmetric games 8.1. Selten's theorem for games with two roles 8.2. Bimatrix games 8.2.1. Dynamics in bimatrix games 8.3. Uncorrelated asymmetry—The Owner-Intruder game 8.4. Correlated asymmetry 8.4.1. Asymmetry in the probability of victory 8.4.2. A game of brood care and desertion 8.4.2.1. Linear version 8.4.2.2. Nonlinear version 8.4.3. Asymmetries in rewards and costs: the asymmetric war of attrition 8.5. Python code 8.6. Further reading 8.7. Exercises 9. Multi-player games 9.1. Multi-player matrix games 9.1.1. Two-strategy games 9.1.2. ESSs for multi-player games 9.1.3. Patterns of ESSs 9.1.4. More on two-strategy, m-player matrix games 9.1.5. Dynamics of multi-player matrix games 9.2. The multi-player war of attrition 9.2.1. The multi-player war of attrition without strategy adjustments 9.2.2. The multi-player war of attrition with strategy adjustments 9.2.3. Multi-player war of attrition with several rewards 9.3. Structures of dependent pairwise games 9.3.1. Knockout contests 9.4. Python code 9.5. Further reading 9.6. Exercises 10. Extensive form games and other concepts in game theory 10.1. Games in extensive form 10.1.1. Key components 10.1.1.1. The game tree 10.1.1.2. The player partition 10.1.1.3. Choices 10.1.1.4. Strategy 10.1.1.5. The payoff function 10.1.2. Backwards induction and sequential equilibria 10.1.3. Games in extensive form and games in normal form 10.2. Perfect, imperfect and incomplete information 10.2.1. Disturbed games 10.2.2. Games in extensive form with imperfect information—The information partition 10.3. Repeated games 10.4. Python code 10.5. Further reading 10.6. Exercises 11. State-based games 11.1. State-based games 11.1.1. Optimal foraging 11.1.2. The general theory of state-based games 11.1.3. A simple foraging game 11.1.4. Evolutionary games based upon state 11.2. A question of size 11.2.1. Setting up the model 11.2.2. ESS analysis 11.2.3. A numerical example 11.3. Life history theory 11.4. Python code 11.5. Further reading 11.6. Exercises 12. Games in finite populations and on graphs 12.1. Finite populations and stochastic games 12.1.1. The Moran process 12.1.2. The xation probability 12.1.3. General Birth-Death processes 12.1.4. The Moran process and discrete replicator dynamics 12.1.5. Fixation and absorption times 12.1.5.1. Exact formulae 12.1.5.2. The diffusion approximation 12.2. Games in finite populations 12.3. Evolution on graphs 12.3.1. The fixed fitness case 12.3.1.1. Regular graphs 12.3.1.2. Selection suppressors and amplifiers 12.3.2. Dynamics and fitness 12.4. Games on graphs 12.4.1. Strong selection models 12.4.1.1. Theoretical results for strong selection 12.4.2. Weak selection models 12.4.2.1. The structure coefficient 12.5. Python code 12.6. Further reading 12.7. Exercises 13. Evolution in structured populations 13.1. Spatial games and cellular automata 13.2. Theoretical developments for modelling general structures 13.3. Evolution in structured populations with multi-player interactions 13.3.1. Basic setup 13.3.2. Fitness 13.3.3. Multi-player games 13.3.4. Evolutionary dynamics 13.3.5. The Territorial Raider model 13.4. More multi-player games 13.4.1. Structure coefficients and multi-player games 13.4.2. Games with variable group sizes 13.5. Evolving population structures 13.5.1. Games with reproducing vertices 13.5.2. Link formation models 13.6. Python code 13.7. Further reading 13.8. Exercises 14. Adaptive dynamics 14.1. Introduction and philosophy 14.2. Fitness functions and the fitness landscape 14.2.1. Taylor expansion of s(y; x) 14.2.2. Adaptive dynamics for matrix games 14.3. Pairwise invasibility and Evolutionarily Singular Strategies 14.3.1. Four key properties of Evolutionarily Singular Strategies 14.3.1.1. Non-invasible strategies 14.3.1.2. When an ess can invade nearby strategies 14.3.1.3. Convergence stability 14.3.1.4. Protected polymorphism 14.3.2. Classi cation of Evolutionarily Singular Strategies 14.3.2.1. Case 5 14.3.2.2. Case 7 14.3.2.3. Case 3—Branching points 14.4. Adaptive dynamics with multiple traits 14.5. The assumptions of adaptive dynamics 14.6. Python code 14.7. Further reading 14.8. Exercises 15. The evolution of cooperation 15.1. Kin selection and inclusive fitness 15.2. Greenbeard genes 15.3. Direct reciprocity: developments of the Prisoner's Dilemma 15.3.1. An error-free environment 15.3.2. An error-prone environment 15.3.3. ESSs in the IPD game 15.3.4. A simple rule for the evolution of cooperation by direct reciprocity 15.3.5. Extortion and the Iterated Prisoner's Dilemma 15.4. Public Goods games 15.4.1. Punishment 15.4.2. General social dilemmas 15.5. Indirect reciprocity and reputation dynamics 15.6. The evolution of cooperation on graphs 15.7. Multi-level selection 15.8. Python code 15.9. Further reading 15.10. Exercises 16. Group living 16.1. The costs and benefits of group living 16.2. Dominance hierarchies: formation and maintenance 16.2.1. Stability and maintenance of dominance hierarchies 16.2.2. Dominance hierarchy formation 16.2.2.1. Winner and loser models 16.2.3. Swiss tournaments 16.3. The enemy without: responses to predators 16.3.1. Setting up the game 16.3.1.1. Modelling scanning for predators 16.3.1.2. payoffs 16.3.2. Analysis of the game 16.4. The enemy within: infanticide and other anti-social behaviour 16.4.1. Infanticide 16.4.2. Other behaviour which negatively affects groups 16.5. Python code 16.6. Further reading 16.7. Exercises 17. Mating games 17.1. Introduction and overview 17.2. Direct conflict 17.2.1. Setting up the model 17.2.1.1. Analysis of a single contest 17.2.1.2. The case of a limited number of contests per season 17.2.2. An unlimited number of contests 17.2.3. Determining rewards and costs 17.3. Indirect conflict and sperm competition 17.3.1. Setting up the model 17.3.1.1. Modelling sperm production 17.3.1.2. Model parameters 17.3.1.3. Modelling fertilisation and payoffs 17.3.2. The ESS if males have no knowledge 17.3.3. The ESS if males have partial knowledge 17.3.4. Summary 17.4. The Battle of the Sexes 17.4.1. Analysis as a bimatrix game 17.4.2. The coyness game 17.4.2.1. The model 17.4.2.2. Fitness 17.4.2.3. Determining the ESS 17.5. Python code 17.6. Further Reading 17.7. Exercises 18. Signalling games 18.1. The theory of signalling games 18.2. Selecting mates: signalling and the handicap principle 18.2.1. Setting up the model 18.2.2. Assumptions about the game parameters 18.2.3. ESSs 18.2.4. A numerical example 18.2.5. Properties of the ESS—honest signalling 18.2.6. Limited options 18.3. Alternative models of costly honest signalling 18.3.1. Index signals 18.3.2. The Pygmalion game: signalling with both costs and constraints 18.3.3. Screening games 18.4. Signalling without cost 18.5. Pollinator signalling games 18.6. Python code 18.7. Further Reading 18.8. Exercises 19. Food competition 19.1. Introduction 19.2. Ideal Free Distribution for a single species 19.2.1. The model 19.3. Ideal Free Distribution for multiple species 19.3.1. The model 19.3.2. Both patches occupied by both species 19.3.3. One patch occupied by one species, another by both 19.3.4. Species on different patches 19.3.5. Species on the same patch 19.4. Distributions at and deviations from the Ideal Free Distribution 19.5. Compartmental models of kleptoparasitism 19.5.1. The model 19.5.2. Analysis 19.5.3. Extensions of the model 19.6. Compartmental models of interference 19.7. Producer-scrounger models 19.7.1. The Finder-Joiner game—the sequential version with complete information 19.7.1.1. The model 19.7.1.2. Analysis 19.7.1.3. Discussion 19.7.2. The Finder-Joiner game—the sequential version with partial information 19.8. Python code 19.9. Further reading 19.10. Exercises 20. Predator-prey and host-parasite interactions 20.1. Game-theoretical predator-prey models 20.1.1. The model 20.1.2. Analysis 20.1.3. Results 20.2. The evolution of defence and signalling 20.2.1. The model 20.2.1.1. Interaction of prey with a predator 20.2.1.2. Payo to an individual prey 20.2.2. Analysis and results 20.2.3. An alternative model 20.2.4. Cheating 20.3. Brood parasitism 20.3.1. The model 20.3.2. Results 20.4. Parasitic wasps and the asymmetric war of attrition 20.4.1. The model 20.4.2. Analysis—evaluating the payoffs 20.4.3. Discussion 20.5. Complex parasite lifecycles 20.5.1. A model of upwards incorporation 20.5.2. Analysis and results 20.6. Search games involving predators and prey 20.6.1. Search games 20.6.2. The model of Gal and Casas 20.6.3. The repeated game 20.6.4. Capture can occur in transit 20.7. Python code 20.8. Further reading 20.9. Exercises 21. Epidemic models 21.1. SIS and SIR models 21.1.1. The SIS epidemic 21.1.1.1. The model 21.1.1.2. Analysis 21.1.1.3. Summary of results 21.1.2. The SIR epidemic 21.1.2.1. The model 21.1.2.2. Analysis and results 21.1.2.3. Some other models 21.1.3. Epidemics on graphs 21.2. The evolution of virulence 21.2.1. An SI model for single epidemics with immigration and death 21.2.1.1. Model and results 21.2.2. An SI model for two epidemics with immigration and death and no superinfection 21.2.2.1. Model and results 21.2.3. Superinfection 21.2.3.1. Model and results 21.3. Viruses and the Prisoner's Dilemma 21.3.1. The model 21.3.2. Results 21.3.3. A real example 21.4. Vaccination models 21.5. Python code 21.6. Further reading 21.7. Exercises 22. Evolutionary cancer modelling 22.1. Modelling tumour growth — an ecological approach to cancer 22.2. A spatial model of cancer evolution 22.3. Cancer therapy as a game-theoretic scenario 22.4. Adaptive therapies 22.5. Python code 22.6. Further reading 22.7. Exercises 23. Conclusions 23.1. Types of evolutionary games used in biology 23.1.1. Classical games, linearity on the left and replicator dynamics 23.1.2. Strategies as a continuous trait and nonlinearity on the left 23.1.3. Departures from infinite, well-mixed populations of identical individuals 23.1.4. More complex interactions and other mathematical complications 23.1.5. Some biological issues 23.1.6. Models of specific behaviours 23.2. What makes a good mathematical model? 23.3. Future developments 23.3.1. Agent-based modelling 23.3.2. Multi-level selection 23.3.3. Unifying timescales 23.3.4. Games in structured populations 23.3.5. Nonlinear games 23.3.6. Asymmetries in populations 23.3.7. What is a payoff? 23.3.8. A more unifed approach to model applications 23.3.9. A more integrated understanding of the role of natural selection 23.3.10. Integrating player and strategy evolution into evolutionary dynamics A. Python Bibliography Index "Preface: Since its inception in the 1960s, evolutionary game theory has become increasingly influential in the modelling of biology, both in terms of mathematical developments and especially in the range of applications. Important biological phenomena, such as the fact that the sex ratio of so many species is close to a half, the evolution of cooperative behaviour and the existence of costly ornaments like the peacock's tail, have been explained using ideas underpinned by game theoretical modelling. The key concept in biological games is the Evolutionarily Stable Strategy (ESS), which resists invasion by all others once it has achieved dominance in the population. This static concept is very powerful, and is the focus of analysis of the majority of models, and while we discuss numerous other mathematical concepts, this is the most important one for our book. For a number of years the authors have been aware that, while there are a number of good books on evolutionary games, a particular type of book that we were looking for did not exist. The catalyst for writing this book was a discussion between Nick Britton and MB on the subject of books in evolutionary game theory. Nick was looking for a book on this subject for the Taylor and Francis Mathematical and Computational Biology book series. After discussing this, the authors decided that this was an opportunity to write the book we had been looking for. The book that we were missing was a wide ranging book covering the major topics of evolutionary game theory, containing both the more abstract mathematical models and a range of mathematical models of real biological situations, and this is the book we have tried to write"--