ESS equations sometimes do not specify an ESS

The equations used to find an evolutionarily stable strategy in the basic game theory model (Maynard Smith, 1974, 1982; Maynard Smith & Price, 1973), and in sexual conflict models (Maynard Smith, 1977; Parker, 1979) do not, in fact, specify an ESS when the expected number of contests entered is not the same for each strategy. This means that the conclusions of many game theory models may be incorrect. This is particularly likely to be true when the mean durations of contests for different strategies are not the same, or when the probability that an individual enters a contest is not the same for all strategies. New ESS equations are developed which incorporate the expected number of contests entered.     

Does frequency-dependent selection optimize fitness?

In evolutionary biology, the axiom that natural selection tends ideally to maximize inclusive fitness of the individual or some other suitable quantity is often advanced (Cody, 1974; Maynard Smith, 1978; Krebs & McCleery, 1984; Houston et al., 1988). Moreover, the evolutionists generally distinguish two situations (Dawkins, 1980; Maynard Smith, 1982): one in which fitness is independent of the frequency of the phenotypes present in the population (frequency-independent selection), and one in which it does depend on this frequency (frequency-dependent selection). This led some authors such as Parker (1984), and more recently Parker & Maynard Smith (1990), to consider "a 2-speed optimization": frequency-independent selection should lead to a "simple optimum" at the end of the selective process, since all the individuals should have the same strategy and the mean fitness of the population should be maximized; frequency-dependent selection, formulated in terms of the theory of games, should lead to a "competitive optimum" even though the "evolutionary stable strategy" (or "ESS"; Maynard Smith & Price, 1973) characterizing the equilibrium "is not the strategy that maximizes fitness in a population sense" (Parker & Maynard Smith, 1990: 30). Our aim in this short communication is to criticize the concept of "competitive optimum" by Parker & Maynard Smith, as well as the general ability of natural selection to "maximize fitness", even in "phenotypic models" (Lloyd, 1977). These models, devoid of genetic constraints since each strategist is assumed to reproduce its own kind, are especially suitable for examining the ideal effect of natural selection.     

Evolutionarily stable strategies and viability selection in mendelian populations

For various genetical structures, including haploid and diploid, one-locus n-alleles, and n-locus additive viability random mating models, natural selection resulting from intrapopulation conflicts between random individuals leads to exactly those genetical equilibria which determine a mixture of strategies evolutionarily stable according to the game theory definition of Maynard Smith and Price (1973).     

John Maynard Smith and the importance of consistency in evolutionary game theory

John Maynard Smith was the founder of evolutionary game theory. He has also been the major influence on the direction of this field, which now pervades behavioural ecology and evolutionary biology. In its original formulation the theory had three components: a set of strategies, a payoff structure, and a concept of evolutionary stability. These three key components are still the basis of the theory, but what is assumed about each component is often different to the original assumptions. We review modern approaches to these components. We emphasis that if a game is considered in isolation, and arbitrary payoffs are assumed, then the payoffs may not be consistent with other components of the system which are not modelled. Modelling the whole system, including not only the focal game, but also the future behaviour of the players and the behaviour of other population members, allows a consistent model to be constructed. We illustrate this in the case of two models of parental care, showing how linking a focal game to other aspects of the system alters what is predicted.     

Evolutionary game dynamics in diploid populations

Selection that influences behaviour can be studied using game theory if individual behavioural success depends on the frequencies of various behavioural types in the population. The evolutionarily stable strategy of J. Maynard Smith and G. R. Price (1973. Nature (London) 246, 15–18) is an equilibrium concept like the solution of a game. The dynamic model of Taylor and Jonker, studied in detail by Zeeman, goes beyond game theory using fitness to cause evolution, perhaps towards an equilibrium. A diploid version of their haploid model is considered and it is found that diploid evolution can be quite different. For example “catastrophic” bifurcations can occur between stable internal polymorphisms when the game matrix entries are changed slowly. A slight drop in food supply may cause extinction. Totally unfit altruistic genotypes can be maintained if they help the rest of the population. The relation of haploid game models to constant selection in diploids is also discussed.     

The Maynard Smith model of sympatric speciation

The paper entitled "Sympatric speciation," which was published by John Maynard Smith in 1966, initiated the development of mathematical models aiming to identify the conditions for sympatric speciation. A part of that paper was devoted to a specific two-locus, two-allele model of sympatric speciation in a population occupying a two-niche system. Maynard Smith provided some initial numerical results on this model. Later, Dickinson and Antonovics (1973) and Caisse and Antonovics (1978) performed more extensive numerical studies on the model. Here, I report analytical results on the haploid version of the Maynard Smith model. I show how the conditions for sympatric and parapatric speciation and the levels of resulting genetic divergence and reproductive isolation are affected by the strength of disruptive selection and nonrandom mating, recombination rate, and the rates of male and female dispersal between the niches.     

Maynard Smith,optimization, and evolution

Maynard Smith’s defenses of adaptationism and of the value of optimization theory in evolutionary biology are both criticized. His defense does not adequately respond to the criticism of adaptationism by Gould and Lewontin. It is also argued here that natural selection cannot be interpreted as an optimization process if the objective function to be optimized is either (i) interpretable as a fitness, or (ii) correlated with the mean population fitness. This result holds even if fitnesses are frequency-independent; the problem is further exacerbated in the frequency-dependent context modeled by evolutionary game theory. However, Eshel and Feldman’s new results on “long-term” evolution may provide some hope for the continuing relevance of the game-theoretic framework. These arguments also demonstrate the irrelevance of attempts by Intelligent Design creationists to use computational limits on optimization algorithms as evidence against evolutionary theory. It is pointed out that adaptation, natural selection, and optimization are not equivalent processes in the context of biological evolution. It is a pleasure to dedicate this paper to the memory of John Maynard Smith. Thanks are due to James Justus and Samir Okasha for comments on an earlier draft.     

Further remarks on Darwinian selection and “altruism”

In a recent note, Maynard Smith (Theor. Pop. Biol., in press) has claimed that there are certain difficulties in applying the method developed by us (Theor. Pop. Biol.14, 268–280) which incorporates the evolution of altruism into the population genetic theory of frequency-dependent selection. Of the four examples presented by Maynard Smith, the case of alarm calls was shown to have a natural expression in our multiplicative framework, and to produce conclusions different from those expected under the usual additive assumptions. We show here that the examples of the sterile worker, and of parental care are both simply expressible in terms of our conditional probability approach. The fourth example, the case of incest taboos, will be discussed elsewhere. Using Maynard Smith's examples important differences between the results for the additive and multiplicative fitness constructions are revealed. It is concluded that the heuristic approach using “inclusive fitness” offers no substantive advantages over exact population genetic modelling.     

The strategy concept and John Maynard Smith’s influence on theoretical biology

Here we argue that the concept of strategies, as it was introduced into biology by John Maynard Smith, is a prime illustration of the four dimensions of theoretical biology in the post-genomic era. These four dimensions are: data analysis and management, mathematical and computational model building and simulation, concept formation and analysis, and theory integration. We argue that all four dimensions of theoretical biology are crucial to future interactions between theoretical and empirical biologists as well as with philosophers of biology.     

Stability in an evolutionary game

A behavior or strategy which is evolutionarily stable must be both optimal and stable. The strategy must be optimal in that it maximizes the expected fitness of all the individuals using it. In addition, the strategy must be resistant to invasion by a mutant. The difference between the Nash solution of game theory and the ESS used in ecology is that the Nash solution only satisfies an optimality criterion and not an evolutionary stability criterion. We extend the ESS definition of Maynard Smith and Price so that it can be applied directly to two-strategy evolutionary games. The concept of a balanced game is introduced, and necessary conditions are derived which are similar to the Nash necessary conditions. The balanced game necessary conditions may be used for direct calculation of ESS candidates. These results are used to examine the optimal flowering time of an annual plant experiencing competition from neighboring plants. The plant competition model is general, and the results may be applied to a wide range of interference competition problems.     

The evolution of information in the major transitions

Maynard Smith and Szathmáry's analysis of the major transitions in evolution was based on changes in the way information is stored, transmitted and interpreted. With the exception of the transition to human linguistic societies, their discussion centred on changes in DNA and the genetic system. We argue that information transmitted by non-genetic means has played a key role in the major transitions, and that new and modified ways of transmitting non-DNA information resulted from them. We compare and attempt to categorise the major transitions, and suggest that the transition from RNA as both gene and enzyme to DNA as genetic material and proteins as enzymes may have been a double one. Unlike Maynard Smith and Szathmáry, we regard the emergence of the nervous system as a major transition. The evolution of a nervous system not only changed the way that information was transmitted between cells and profoundly altered the nature of the individuals in which it was present, it also led to a new type of heredity-social and cultural heredity-based on the transmission of behaviourally acquired information.     

Strong evolutionary equilibrium and the war of attrition

In developing the concept of an evolutionarily stable strategy, Maynard Smith proposed formal conditions for stability. These conditions have since been shown to be neither necessary nor sufficient for evolutionary stability in finite populations. This paper provides a strong stability condition which is sensitive to the population size. It is then demonstrated that in the war of attrition with uncertain rewards there is a unique “strong evolutionary equilibrium” strategy. As the population becomes large this is shown to approach the solution strategy proposed by Bishop, Cannings and Maynard Smith.The analysis is then extended to wars of attrition between different populations. It is concluded that for such contests there is a whole family of potential strong evolutionary equilibria.     

Evolutionary equilibrium strategies.

This paper re-examines the concept of evolutionary stability proposed by Maynard Smith.For any finite population it is shown that a strategy which is stable in the sense of Maynard Smith may have a lower fitness than a mutant strategy regardless of the proportion of contestants using the latter.Two alternative concepts of evolutionary stability are then proposed. A strategy is described as being strongly stable if no mutant is able to invade because of its higher fitness and weakly stable if it has a higher fitness whenever the contestants using any particular mutant strategy become sufficiently numerous.For the “war of attrition” between contestants of a given species Maynard Smith and others have argued that the evolutionarily stable strategy is for a contestant to bid (for food or territory) by attempting to wait out its opponent according to an exponential mixed strategy. This paper establishes that such a strategy is only weakly stable and that for any number n there exists a mutant strategy with a higher fitness until the number of mutants in the population exceeds n.The final section reconsiders the stability issue when a natural informational asymmetry is introduced. Each contestant is assumed to be uncertain as to the value its opponent places on the object over which they are competing. In contrast to the symmetric case it is shown that there is a strategy which is strongly stable with respect to any feasible mutant as long as the population is sufficiently large.     

Modeling social and evolutionary games

When game theory was introduced to biology, the components of classic game theory models were replaced with elements more befitting evolutionary phenomena. The actions of intelligent agents are replaced by phenotypic traits; utility is replaced by fitness; rational deliberation is replaced by natural selection. In this paper, I argue that this classic conception of comprehensive reapplication is misleading, for it overemphasizes the discontinuity between human behavior and evolved traits. Explicitly considering the representational roles of evolutionary game theory brings to attention areas of overlap that are often neglected, and so a range of evolutionary possibilities that are often overlooked. The clarifications this analysis provides are well illustrated by-and particularly valuable for-game theoretic treatments of the evolution of social behavior.     

Coevolution of competing species: ecological character displacement

Character displacement of competing species is studied. A model, originally developed by MacArthur and Levins (Proc. Natl. Acad. Sci. USA 51 (1964), 1207-1210) and further analyzed by Lawlor and Maynard Smith (Amer. Nat. 110 (1976), 70-99), has been reanalyzed. In the present paper, a more formally correct analysis of the MacArthur-Levins model is provided. A standard population genetics approach to sexually reproducing populations is adopted. The same conclusion as proposed by Lawlor and Maynard Smith emerges; competition can lead only to character divergence. In our analysis we either require that allopatrically evolved consumer populations must be able to coexist at an ecologically stable equilibrium (hence, we require mutual invasibility), or consider the feasibility of allopatric equilibria.     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.     

Signalling among relatives. I. Is costly signalling too costly?

Zahavi''s handicap principle,originally proposed as an explanation for sexual selection ofelaborate male traits, suggests that a sufficient cost to dishonest signals can outweigh the rewards of deception and allow individuals to communicate honestly. Maynard Smith (1991) and Johnstone and Grafen (1992) introduce the Sir Philip Sidney game in order to extend the handicap principle to interactions among related individuals, and to demonstrate that stable costly signalling systems can exist among relatives.In this paper we demonstrate that despite the benefits associated with honest information transfer, the costs incurred in a stable costly signalling system may leave all participants worse off than they would be in a system with no signalling at all. In both the discrete and continuous forms of the Sir Philip Sidney game, there exist conditions under which costly signalling among relatives, while stable, is so costly that it is disadvantageous compared with no signalling at all. We determine the factors which dictate signal cost and signal benefit in a generalized version of this game, and explain how signal cost can exceed signal value. Such results raise concerns about theevolutionary pathways which could have led to the existence of signalling equilibria in nature. The paper stresses the importance of comparing signalling equilibria with other possible strategies, beforedrawing conclusions regarding the optimality of signalling.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.