Evolution and the Theory of Games
John Maynard Smith
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The last decade has seen a steady increase in the application of concepts from the theory of games to the study of evolution. Fields as diverse as sex ratio theory, animal distribution, contest behaviour and reciprocal altruism have contributed to what is now emerging as a universal way of thinking about phenotypic evolution. This book attempts to present these ideas in a coherent form. It is addressed primarily to biologists. I have therefore been more concerned to explain and to illustrate how the theory can be applied to biological problems than to present formal mathematical proofs – a task for which I am, in any case, ill equipped. Some idea of how the mathematical side of the subject has developed is given in the appendixes.
I hope the book will also be of some interest to game theorists. Paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed. There are two reasons for this. First, the theory requires that the values of different outcomes (for example, financial rewards, the risks of death and the pleasures of a clear conscience) be measured on a single scale. In human applications, this measure is provided by ‘utility’ – a somewhat artificial and uncomfortable concept: in biology, Darwinian fitness provides a natural and genuinely one-dimensional scale. Secondly, and more importantly, in seeking the solution of a game, the concept of human rationality is replaced by that of evolutionary stability. The advantage here is that there are good theoretical reasons to expect populations to evolve to stable states, whereas there are grounds for doubting whether human beings always behave rationally.
This book is about a method of modelling evolution, rather than about any specific problem to which the method can be applied. In this chapter, I discuss the range of application of the method and some of the limitations, and, more generally, the role of models in science.
Evolutionary game theory is a way of thinking about evolution at the phenotypic level when the fitnesses of particular phenotypes depend on their frequencies in the population. Compare, for example, the evolution of wing form in soaring birds and of dispersal behaviour in the same birds. To understand wing form it would be necessary to know about the atmospheric conditions in which the birds live and about the way in which lift and drag forces vary with wing shape. One would also have to take into account the constraints imposed by the fact that birds’ wings are made of feathers – the constraints would be different for a bat or a pterosaur. It would not be necessary, however, to allow for the behaviour of other members of the population. In contrast, the evolution of dispersal depends critically on how other conspecifics are behaving, because dispersal is concerned with finding suitable mates, avoiding competition for resources, joint protection against predators, and so on.
In the case of wing form, then, we want to understand why selection has favoured particular phenotypes. The appropriate mathematical tool is optimisation theory. We are faced with the problem of deciding what particular features (e.g. a high lift:drag ratio, a small turning circle) contribute to fitness, but not with the special difficulties which arise when success depends on what others are doing. It is in the latter context that game theory becomes relevant.
The theory of games was first formalised by Von Neumann & Morgenstern (1953) in reference to human economic behaviour. Since that time, the theory has undergone extensive development; Luce & Raiffa (1957) give an excellent introduction. Sensibly enough, a central assumption of classical game theory is that the players will behave rationally, and according to some criterion of self-interest. Such an assumption would clearly be out of place in an evolutionary context. Instead, the criterion of rationality is replaced by that of population dynamics and stability, and the criterion of self-interest by Darwinian fitness. The central assumptions of evolutionary game theory are set out in Chapter 2. They lead to a new type of ‘solution’ to a game, the ‘evolutionarily stable strategy’ or ESS.
Game theory concepts were first explicitly applied in evolutionary biology by Lewontin (1961). His approach, however, was to picture a species as playing a game against nature, and to seek strategies which minimised the probability of extinction. A similar line has been taken by Slobodkin & Rapoport (1974). In contrast, here we picture members of a population as playing games against each other, and consider the population dynamics and equilibria which can arise. This method of thinking was foreshadowed by Fisher (1930), before the birth of game theory, in his ideas about the evolution of the sex ratio and about sexual selection (see p. 43). The first explicit use of game theory terminology in this context was by Hamilton (1967), who sought for an ‘unbeatable strategy’ for the sex ratio when there is local competition for mates. Hamilton’s ‘unbeatable strategy’ is essentially the same as an ESS as defined by Maynard Smith & Price (1973).
Most of the applications of evolutionary game theory in this book are directed towards animal contests. The other main area so far has been the problem of sexual allocation – i.e. the sex ratio, parental investment, resource allocation in hermaphrodites etc. I have said rather little on those topics because they are treated at length in a book in preparation by Dr Eric Charnov (1982). Other applications include interspecific competition for resources (Lawlor & Maynard Smith, 1976), animal dispersal (Hamilton & May, 1977), and plant growth and reproduction (Mirmirani & Oster, 1978).
The plan of the book is as follows. In Chapter 2 I describe the basic method whereby game theory can be applied to evolutionary problems. In fact, two different models are considered. The first is that originally proposed by Maynard Smith & Price (I973) to analyse pairwise contests between animals. Although often appropriate for the analysis of fighting behaviour, this model is too restrictive to apply to all cases where fitnesses are frequency-dependent. A second, extended, model is therefore described which can be used when an individual interacts, not with a single opponent at a time, but with some group of other individuals, or with some average property of the population as a whole.
Chapters 3 to 5 deal with other mainly theoretical issues. Chapter 3 analyses the ‘war of attrition,’ whose characteristic feature is that animals can choose from a continuously distributed set of strategies, rather than from a set of discrete alternatives. In chapter 4, I consider the relationship between game theory models and those of population genetics, and in Chapter 5, the relation between evolution and learning.
Chapters 6 to 10 are concerned with applying the theoretical ideas to field data. My aim here has been to indicate as clearly as possible the different kinds of selective explanation of behaviour that are possible and the kinds of information which are needed if we are to distinguish between them. For most of the examples discussed there are important questions still to be answered. The game-theoretic approach, however, does provide a framework within which a wide range of phenomena, from egg-trading to anisogamy, can be discussed. Perhaps more important, it draws attention to the need for particular kinds of data. In return, the field data raise theoretical problems which have yet to be solved.
The last three chapters are more speculative. Chapter 11 is concerned with how game theory might be applied to the evolution of life history strategies. The particular model put forward, suggested by the evolution of polygynous mammals, is of a rather special and limited kind, but may encourage others to attempt a more general treatment. In Chapter 12, I discuss what may be the most difficult theoretical issue in evolutionary game theory – the transfer of information during contests. Territorial behaviour is discussed in this chapter, because of the theoretical possibility that information transfer will be favoured by selection when the resource being contested is divisible. Finally, Chapter 13 discusses the evolution of cooperation in a game-theoretic context.
John Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982, pp. vii-3.
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