Mark Broom is a professor of mathematics at City, University of London. For over 30 years, he has carried out mathematical research in game theory applied to biology. His major research themes include multi-player games, patterns of evolutionarily stable strategies, models of parasitic behavior (especially kleptoparasitism), the evolution of defence and signalling, and evolutionary processes in structured populations. He earned his PhD in mathematics from the University of Sheffield. Jan Rychtář□is a professor of mathematics at Virginia Commonwealth University. Prior to joining VCU, he was a Professor at UNC Greensboro. He works on game theoretical models and mathematical models of kleptoparasitism. His recent research interests include mathematical biology and game theory. He earned his PhD in mathematics from the University of Alberta.

"It is hard to imagine that the book by Broom & Rychtar is already a decade old, it stills feels fresh! However, in a second edition, the authors have now successfully addressed several novel topics that have developed rapidly in these past 10 years, such as the evolution of cancer or vaccination games. On top of the excellent structure of the original edition of this book, including great exercises, the authors have now included python code and pointers to relevant packages. This is an excellent way to make a new generation of game theorists familiar with the field and at the same time allow them a much more interactive experience! The new edition of this wonderful book proves that evolutionary game theory is alive and kicking!"– Arne Traulsen, Director, Max Planck Institute for Evolutionary Biology, Germany"The second edition of Game-Theoretical Models in Biology shows the tremendous development and applications that this theory has undergone since its inception fifty years ago. That fact that this theory is still undergoing development and finding new applications is evidenced by the fact that the authors have included completely new chapters reflecting, for example, recent applications in the study of cancer evolution or evolution on graphs in the second edition. The book is suitable both for students of mathematical disciplines, to whom it will show the strong application of mathematics in biology, and for students of biological disciplines, who want to gain a solid theoretical basis for the description of eco-evolutionary laws. It is the most comprehensive treatise on the applications of evolutionary game theory in evolutionary and population biology. "– Vlastimil Křivan, University of South Bohemia, Czech Republic"If you want a solid foundation in the topic of game theory in biology, then work your way through this book. It is an authoritative account of the basics, introduces many important biological applications, and has a plethora of insights. Really excellent."– John McNamara, Emeritus Professor at University of Bristol, United Kingdom"In this update of their 2013 book, Broom and Rychtar continue to provide a valuable resource to any researcher interested in evolutionary games. Readers of the original book will especially appreciate the new chapters/sections on recent developments and applications in the field as well as the expanded bibliography. The book will also serve as an excellent self-contained text, with an extensive set of exercises in each chapter, for students at a graduate or senior undergraduate level studying game-theoretic applications to biology"– Ross Cressman, Wilfrid Laurier University, Canada

• 1. Introduction. 1.1. The History of Evolutionary Games. 1.2. The Key Mathematical Developments. 1.3. The Range of Applications. 1.4. Reading this Book. 2. What is a Game? 2.1. Key Game Elements. 2.2. Games in Biological Settings. 2.3. Further Reading. 2.4. Exercises. 3. Two Approaches to Game Analysis. 3.1. The Dynamical Approach. 3.2. The Static Approach — ESS. 3.3. Dynamics Versus Statics. 3.4. Python Code. 3.5 Further Reading. 3.6 Exercises. 4. Some Classical Games. 4.1. The Hawk-Dove Game. 4.2. The Prisoner’s Dilemma. 4.3. The War of Attrition. 4.4. The Sex Ratio Game. 4.5. Python Code. 4.6. Further Reading. 4.7. Exercises. 5. The Underlying Biology. 5.1. Darwin and Natural Selection. 5.2. Genetics. 5.3. Games Involving Genetics. 5.4. Fitness, Strategies and Players. 5.5. Selfish Genes: How can Non-Beneficial Genes Propagate? 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. 6.2. ESSs in a 2 × 2 Matrix Game. 6.3. Haigh’s Procedure to Locate all ESSs. 6.4. ESSs in a 3 × 3 Matrix Game. 6.5. Patterns of ESSs 6.6. Extensions to the Hawk-Dove Game. 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. 7.3. Nonlinearity Due to Non-Constant Interaction Rates. 7.4. Nonlinearity due to Games with Time Constraints. 7.5. Nonlinearity in the Strategy of the Focal Player. 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. 8.3. Uncorrelated Asymmetry—The Owner-Intruder Game. 8.4. Correlated Asymmetry. 8.7. Python Code. 8.8. Further Reading. 8.9. Exercises. 9. Multi-player Games. 9.1. Multi-player Matrix Games. 9.2. The multi-player War of Attrition. 9.3 Structures of Dependent Pairwise Games. 9.7. Python Code. 9.8. Further Reading. 9.9. Exercises. 10. Extensive Form Games and other Concepts in Game Theory. 10.1. Games in Extensive Form. 10.2. Perfect, imperfect and incomplete information. 10.3. Repeated games. 10.4. Python Code. 10.5. Further Reading. 10.6. Exercises. 11. State-based Games. 11.1. State-based Games. 11.2. A Question of Size. 11.3. Life History Theory. 11.7. Python Code. 11.8. Further Reading. 11.9. Exercises. 12. Games in Finite Populations and on Graphs. 12.1. Finite Populations and Stochastic Games. 12.2. Games in Finite Populations. 12.3. Evolution on Graphs. 12.4 Games on Graphs. 12.7. Python Code. 12.8. Further Reading. 12.9. 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. 13.4. More Multi-Player Games. 13.5. Evolving Population Structures. 13.7. Python Code. 13.8. Further Reading. 13.9. Exercises. 14. Adaptive Dynamics. 14.1. Introduction and Philosophy. 14.2. Fitness Functions and the Fitness Landscape. 14.3. Pairwise Invasibility and Evolutionarily Singular Strategies. 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. 15.4. Public Goods Games. 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. 16.3. The Enemy without: Responses to Predators. 16.4. The Enemy Within: Infanticide and Other anti-social Behaviour. 16.5. Python Code. 16.6. Further Reading. 16.7. Exercises. 17. Mating Games. 17.1. Introduction and Overview. 17.2. Direct Conflict. 17.3. Indirect Conflict and Sperm Competition. 17.4. The Battle of the Sexes. 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. 18.3. Alternative Models of Costly Honest Signalling. 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. 19.3. Ideal Free Distribution for Multiple Species. 19.4. Distributions at and Deviations from the Ideal Free Distribution. 19.5. Compartmental Models of kleptoparasitism. 19.6. Compartmental Models of Interference. 19.7. Producer-scrounger Models. 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. 20.2. The Evolution of Defence and Signalling. 20.3. Brood Parasitism. 20.4. Parasitic Wasps and the Asymmetric war of Attrition. 20.5. Complex Parasite Lifecycles. 20.6. Search Games Involving Predators and Prey. 20.7. Python Code. 20.8. Further Reading. 20.9. Exercises. 21. Epidemic models. 21.1. SIS and SIR models. 21.2. The Evolution of virulence. 21.3. Viruses and the Prisoner’s Dilemma. 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. 23.2. What Makes a Good Mathematical Model? 23.3. Future Developments. A. Python.

Mark Broom is a professor of mathematics at City, University of London. For over 30 years, he has carried out mathematical research in game theory applied to biology. His major research themes include multi-player games, patterns of evolutionarily stable strategies, models of parasitic behavior (especially kleptoparasitism), the evolution of defence and signalling, and evolutionary processes in structured populations. He earned his PhD in mathematics from the University of Sheffield. Jan Rychtář□is a professor of mathematics at Virginia Commonwealth University. Prior to joining VCU, he was a Professor at UNC Greensboro. He works on game theoretical models and mathematical models of kleptoparasitism. His recent research interests include mathematical biology and game theory. He earned his PhD in mathematics from the University of Alberta.

"It is hard to imagine that the book by Broom & Rychtar is already a decade old, it stills feels fresh! However, in a second edition, the authors have now successfully addressed several novel topics that have developed rapidly in these past 10 years, such as the evolution of cancer or vaccination games. On top of the excellent structure of the original edition of this book, including great exercises, the authors have now included python code and pointers to relevant packages. This is an excellent way to make a new generation of game theorists familiar with the field and at the same time allow them a much more interactive experience! The new edition of this wonderful book proves that evolutionary game theory is alive and kicking!"– Arne Traulsen, Director, Max Planck Institute for Evolutionary Biology, Germany"The second edition of Game-Theoretical Models in Biology shows the tremendous development and applications that this theory has undergone since its inception fifty years ago. That fact that this theory is still undergoing development and finding new applications is evidenced by the fact that the authors have included completely new chapters reflecting, for example, recent applications in the study of cancer evolution or evolution on graphs in the second edition. The book is suitable both for students of mathematical disciplines, to whom it will show the strong application of mathematics in biology, and for students of biological disciplines, who want to gain a solid theoretical basis for the description of eco-evolutionary laws. It is the most comprehensive treatise on the applications of evolutionary game theory in evolutionary and population biology. "– Vlastimil Křivan, University of South Bohemia, Czech Republic"If you want a solid foundation in the topic of game theory in biology, then work your way through this book. It is an authoritative account of the basics, introduces many important biological applications, and has a plethora of insights. Really excellent."– John McNamara, Emeritus Professor at University of Bristol, United Kingdom"In this update of their 2013 book, Broom and Rychtar continue to provide a valuable resource to any researcher interested in evolutionary games. Readers of the original book will especially appreciate the new chapters/sections on recent developments and applications in the field as well as the expanded bibliography. The book will also serve as an excellent self-contained text, with an extensive set of exercises in each chapter, for students at a graduate or senior undergraduate level studying game-theoretic applications to biology"– Ross Cressman, Wilfrid Laurier University, Canada

• 1. Introduction. 1.1. The History of Evolutionary Games. 1.2. The Key Mathematical Developments. 1.3. The Range of Applications. 1.4. Reading this Book. 2. What is a Game? 2.1. Key Game Elements. 2.2. Games in Biological Settings. 2.3. Further Reading. 2.4. Exercises. 3. Two Approaches to Game Analysis. 3.1. The Dynamical Approach. 3.2. The Static Approach — ESS. 3.3. Dynamics Versus Statics. 3.4. Python Code. 3.5 Further Reading. 3.6 Exercises. 4. Some Classical Games. 4.1. The Hawk-Dove Game. 4.2. The Prisoner’s Dilemma. 4.3. The War of Attrition. 4.4. The Sex Ratio Game. 4.5. Python Code. 4.6. Further Reading. 4.7. Exercises. 5. The Underlying Biology. 5.1. Darwin and Natural Selection. 5.2. Genetics. 5.3. Games Involving Genetics. 5.4. Fitness, Strategies and Players. 5.5. Selfish Genes: How can Non-Beneficial Genes Propagate? 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. 6.2. ESSs in a 2 × 2 Matrix Game. 6.3. Haigh’s Procedure to Locate all ESSs. 6.4. ESSs in a 3 × 3 Matrix Game. 6.5. Patterns of ESSs 6.6. Extensions to the Hawk-Dove Game. 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. 7.3. Nonlinearity Due to Non-Constant Interaction Rates. 7.4. Nonlinearity due to Games with Time Constraints. 7.5. Nonlinearity in the Strategy of the Focal Player. 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. 8.3. Uncorrelated Asymmetry—The Owner-Intruder Game. 8.4. Correlated Asymmetry. 8.7. Python Code. 8.8. Further Reading. 8.9. Exercises. 9. Multi-player Games. 9.1. Multi-player Matrix Games. 9.2. The multi-player War of Attrition. 9.3 Structures of Dependent Pairwise Games. 9.7. Python Code. 9.8. Further Reading. 9.9. Exercises. 10. Extensive Form Games and other Concepts in Game Theory. 10.1. Games in Extensive Form. 10.2. Perfect, imperfect and incomplete information. 10.3. Repeated games. 10.4. Python Code. 10.5. Further Reading. 10.6. Exercises. 11. State-based Games. 11.1. State-based Games. 11.2. A Question of Size. 11.3. Life History Theory. 11.7. Python Code. 11.8. Further Reading. 11.9. Exercises. 12. Games in Finite Populations and on Graphs. 12.1. Finite Populations and Stochastic Games. 12.2. Games in Finite Populations. 12.3. Evolution on Graphs. 12.4 Games on Graphs. 12.7. Python Code. 12.8. Further Reading. 12.9. 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. 13.4. More Multi-Player Games. 13.5. Evolving Population Structures. 13.7. Python Code. 13.8. Further Reading. 13.9. Exercises. 14. Adaptive Dynamics. 14.1. Introduction and Philosophy. 14.2. Fitness Functions and the Fitness Landscape. 14.3. Pairwise Invasibility and Evolutionarily Singular Strategies. 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. 15.4. Public Goods Games. 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. 16.3. The Enemy without: Responses to Predators. 16.4. The Enemy Within: Infanticide and Other anti-social Behaviour. 16.5. Python Code. 16.6. Further Reading. 16.7. Exercises. 17. Mating Games. 17.1. Introduction and Overview. 17.2. Direct Conflict. 17.3. Indirect Conflict and Sperm Competition. 17.4. The Battle of the Sexes. 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. 18.3. Alternative Models of Costly Honest Signalling. 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. 19.3. Ideal Free Distribution for Multiple Species. 19.4. Distributions at and Deviations from the Ideal Free Distribution. 19.5. Compartmental Models of kleptoparasitism. 19.6. Compartmental Models of Interference. 19.7. Producer-scrounger Models. 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. 20.2. The Evolution of Defence and Signalling. 20.3. Brood Parasitism. 20.4. Parasitic Wasps and the Asymmetric war of Attrition. 20.5. Complex Parasite Lifecycles. 20.6. Search Games Involving Predators and Prey. 20.7. Python Code. 20.8. Further Reading. 20.9. Exercises. 21. Epidemic models. 21.1. SIS and SIR models. 21.2. The Evolution of virulence. 21.3. Viruses and the Prisoner’s Dilemma. 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. 23.2. What Makes a Good Mathematical Model? 23.3. Future Developments. A. Python.