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This book is an introduction to game theory from a mathematical perspective.
It is intended to be a first course for undergraduate students of mathematics,
but I also hope that it will contain something of interest to advanced students
or researchers in biology and economics who often encounter the basics of game
theory informally via relevant applications. In view of the intended audience,
the examples used in this book are generally abstract problems so that the
reader is not forced to learn a great deal of a subject – either biology or economics
– that may be unfamiliar. Where a context is given, these are usually
“classical” problems of the subject area and are, I hope, easy enough to follow.
The prerequisites are generally modest. Apart from a familiarity with (or
a willingness to learn) the concepts of a proof and some mathematical notation,
the main requirement is an elementary understanding of probability. A
familiarity with basic calculus would be useful for Chapter 6 and some parts of
Chapters 1 and 8. The basic ideas of simple ordinary differential equations are
required in Chapter 9 and, towards the end of that chapter, some familiarity
with matrices would be an advantage – although the relevant ideas are briefly
described in an appendix.
I have tried to provide a unified account of single-person decision problems
(“games against nature”) as well as both classical and evolutionary game theory,
whereas most textbooks cover only one of these. There are two immediate
consequences of this broad approach. First, many interesting topics are left out.
However, I hope that this book will provide a good foundation for further study
and that the books suggested for further reading at the end of this volume will
go some way to filling the gaps. Second, the notation and terminology used
may be different in places from that which is commonly used in each of the
three separate areas. In this book, I have tried to use similar (combinations of)
Part I. Decisions
1. Simple Decision Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Optimisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Making Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Modelling Rational Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Modelling Natural Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Optimal Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2. Simple Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Strategic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Randomising Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Optimal Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3. Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 State-dependent Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Stochastic Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Optimal Strategies for Finite Processes . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Infinite-horizon Markov Decision Processes . . . . . . . . . . . . . . . . . . 48
3.6 Optimal Strategies for Infinite Processes . . . . . . . . . . . . . . . . . . . . . 50
3.7 Policy Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Part II. Interaction
6. Games with Continuous Strategy Sets . . . . . . . . . . . . . . . . . . . . . . 107
6.1 Infinite Strategy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 The Cournot Duopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 The Stackelberg Duopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 War of Attrition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7. Infinite Dynamic Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 The Iterated Prisoners’ Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3 Subgame Perfection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Folk Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.5 Stochastic Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Part III. Evolution
8. Population Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.1 Evolutionary Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.2 Evolutionarily Stable Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Games Against the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4 Pairwise Contest Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.5 ESSs and Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.6 Asymmetric Pairwise Contests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.7 Existence of ESSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
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