(from Cambridge)
Evolutionarily stable strategies and nonequilibrium
Games Theory was developed by John von Neumann and Oskar Morgenstern (1944), although the French mathematician Antoine Augustin Cournot studied some aspects (1838, later further developed by John Forbes Nash) in the nineteenth century. Its most important contribution to evolutionary biology is the concept of evolutionarily stable strategy (ESS). It is central to modern evolutionary ecology, and Richard Dawkins (1976) suggests that it may be “one of the most important advances in evolutionary theory since Darwin”. It was introduced into ecology by John Maynard Smith and George R. Price (1973). It can be derived from the concept of Nash Equilibrium (John Nash 1950), according to which none of a number of players in a game can gain by changing her/his strategy unilaterally. John Maynard Smith (1982) gave a detailed account of applications of Game Theory to evolutionary theory including ESS. However, parts of his book rely heavily on mathematics. Richard Dawkins’ (1976) The Selfish Gene contains a discussion of ESS and many examples, clearly explained without any mathematics.
According to Maynard Smith (1982), “An ‘ESS’ or ‘evolutionarily stable strategy’ is a strategy such that, if all the members of a population adopt it, no mutant strategy can invade”. A strategy is a genetically determined behavioural “policy” (“course of action”). There may be more than one ESS for a population, and the type(s) of ESS depend on many characteristics of the members of a population, such as the genetic relatedness of the members in the population, population size, whether members of a population can learn from previous experience, whether populations reproduce asexually or sexually, whether contests are symmetric or asymmetric, etc. A symmetric game is one in which the adversaries start in similar situations and can choose the same strategies with the same potential payoffs (the changes in reproductive success due to the strategy). The game using dove-hawk strategies discussed below is an example of a symmetric game.
It is important to realize that an ESS is not necessarily a strategy that is “best” for all the members of the population, i.e. guarantees the greatest fitness (reproductive success) for them in the long term. The reason is that genes (any proportion of genetic material potentially lasting long enough for natural selection to act on it as a unit) have no “foresight”. They are selected on the basis of the present conditions in their environment.
The following very simple theoretical example (discussed by Maynard Smith and Dawkins) demonstrates the principle of an ESS. We assume that only two strategies are possible, i.e. hawk (fight as hard as possible, retreat only when badly hurt) and dove (threaten only in a mild way, never hurt the adversary). We further assume that individuals have not learned from previous experience, i.e. they cannot tell in advance who is dove and who is hawk. In an initial population entirely consisting of doves, all doves do well, however introduction of a single mutant hawk puts all the others at a disadvantage, it always beats them and the hawk gene spreads through the population. But a hawk now, with increasing frequency, encounters other hawks and gets hurt when fighting. Consequently, even a single dove is at an advantage, because it will never get hurt, and the dove gene will now become more common. A stable ratio hawks/doves is reached when the average payoff for hawks equals that for doves. There may be oscillations around the stable point, but they are not necessarily large.
Actual examples for animals are much more complex. Usually there are not two, but – in addition – other possible strategies. Important strategies are the so-called conditional strategies (strategy depends on the behaviour of the adversary). Examples are that of a retaliator (behaves like a hawk when he meets another hawk, like a dove when he meets another dove), bully (treats everybody in a hawk-like spirit, but retreats when treated as a hawk himself), prober-retaliator (usually behaves like a retaliator, but occasionally escalates the fight). For a mixture of these five possible strategies, reasoning like that used for the hawk/dove strategy leads to the conclusion that a mixture of retaliators and prober-retaliators tends to become dominant, i.e., is an evolutionarily stable strategy. And Dawkins suggests that this is indeed “often” (approximately at least) the case in nature. – It is important to re-emphasize that an ESS is not necessarily the “best” strategy for all. For example, a pure dove strategy might be better, because it involves less “costs” invested in fighting, but genes which are the units on which selection acts, cannot conspire to set up the best of all possible worlds, they have no foresight, and they are therefore exposed to the possibility of invasion by mutants which behave badly, that is, not as doves.
Other possible strategies are found, for example, in species which never fight seriously, they posture only. The individual with the greatest patience wins (“war of attrition”), and the ESS is that each of the adversaries goes on posturing for an unpredictable time, the time spent on posturing depending on the value of the resource.
Most contests in nature are asymmetric, the adversaries differ for example in size, strength, fighting spirit, sex, or owner vs. not-owner of a territory. In the last case, the ESS is such that the owner usually wins (a well researched example is that of a freshwater fish, the stickleback).
The concept of evolutionarily stable strategy is analogous to those of developmentally stable strategy (DSS) and culturally stable strategy (CSS) (Dawkins). In a CSS, transmission of information is not via genes, but by cultural inheritance from one generation to the next. In a DSS, transmission of information is by learning within a generation. For illustrating a DSS, Maynard-Smith uses the example of a pair of pigs in an experimental “Skinner box”, studied by Baldwin and Meese. In the box, pressing down a lever at one end released food from a dispenser at the other. The only stable behaviour pattern that established itself was as follows: the dominant pig pressed the bar and then rushed to the other end to feed, while the subordinate pig fed at the dispenser until the dominant pig arrived, and then moved “politely” away. The strategy was stable, even when the amount of food was such that the subordinate pig got more of it. The only other strategy, i.e., the subordinate pig pressing the bar, was not stable because the dominant pig would prevent it from eating.
We have to address the question of how common evolutionarily stable strategies really are in nature. As pointed out by Dawkins, we know little about the actual costs and benefits involved in calculating the payoffs for an ESS. Therefore, it is likely that many empirical examples that suggest an ESS are at best just that, a suggestion. Which factors determine that an ESS can be established? We consider five important aspects, (1) stability of the environment, (2) nonlinear dynamics, (3) interspecific ESS‘s, (4) speed of evolution, and (5) genetic constraints.
(1) An evolutionarily stable state of a population is defined by Maynard Smith as a genetic composition which is “restored by selection after a disturbance, provided the disturbance it not too large”. This shows that establishment of evolutionarily stable states and strategies critically depend, in addition to the degree of interaction between contestants and the time necessary for establishment of an ESS, on the stability of environmental conditions. In a stable world in which equilibrium is the rule, ESS’s can be expected to be much more common than in an unstable world, in which nonequilibrium prevails. In other words, it is to be expected that frequent and drastic abiotic and biotic changes in the environment which affect the fitness (reproductive success) of potential contestants in evolutionary “games”, will make it more difficult to establish evolutionarily stable strategies, because the establishment of an ESS cannot keep up with the changes (see also Rohde 2005, pp.10, 13 and 34). If the establishment of an ESS takes a long time, even a single strong environmental disruption with long lasting effects may make it impossible for an ESS to develop. An established ESS may be affected by environmental instability for example by reducing population size so much that encounters with possible contestants are radically reduced, facilitating the invasion by mutants which are less fit, on a more or less random basis. Alternatively, a strategy stable under certain conditions may become less stable, permitting establishment of a different ESS. For example, it may well be that in the “war of attrition” mentioned above, the strategy of posturing in an unpredictable way and for a duration depending on the value of the resource, is replaced by one which demands greater aggressiveness, because resources are severely depleted.
(2) Even long-term constant environmental conditions do not necessarily guarantee establishments of ESS’s. For instance, much theoretical work has gone into investigating the evolution of sexually selected traits, of which male ornaments and elaborate behavioural displays are the best known examples (discussion in one of the most recent papers on the topic, Sander van Doorn and Weissing 2006, further references therein). According to one hypothesis, male ornaments indicate to the female the good quality of his genes (review in Maynard Smith 1991). However, whereas females’ interests are served best by reliable ornaments, males may cheat, i.e., they may attract females even in the absence of good genes. This discrepancy may lead to complicated evolutionary dynamics, many aspects of which are not fully understood. Indeed, predictions depend to a large degree on the model used and the assumptions made. In the model used by Sanders van Doorn and Weissing, sexual conflict prevents establishment of an ESS. Equilibrium assumptions are therefore not necessarily a reliable basis for predicting the outcome of sexual selection. – More generally, as shown by Abrams et al. (1993), evolution may even cause divergence from an ESS, that is, result in characters that minimize fitness and prevent evolution towards characters that maximize fitness. Also, the extensive computer simulations of Wolfram (2002) have shown that in all the systems (programs) used, relatively short “instructions” led to complex patterns. The same may apply to genetic programs, i.e., the possibility must be considered that single or few mutations will frequently lead to complex characters affecting fitness, making the establishment of long-lasting stable equilibria more difficult (Rohde 2005, pp.186-188).
(3) Most examples of animal contests given in the literature refer to intraspecific contests, leading for instance to the establishment of territories and intraspecific communication. How common are ESS’s in contests between populations of different species?
As pointed out by Dawkins, individuals belonging to the same species compete more strongly for resources than individuals belonging to different species. He gives as an example European birds: robins defend territories against other robins, but not against great tits. Rohde (2005) has shown that interspecific competition for resources occurs, but in many cases has little evolutionary significance. For example, many parasite species, sometimes with great infection intensities, may be found on the gills of one species and even one individual of freshwater or marine fish. Different species live intermingled in the same microhabitat on the gills and use the same food, they segregate only when they meet individuals of a species belonging to the same genus (congenerics), not because they compete for resources, but because selection has “taught” them to avoid congenerics in order to prevent hybridisation: interspecific hybridisation leads to reproductive failure and reduces fitness of the population. More generally, the outcome of “contests” between species is very often unpredictable, as shown by theoretical and experimental studies which demonstrated occurrence of chaos and unpredictability for example in multispecies contests of plankton species (e.g., Huisman J. and Weissing F.J. 2001; Rohde, 2005, pp. 65-67). Furthermore, the prevalence of vacant niches in many ecosystems (Rohde 2005, pp. 39-48) reduces the significance of interspecific competition and with it the likelihood that an ESS can evolve. On the other hand, competition may lead to nonequilibrium dynamics, also preventing establishment of ESS’s (see appendix “Plankton, a paradox resolved”). Our conclusion, then, is that ESS’s that “regulate” interactions between populations of different species are not likely. This means that it is also very unlikely that community “structure” is determined by interspecific competition which has led to an ESS. “Structure” is unstable in an evolutionary sense and largely determined by evolutionary “accidents”. Various statistical tests applied to parasite communities have demonstrated that this is indeed the case. The same is likely for communities of many other animals (examples in Rohde 2005). – These conclusions apply to competing species, but not to predator-prey or host-parasite interactions, which may be very intense and, if active over many generations, may have significant effects on community structure (examples in Dawkins).
(4) Rohde (1992) has proposed that the greater species richness of animals and plants in tropical habitats can be explained by an accelerated evolution there, due to direct temperature effects on generation times, mutation rates and speed of selection. Much subsequent work has provided support for this hypothesis (Rohde 2005, pp.159-165; and appendix on “New evidence for the hypothesis of effective evolutionary time”). Applied to the establishment of ESS’s, it means that such strategies will develop faster in the tropics than elsewhere, provided conditions are otherwise the same, although this does not necessarily imply that the relative number of ESS’s is greater (see also Rohde 2005, pp.186-188).
(5) Weissing (1996, further references therein) and others have pointed out that genetic constraints may prevent maximization of fitness and hence the establishment of an ESS. However, although such constraints may be important in the short-term, they may often disappear in the long-term as the result of continuing influx of mutations. Weissing concludes that (a) long-term stability (Nash strategies) can be expected only in populations at local fitness optima, (b) that “in monomorphic populations, evolutionary stability is necessary and sufficient to ensure long-term dynamic stability”, and (c) that long-term stability can also be expected in non-linear frequency dependent selection, for multiple loci, and for quite general mating systems. Nevertheless, game theoretical characterization of stability in polymorphic populations is probably not possible; many systems do not attain long-term stable equilibria; and even if such equilibria exist, it is not understood whether and how a series of gene substitution events can explain them.
In summary, evolutionarily stable strategies are most likely in populations of animals belonging to the same species, when conditions remain relatively unchanged, and in the tropics. They are likely to structure predator-prey and host-parasite systems, but less likely to structure multi-species communities by interspecific competition, when conditions tend to change significantly over relatively short time spans, and at high latitudes. Even under long-term environmental stability, nonlinear evolutionary dynamics may prevent establishment of ESS’s and may even lead away from them.
Acknowledgement: I wish to thank Dietrich Stauffer for critical comments on an earlier version of this appendix.
References
Abrams, P.A., Matsuda, H. and Harada, Y. (1993). Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evolutionary Ecology 7, 465-487.
Dawkins, R. (1976). The Selfish Gene. Oxford, Oxford University Press.
Huisman J. and Weissing F.J. (2001). Fundamental unpredictability in multispecies competition. American Naturalist 157, 488-494.
Maynard Smith, J. and Price, G.R. (1973). The Logic of Animal Conflict. Nature 246, 15-18.
Maynard Smith, J. (1982). Evolution and the Theory of Games. Cambridge, Cambridge University Press.
Maynard Smith, J. (1991). Theories of sexual selection. Trends in Ecology and Evolution 6, 146-151.
Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36, 48-49.
Rohde, K. (1992). Latitudinal gradients in species diversity: the search for the primary cause. Oikos 65, 514-527.
Rohde, K. (2005). Nonequilibrium Ecology. Cambridge University Press, Cambridge.
Sander van Doorn, G. and Weissing, F. J. (2006). Sexual conflict and the evolution of female preferences for indicators of male quality. American Naturalist 168, 742-757.
von Neumann, J. and Morgenstern, O. (1944). Theory of Games and Economic Behaviour. Princeton University Press, Princeton, N.J.
Weissing, F.J. (1996). Genetic versus phenotypic models of selection: Can genetics be neglected in a long-term perspective? Journal of Mathematical Biology 34, 533-555.
Wolfram, S. (2002). A New Kind of Science. Champaign, Il., Wolfram Media Inc.
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