Game-Theoretical Models in Biology

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