One of the many curious discoveries made, earlier this century, by ethologists such as Konrad Lorenz and Niko Tinbergen was that fighting in animals is restrained and, as they called it, ‘ritualised’. Animal contests, over such valuable resources as food, territory or mates, almost resemble tournaments, which pass through a regular series of harmless stages, before one animal emerges as the winner, and the other retreats unharmed as the loser. Take the cichlid fish Cichlasoma biocellatum, whose contests are described by Lorenz in his book On Aggression. A fight between two males passes through three main stages, at any of which one of the contestants may back out. They start with broadside displays, move on to tail beating, and then to harmless mouth fighting, in which the pair grip and pull each other by the mouth. The rules of the contest are, according to Lorenz, strictly obeyed. Each fish only moves on to the next stage when the other is ready. More strikingly still, if one of the fish finds itself in possession of a temporary but irregular advantage, it will not unfairly press it home. In Lorenz’s own words, ‘one of them may be inclined to go on to mouth-pulling a few seconds before the other one. He now turns from his broadside position and thrusts with open jaws at his rival who, however, continues his broadsides threatening, so that his unprotected flank is presented to the teeth of his enemy. But the aggressor never takes advantage of this; he always stops his thrust before his teeth have touched the skin of his adversary.’ Behaviour does not come much more gentlemanly than that.
Ritualised animal contests were not known before the 20th century. If Darwin had known of it, we can be quite sure that, with his interest in animal analogies of human morality, he would have greedily transcribed it into The Descent of Man. But the several sections of that work which are concerned with the ‘law of battle’, while containing an impressive list of carnage, assembled from a variety of anecdotal sources, are unrelieved by even the smallest hint of restraint on the animal battlefield.
In fact, it is still a matter of controversy whether, in particular cases, fighting is restrained or unrestrained. Lorenz, having discovered restrained fighting, has probably exaggerated its importance. But it is only its frequency that is controversial, not its existence: restrained fighting, for some species at least, is a fact of nature. This being so, it remains to ask why.
For Lorenz, there was no theoretical puzzle. It was just another illustration of the general principle that animals behave for ‘the good of the species’. If the members of a species do not hurt each other, then the species as a whole is better-off. But even as Lorenz was writing On Aggression, at the beginning of the Sixties, his ‘good of the species’ thinking was going out of date. Good of the species was to be replaced by good of the individual. That revolution has now penetrated into almost all departments of biology, and Lorenz himself, in his home in Altenberg, is one of the few to remain innocent of the change that has taken place.
The immediate consequence of the principle of individual selfishness was that animal contests ceased to illustrate a principle and became, instead, a paradox. Just what are those fish up to, as they refuse to take advantage of what good fortune has given them? If there is an advantage in winning, then animals should fight to win, regardless of rules and the health of their opponents.
Apparently, however, they do not. Which brings us, at last, to John Maynard Smith. It was in an attempt, ten years ago, to show how the theory of individual selfishness could explain the restrained nature of animal contests that he introduced game theory to evolutionary biology. The explanation so perfectly illustrates his method, being a simple model which immediately illuminates a wide range of biological phenomena, that it is worth considering in detail. In a short verbal summary of a mathematical model, there is bound to be some simplification, and some assumptions will be left unspecified; but we can examine its main features. The model, like any in the theory of games, supposes that animals can adopt one of a list of possible ‘strategies’: a ‘strategy’, as Maynard Smith defines it, is ‘a behavioural phenotype, i.e. it is a specification of what an animal will do in any situation in which it may find itself’. For the simplest model of fighting, he considers two such strategies, called ‘hawk’ and ‘dove’. A hawk fights dangerously; a dove fights only in ritual form, and retreats from a hawk. The question, then, is: why are all animals not hawkish?
In a population of doves, a hawk would win all its fights; it would therefore be favoured by natural selection, leave more offspring, and the hawk strategy would increase in frequency in the population. But consider what will happen to the hawks when there are many of them in the population. Instead of fighting all the time against doves, they now have to fight against each other as well. When a hawk meets a hawk, the real biffing begins, and injuries result: the advantage to being a hawk will have decreased. If the cost of injury is greater than the benefit of winning a fight, the doves, who run away before being injured, will start to do better than hawks.
The result of the model, therefore, is that the population settles down to an intermediate equilibrium, with some mixture of hawks and doves. If hawks are in high frequency, doves will increase: if doves are in high frequency, hawks will increase. Some mixture, whose exact proportions depend on the exact costs and benefits of injury and winning, will be stable in evolution. That is the mixture which Maynard Smith calls the ‘evolutionarily stable strategy’ (or ESS) for the model. The exact ESS is simple to calculate by the rule that the different strategies must, at equilibrium, be doing equally well: if one was doing better than the other, it would increase in frequency, so the population would not be stable.
The main point of the hawk-dove game is that it shows, in principle, how the natural selection of individually selfish behaviour can lead to a population of animals that show, as real animals do, some restraint when fighting. But there are other points too. One is that the evolutionarily stable strategy is not necessarily the strategy that is best for the welfare of the species. The species would do best if all individuals were doves, because doves never injure each other. The evolutionarily stable strategy, however, permits some unrestrained hawkery. So at the equilibrium point each individual does, on average, worse than if they were all doves. But natural selection cannot produce a population of doves: the hawk strategy could invade it.
The theory of games, in evolutionary biology, is not merely an explanation of restrained fighting. It is not even a particular theory at all: it is a method. Like all methods, it has its own particular set of appropriate problems. The theory of games is appropriate for any case that possesses a certain abstract feature, which can be expressed informally by saying that the advantage to a strategy depends on what the other animals in the population are doing. More formally, it is what biologists call ‘frequency-dependent fitness’. The hawk-dove game, for example, has it: the advantage to (or ‘fitness’ of) being a hawk depends on the frequency of hawks in the population. More particularly, the model possesses negatively frequency-dependent fitnesses: as hawks become more frequent, the advantage to being a hawk decreases. Now, whenever the circumstances in nature are such that the advantage of a strategy decreases as its frequency increases, the theory of games is the appropriate method of analysis. Circumstances of this kind are common, and they are especially common in social behaviour.
Maynard Smith discusses many different models in Evolution and the Theory of Games. If we consider them as a whole, the method appears, in Maynard Smith’s hands, to have two great merits. One is that, in setting up the model, only the essence of the problem, and nothing else, is included; the other is that the models are relatively easy to solve. His purpose, he says, is ‘to identify the selective forces responsible for the evolution of particular traits or groups of traits’. And how successful he is! Each model, whether of the evolution of sexual behaviour, of territorial behaviour, of honesty and bluff in communication, of co-operative behaviour, or of fighting, contains a delightfully simple set of strategies, which illuminate the biological costs and benefits without being cluttered up by the kind of complications that biologists are so good at confusing themselves with. Once the strategies have been identified, the evolutionarily stable strategy, or set of strategies, can usually be calculated. Not only can the models be solved, which is a distinct advantage over many biological models, but the solutions can be compared with nature. Most models in biology are too far removed from real life for this to be possible. But the simple prediction that the different strategies of animals must be doing equally well is one that can be tested. The obvious relevance, ease of manipulation and testability of the models combine to make Maynard Smith’s version of game theory one of the most important new ideas to have appeared in evolutionary biology during the last decade.
Maynard Smith is not the first to use the theory of games in evolutionary biology. In its origins, indeed, game theory is not a theory of evolution at all, but a theory of human economic behaviour. But Maynard Smith has been much more successful than his predecessors, for the reason that his system has a sounder theoretical base. The analogies of natural selection and the ESS in the original theory of games were rational choice and something called the ‘minimax’ strategy. But while animal behaviour is known to be produced by natural selection, and (with certain assumptions) natural selection is known to produce ESSs, it is far from clear that human economic behaviour is produced by rational choice, or that rational choice produces minimax strategists.
The language of game theory, together with its origins in a theory of human behaviour, invite, and often receive, a misunderstanding. When biologists talk about the ‘strategies’ or ‘decisions’ of animals they do not mean that animals consciously calculate what to do. But, as Maynard Smith says, ‘I have deliberately used human analogies to suggest hypotheses. Of course, students of animal behaviour do this all the time, and I am no exception. The proper procedure, however, is to reformulate the hypothesis, once found, in terms of a population model; if this cannot be done, the hypothesis is inadmissible.’ There is absolutely nothing wrong with this. The logical structure of the theory of natural selection is such that animals really do evolve much as if they were making rational decisions. Biologists translate easily between rational analogies and natural selection. But some outside commentators do not: and after failing to recognise the sound logic of the biologist’s method, and the peculiar language that results, they often end up by sending down some such condescending advice as that ‘the biologists should be careful with their analogies.’ Evolutionary biologists, however, are used to receiving condescending advice from outsiders, especially philosophers and journalists, and invariably ignore it.
The first decade of research on game theory has been an industrious time. It has been exciting, as Maynard Smith remarks, ‘gradually to appreciate the way in which ... various lines of thought have converged, and to recognise the range of application of game theory in evolutionary biology’. Maynard Smith himself, from his department in Sussex, has done or directed much of the work, although increasingly helped by colonies elsewhere in this country, and in Germany and Canada.
There is a great gulf in most sciences between equation-scribblers and fact-collectors, which attracts, into delicious tourist resorts, a regular series of ponderous and utterly inconsequential bridge-building conferences. Maynard Smith is one of the few solid bridges across that gulf; and he is just the kind of bridge that is needed, since he reliably directs field-workers to interesting questions, and keeps the mathematicians’ feet on the ground. These, too, are the virtues of Evolution and the Theory of Games. Much of the material that it covers has been published before, but mainly in the forbidding pages of the Journal of Theoretical Biology. The work is now made available to a wider audience, although mainly, I imagine, of biologists.
It is at this point that I would enter one reservation. While Maynard Smith’s facts are controlled, his equations are not. The formidable mathematical apparatus is only imperfectly confined to the appendices, from which equations keep on breaking out to menace the lightly-armed reader. I would only warn the professor that the advantage of equations, like the strategy of hawks, is frequency-dependent. Richard Dawkins may have correctly said, in his review of the book, that every Serengeti landrover should have one: but when the field biologist puts down his binoculars and takes up his copy of Maynard Smith he may not like what he finds. After the first dozen equations, he will declare that enough is enough: then he may sew up the offensive material in the skin of a wild beast, as Nero did with the Christians, and throw it to the lions.