Evolution and the Theory of Games
John Maynard Smith
PART 1 PART 2
The rest of this introductory chapter discusses some more general issues concerned with the application of game theory to evolution. Those with no taste for philosophical arguments are advised to skip this section, or to treat it as a postscript rather than an introduction; the rest of the book should make sense without it.
First, a word has to be said as to why one should use a game theory model when a classical population genetics model more precisely represents biological reality. The answer is that the two types of model are useful in different circumstances. When thinking about the evolution either of wing shape or of dispersal behaviour it is most unlikely that one would have any detailed knowledge of the genetic basis of variation in the trait. It would, however, be reasonable to suppose that there is some additive genetic variance, because artificial selection experiments have almost always revealed such variance in outbred sexual populations for any selected trait (for a rare exception, see Maynard Smith & Sondhi, 1960). The basic assumption of evolutionary game theory – that like begets like – corresponds to what we actually know about heredity in most cases. To analyse a more detailed genetic model would be out of place. For example, it is relevant to the evolution of wing form that the shape which generates a given lift for the minimum induced drag is an elliptical one. If someone were to say ‘Maybe, but how do you know that a bird with an elliptical wing is not a genetic heterozygote which cannot breed true?,’ he would rightly be regarded as unreasonable.
There are, of course, contests in which population genetic models become necessary. These are discussed in more detail in Chapter 4. Essentially, they are cases in which the centre of interest concerns the genetic variability of the population. Although game theory can sometimes point to situations in which genetic polymorphism can be maintained by frequency-dependent selection, such cases call for proper genetic analysis. Essentially, game theory models are appropriate when one wants to know what phenotypes will evolve, and when it is reasonable to assume the presence of additive genetic variance. Rather surprisingly, game theory methods have proved to be particularly effective in analysing phenotypes (e.g. sex ratio, resource allocation to male and female functions in hermaphrodites) which are themselves relevant to sexual reproduction; all that is required is that the phenotype itself be heritable. The point is also discussed in Chapter 4.
Two further criticisms which can be made of optimisation and game theory models are, first, that it is misleading to think of animals optimising and, secondly, that in any case animals are constrained developmentally and hence unable to reach an optimum. On the first point, optimisation models are certainly misleading if they lead people to think that animals consciously optimise their fitness; they are still more misleading if they lead people to suppose that natural selection leads to the evolution of characteristics which are optimal for the survival of the species. But there is no reason why the models should be so interpreted. An analogy with physical theory should make this point clear. When calculating the path of a ray of light between two points, A and B, after reflection or refraction, it is sometimes convenient to make use of the fact that the light follows that path which minimises the time taken to reach B. It is a simple consequence of the laws of physics that this should be so; no-one supposes that the ray of light setting out from A calculates the quickest route to B. Similarly, it can be a simple consequence of the laws of population genetics that, at equilibrium, certain quantities are maximised. If so, it is simplest to find the equilibrium state by performing the maximisation. Nothing is implied about intention, and nothing is asserted about whether or not the equilibrium state will favour species survival.
On the subject of developmental constraints (see, for example, Gould & Lewontin, 1979), I think there is some misunderstanding. Whenever an optimisation or game-theoretic analysis is performed, an essential feature of the analysis is a specification of the set of possible phenotypes from among which the optimum is to be found. This specification is identical to a description of developmental constraints. I could reasonably claim that by introducing game theory methods I have drawn attention to developmental constraints by insisting that they be specified. Rather than make this claim, I will instead admit that although I can see no theoretical justification for Gould & Lewontin’s criticism of the ‘adaptationist programme,’ I can see some practical force to it. This is that, in practice, too much effort is put into seeking an optimum and not enough into defining the phenotype set. In the Hawk-Dove game (p. 11), for example, considerable sophistication has been devoted to analysing the game, but the strategy set is ridiculously naive.
My reply to this complaint would be that it wrongly identifies the purpose of the Hawk-Dove game, which is not to represent any specific animal example, but to reveal the logical possibilities (for example, the likelihood of mixed strategies) inherent in all contest situations. When confronted with specific cases, much more care must be taken in establishing the strategy set. It is interesting, as an example, that in analysing competition between female digger wasps (p. 74), Brockmann, Grafen & Dawkins (1979) were at first unsuccessful because they wrongly determined the alternative strategies available to the wasps.
There is, however, a wider conflict between the developmental and the evolutionary points of view. After the publication of Darwin’s Origin of Species, but before the general acceptance of Weismann’s views, problems of evolution and development were inextricably bound up with one another. One consequence of Weismann’s concept of the separation of germ line and soma was to make it possible to understand genetics, and hence evolution, without understanding development. In the short run this was an immensely valuable contribution, because the problems of heredity proved to be soluble, whereas those of development apparently were not. The long-term consequences have been less happy, because most biologists have been led to suppose either that the problems of development are not worth bothering with, or that they can be solved by a simple extension of the molecular biology approach which is being so triumphant in genetics.
My own view is that development remains one of the most important problems of biology, and that we shall need new concepts before we can understand it. It is comforting, meanwhile, that Weismann was right. We can progress towards understanding the evolution of adaptations without understanding how the relevant structures develop. Hence, if the complaint against the ‘adaptationist programme’ is that it distracts attention from developmental biology, I have some sympathy. Development is important and little understood, and ought to be studied. If, however, the complaint is that adaptation cannot (rather than ought not to) be studied without an understanding of developmental constraints, I am much less ready to agree.
The disagreement, if there is one, is empirical rather than theoretical – it is a disagreement about what the world is like. Thus, I am sure, Gould and Lewontin would agree with me that natural selection does bring about some degree of adaptive fit between organisms and their environments, and I would agree with them that there are limits to the kinds of organisms which can develop. We may disagree, though, about the relative importance of these two statements. Suppose, for example, that only two kinds of wings could ever develop – rectangular and triangular. Natural selection would probably favour the former in vultures and the latter in falcons. But if one asked ‘Why are birds’ wings the shapes they are?,’ the answer would have to be couched primarily in terms of developmental constraints. If, on the other hand, almost any shape of wing can develop, then the actual shape, down to its finest details, may be explicable in selective terms.
Biologists differ about which of these pictures is nearer the truth. My own position is intermediate. Clearly, not all variations are equally likely for a given species. This fact was well understood by Darwin, and was familiar to me when I was an undergraduate under the term ‘Vavilov’s law of homologous variation’ (Spurway, 1949; Maynard Smith, 1958). In some cases, the possible range of phenotypic variation may be quite sharply circumscribed; for example, Raup (1966) has shown that the shapes of gastropod shells can be described by a single mathematical expression, with only three parameters free to vary. Further, the processes of development seem to be remarkably conservative in evolution, so that the evolution of legs, wings and flippers among the mammals has been achieved by varying the relative sizes and, to some extent, numbers of parts rather than by altering the basic pattern, or bauplan.
It follows from this that, when thinking about the evolution of any particular group, we must keep in mind the constraints which development is likely to place on variation. Looking at existing mammals, however, makes it clear that the constraint of maintaining a particular basic structure does not prevent the evolution of an extraordinary range of functional adaptations. It would be a mistake to take a religious attitude towards bauplans, or to regard them as revealing some universal laws of form. Our ancestors first evolved a notochord, segnented muscles and two pairs of fins as adaptations for swimming, and not because they were conforming to a law of form. As Darwin remarked in the Origin, the ‘Unity of Type’ is important, but it is subordinate to the ‘conditions of existence’ because the ‘Type’ was once an organism which evolved to meet particular conditions of existence.
An obvious weakness of the game-theoretic approach to evolution is that it places great emphasis on equilibrium states, whereas evolution is a process of continuous, or at least periodic, change. The same criticism can be levelled at the emphasis on equilibria in population genetics. It is, of course, mathematically easier to analyse equilibria than trajectories of change. There are, however, two situations in which game theory models force us to think about change as well as constancy. The first is that a game may not have an ESS, and hence the population cycles indefinitely. On the whole, symmetrical games with no ESS seem biologically rather implausible. They necessarily imply more than two pure strategies (see Appendix D), and usually have the property that A beats B, B beats C and C beats A. Asymmetric games, on the other hand, very readily give rise to indefinite cyclical behaviour (see Appendix J). Although it is hard to point to examples, perhaps because of the long time-scales involved, the prediction is so clear that it would be odd if examples are not found.
The second situation in which a game theory model obliges one to think of change rather than constancy is when, as is often the case, a game has more than one ESS. Then, in order to account for the present state of a population, one has to allow for initial conditions – that is, for the state of the ancestral population. This is particularly clear in the analysis of parental care (p. 126).
Evolution is a historical process; it is a unique sequence of events. This raises special difficulties in formulating and testing scientific theories, but I do not think the diffculties are insuperable. There are two kinds of theories which can be proposed: general theories which say something about the mechanisms underlying the whole process, and specific theories accounting for particular events. Examples of general theories are ‘all previous history is the history of class struggle,’ and ‘evolution is the result of the natural selection of variations which in their origin are non-adaptive.’ Evolutionary game theory is not of this kind. It assumes that evolutionary change is caused by natural selection within populations. Rather, game theory is an aid to formulating theories of the second kind; that is, theories to account for particular evolutionary events. More precisely, it is concerned with theories which claim to identify the selective forces responsible for the evolution of particular traits or groups of traits.
It has sometimes been argued that theories of this second, specific, kind are untestable, because it is impossible to run the historical process again with some one factor changed, to see whether the result is different. This misses the point that any causal explanation makes assumptions which can be tested. For example, in his The Revolt of the Netherlands, Geyl (1949) discusses why it was that the northern part of the Netherlands achieved independence when the south did not. The most commonly held explanation had been that the population of the north were mainly Protestant and of the south Catholic. Geyl shows that this explanation is wrong, because at the outbreak of the revolt, the proportion of Catholics did not differ between the two regions. Hypotheses about the causes of particular evolutionary events are likewise falsifiable. For example, the hypothesis that size dimorphism in the primates evolved because it reduces ecological competition between mates is almost certainly false, because dimorphism is large in polygynous and promiscuous mammals and absent in monogamous ones (Clutton-Brock, Harvey & Rudder, 1977); the hypothesis may be correct for some bird groups (Selander, 1972).
I think it would be a mistake, however, to stick too rigidly to the criterion of falsifiability when judging theories in population biology. For example, Volterra’s equations for the dynamics of a predator and prey species are hardly falsifiable. In a sense they are manifestly false, since they make no allowance for age structure, for spatial distribution, or for many other necessary features of real situations. Their merit is to show that even the simplest possible model of such an interaction leads to sustained oscillation – a conclusion it would have been hard to reach by purely verbal reasoning. If, however, one were to apply this idea in a particular case, and propose, for example, that the oscillations in numbers of Canadian fur-bearing mammals is driven by the interactions between hare and lynx, that would be an empirically falsifiable hypothesis.
Thus there is a contrast between simple models, which are not testable but which may be of heuristic value, and applications of those models to the real world, when testability is an essential requirement.
John Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982, pp. 3-9.
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