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Game Theory: Decisions, Interaction and Evolution (Springer Undergraduate Mathematics Series)

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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.

This introduction to game theory is intended as a first course for undergraduate students of mathematics, but it also contains material which will be of interest to advanced students or researchers in biology and economics. The outstanding feature of the book is that it provides a unified account of three types of decision problem: Situations involving a single decision-maker: in which a sequence of choices is to be made in "a game against nature". This introduces the basic ideas of optimality and decision processes. Classical game theory: in which the interactions of two or more decision-makers are considered. This leads to the concept of the Nash equilibrium. Evolutionary game theory: in which the changing structure of a population of interacting decision makers is considered. This leads to the ideas of evolutionarily stable strategies and replicator dynamics. An understanding of basic calculus and probability is assumed but no prior knowledge of game theory is required.

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