Game-Theoretical Models in Biology

623 Pages·2022·13.888 MB·English

by  Mark Broom, Jan Rychtář

Game-Theoretical
Models in Biology
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 evolu-
tionarily 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 exam-
ple, 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 new chapter
on evolutionary models of cancer. There are also new sections on games with time con-
straints 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.
• Presents a wide range of biological applications of game theory
• Suitable for researchers and professionals in mathematical biology and the life sci-
ences, and as a text for postgraduate courses in mathematical biology
• Provides numerous examples, exercises and Python code.
Chapman & Hall/CRC Mathematical Biology Series
Series Editors:
Ruth Baker, Mark Broom, Adam Kleczkowski, Doron Levy, Sergei Petrovskiy
About the Series
This series aims to capture new developments in mathematical biology, as well
as high-quality work summarizing or contributing to more established topics.
Publishing a broad range of textbooks, reference works, and handbooks, the series
is designed to appeal to students, researchers, and professionals in mathematical
biology. We will consider proposals on all topics and applications within the
field, including but not limited to stochastic modelling, differential equation
modelling, dynamical systems, game theory, machine learning, data science,
evolutionary biology, cell biology, oncology, epidemiology, ecology and more.
An Introduction to Physical Oncology
How Mechanistic Mathematical Modeling Can Improve Cancer Therapy
Outcomes
Vittorio Cristini, Eugene Koay, Zhihui Wang
Introduction to Mathematical Oncology
Yang Kuang, John D. Nagy, Steffen E. Eikenberry
Stochastic Dynamics for Systems Biology
Christian Mazza, Michel Benaim
Systems Biology
Mathematical Modeling and Model Analysis
Andreas Kremling
Cellular Potts Models
Multiscale Extensions and Biological Applications
Marco Scianna, Luigi Preziosi
Quantitative Biology
From Molecular to Cellular Systems
Edited By Michael E. Wall
Game-Theoretical Models in Biology, Second Edition
Mark Broom, Jan Rychtář
For more information about the series, visit: https://www.routledge.com/
Chapman--HallCRC-Mathematical-Biology-Series/book-series/CRCMBS
Game-Theoretical
Models in Biology
Second Edition
Mark Broom
City, University London, UK
Jan Rychtář
Virginia Commonwealth University, USA
Second edition published 2022
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
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© 2022 Taylor & Francis, LLC
First edition published by CRC Press 2013
CRC Press is an imprint of Taylor & Francis Group, LLC
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ISBN: 978-0-367-45668-9 (hbk)
ISBN: 978-1-032-30870-8 (pbk)
ISBN: 978-1-003-02468-2 (ebk)
DOI: 10.1201/9781003024682
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Publisher’s note: This book has been prepared from a camera-ready copy provided by the authors.
To Monica and Dewey and Walter

Contents
Preface xix
Authors xxiii
1 Introduction 1
1.1 The history of evolutionary games . . . . . . . . . . . . . . 1
1.1.1 Early game playing and strategic decisions . . . . . 2
1.1.2 The birth of modern game theory . . . . . . . . . . 4
1.1.3 The beginnings of evolutionary games . . . . . . . . 5
1.2 The key mathematical developments . . . . . . . . . . . . . 7
1.2.1 Static games . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Dynamic games . . . . . . . . . . . . . . . . . . . . 9
1.3 The range of applications . . . . . . . . . . . . . . . . . . . 10
1.4 Reading this book . . . . . . . . . . . . . . . . . . . . . . . 12
2 What is a game? 13
2.1 Key game elements . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Players . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Strategies . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2.1 Pure strategies . . . . . . . . . . . . . . . . 15
2.1.2.2 Mixed strategies . . . . . . . . . . . . . . . 16
2.1.2.3 Pure or mixed strategies? . . . . . . . . . . 17
2.1.3 Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3.1 Representation of payoffs by matrices . . . 19
2.1.3.2 Contests between mixed strategists . . . . 20
2.1.3.3 Generic payoffs . . . . . . . . . . . . . . . 21
2.1.4 Games in normal form . . . . . . . . . . . . . . . . . 23
2.2 Games in biological settings . . . . . . . . . . . . . . . . . . 23
2.2.1 Representing the population . . . . . . . . . . . . . 24
2.2.2 Payoffs in matrix games . . . . . . . . . . . . . . . . 26
2.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
vii
viii Contents
3 Two approaches to game analysis 29
3.1 The dynamical approach . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Replicator dynamics . . . . . . . . . . . . . . . . . . 29
3.1.1.1 Discrete replicator dynamics . . . . . . . . 29
3.1.1.2 Continuous replicator dynamics . . . . . . 30
3.1.2 Adaptive dynamics. . . . . . . . . . . . . . . . . . . 31
3.1.3 Other dynamics . . . . . . . . . . . . . . . . . . . . 32
3.1.4 Timescales in evolution . . . . . . . . . . . . . . . . 33
3.2 The static approach — ESS . . . . . . . . . . . . . . . . . . 34
3.2.1 Nash equilibria . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Evolutionarily Stable Strategies . . . . . . . . . . . 37
3.2.2.1 ESSs for matrix games . . . . . . . . . . . 38
3.2.3 Polymorphic versus monomorphic populations . . . 39
3.2.4 Stability of Nash equilibria and of ESSs . . . . . . . 41
3.3 Dynamics versus statics . . . . . . . . . . . . . . . . . . . . 42
3.3.1 ESS and replicator dynamics in matrix games. . . . 43
3.3.2 Replicator dynamics and finite populations . . . . . 44
3.4 Python code . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Some classical games 49
4.1 The Hawk-Dove game . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 The underlying conflict situation . . . . . . . . . . . 49
4.1.2 The mathematical model . . . . . . . . . . . . . . . 50
4.1.3 Mathematical analysis . . . . . . . . . . . . . . . . . 50
4.1.4 An adjusted Hawk-Dove game . . . . . . . . . . . . 51
4.1.5 Replicator dynamics in the Hawk-Dove game . . . . 51
4.1.6 Polymorphic mixture versus mixed strategy . . . . . 51
4.2 The Prisoner’s Dilemma . . . . . . . . . . . . . . . . . . . . 53
4.2.1 The underlying conflict situation . . . . . . . . . . . 54
4.2.2 The mathematical model . . . . . . . . . . . . . . . 54
4.2.3 Mathematical analysis . . . . . . . . . . . . . . . . . 55
4.2.4 Interpretation of the results . . . . . . . . . . . . . . 55
4.2.5 The IPD, computer tournaments and Tit for Tat . . 56
4.3 The war of attrition . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 The underlying conflict situation . . . . . . . . . . . 58
4.3.2 The mathematical model . . . . . . . . . . . . . . . 59
4.3.3 Mathematical analysis . . . . . . . . . . . . . . . . . 59
4.3.4 Some remarks on the above analysis and results . . 61
4.3.5 A war of attrition game with limited contest duration 61
4.3.6 A war of attrition with finite strategies . . . . . . . 62
4.3.7 The asymmetric war of attrition . . . . . . . . . . . 63
4.4 The sex ratio game . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 The underlying conflict situation . . . . . . . . . . . 64
Contents ix
4.4.2 The mathematical model . . . . . . . . . . . . . . . 64
4.4.3 Mathematical analysis . . . . . . . . . . . . . . . . . 65
4.5 Python code . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 The underlying biology 73
5.1 Darwin and natural selection . . . . . . . . . . . . . . . . . 73
5.2 Genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 Hardy-Weinberg equilibrium . . . . . . . . . . . . . 77
5.2.2 Genotypes with different fitnesses . . . . . . . . . . 79
5.3 Games involving genetics . . . . . . . . . . . . . . . . . . . 81
5.3.1 Genetic version of the Hawk-Dove game . . . . . . . 82
5.3.2 A rationale for symmetric games . . . . . . . . . . . 82
5.3.3 Restricted repertoire and the streetcar theory . . . . 83
5.4 Fitness, strategies and players . . . . . . . . . . . . . . . . . 84
5.4.1 Fitness 1 . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.2 Fitness 2 . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.3 Fitness 3 . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.4 Fitness 4 . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.5 Fitness 5 . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.6 Further considerations . . . . . . . . . . . . . . . . . 86
5.5 Selfish genes: How can non-beneficial genes propagate? . . . 86
5.5.1 Genetic hitchhiking . . . . . . . . . . . . . . . . . . 87
5.5.2 Selfish genes . . . . . . . . . . . . . . . . . . . . . . 88
5.5.3 Memes and cultural evolution . . . . . . . . . . . . . 89
5.5.4 Selection at the level of the cell . . . . . . . . . . . . 90
5.6 The role of simple mathematical models . . . . . . . . . . . 90
5.7 Python code . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Matrix games 97
6.1 Properties of ESSs . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.1 An equivalent definition of an ESS . . . . . . . . . . 97
6.1.2 A uniform invasion barrier . . . . . . . . . . . . . . 98
6.1.3 Local superiority of an ESS . . . . . . . . . . . . . . 100
6.1.4 ESS supports and the Bishop-Cannings theorem . . 101
6.2 ESSs in a 2 2 matrix game . . . . . . . . . . . . . . . . . 103
×
6.3 Haigh’s procedure to locate all ESSs . . . . . . . . . . . . . 105
6.4 ESSs in a 3 3 matrix game . . . . . . . . . . . . . . . . . 107
×
6.4.1 Pure strategies . . . . . . . . . . . . . . . . . . . . . 107
6.4.2 A mixture of two strategies . . . . . . . . . . . . . . 108
6.4.3 Internal ESSs . . . . . . . . . . . . . . . . . . . . . . 108
6.4.4 No ESS . . . . . . . . . . . . . . . . . . . . . . . . . 109