Efficiency in evolutionary games: Darwin, Nash and the secret handshake

This paper considers any evolutionary game possessing several evolutionarily stable strategies, or ESSs, with differing payoffs. A mutant is introduced which will "destroy" any ESS which yields a lower payoff than another. This mutant possesses a costless signal and also conditions on the presence of this signal in each opponent. The mutant then can protect itself against a population playing an inefficient ESS by matching this against these non-signalers. At the same time, the mutants can achieve the more efficient ESS against the signaling mutant population itself. This construction is illustrated by means of the simplest possible example, a co-ordination game. The one-shot prisoner's dilemma is used to illustrate how a superior outcome which is not induced by an ESS may be temporarily but not permanently attained. In the case of the repeated prisoner's dilemma, the present argument seems to render the "evolution of co-operation" ultimately inevitable.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

Evolutionary stable sets in mixed-strategist models

Evolutionary stable sets are used as an extension of the concept of an evolutionarily stable strategy (ESS). They have, as sets, essentially the same properties as ordinary ESSs. Here, ES sets are applied to the characterization of what will happen in an asexual population of mixed-strategists under frequency-dependent selection. Such a population will tend to establish some state, usually not a unique one, that belongs to an ES set. For an important class of widely used mixed-strategist models, ES sets are found to comprise just those population states that allow the possible behavioural acts to be equally successful, or, to put it more precisely, that establish an evolutionarily stable population strategy.     

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.     

Evolutionarily stable sets in symmetric extensive two-person games.

Evolutionarily stable (ES) sets are characterized for evolutionary games in extensive form. It is shown that, for the normal form of games involving informational symmetries or repeated play, the standard approach of determining evolutionarily stable strategies (ESSs) often fails to predict the evolutionary outcome. The dynamic stability of ES sets is proved in both the pure strategy and mixed strategy models. ES sets are shown to also generalize the notion of direct ESSs (an earlier attempt to apply ESS theory to extensive games). The theory is illustrated by three examples of biological games in extensive form.     

Simulating natural selection as a culling mechanism on finite populations with the hawk-dove game

The behaviors of individuals and species are often explained in terms of evolutionary stable strategies (ESSs). The analysis of ESSs determines which, if any, combinations of behaviors cannot be invaded by alternative strategies. Two assumptions required to generate an ESS (i.e., an infinite population and payoffs described only on the average) do not hold under natural conditions. Previous experiments indicated that under more realistic conditions of finite populations and stochastic payoffs, populations may evolve in trajectories that are unrelated to an ESS, even in very simple games. The simulations offered here extend earlier research by employing truncation selection with random parental selection in a hawk-dove game. Payoffs are determined in pairwise contests using either the expected outcome, or the result of a random variable. In each case, however, the mean fraction of hawks over many generations and across many independent trials does not conform to the expected ESS. Implications of these results and philosophical underpinnings of ESS theory are offered.     

Bounds on the number of ESSs of a matrix game

It is well known that for any evolutionary game there may be more than one evolutionarily stable strategy (ESS). In general, the more ESSs there are, the more difficult it is to work out how the population will behave (unless there are no ESSs at all). If a matrix game has an ESS which allows all possible pure strategies to be played, referred to as an internal ESS, then no other ESS can exist. In fact, the number of ESSs possible is highly dependent upon how many of the pure strategies each allow to be played, their support size. It is shown that if alpha is the ratio of the mean support size to the number of pure strategies n, then as n tends to infinity the greatest number of ESSs can be represented by a continuous function f(alpha) with useful regularity properties, and bounds are found for both f(alpha) and the value alpha(*), where it attains its maximum. Thus we can obtain a limit on the complexity of any particular system as a function of its mean support size.     

The Napoleon Complex: why smaller males pick fights

Does it ever pay for smaller animals to initiate fights even when they are likely to lose? Asymmetry in payoffs between opponents or a suboptimal strategy resulting from likely losers misperceiving themselves as likely winners have both been proposed as possible explanations for the aggressive behavior of smaller males. The model presented here suggests that in some cases, even without a payoff asymmetry and allowing for only a small error in perception, likely losers are expected to attack first. If the value of the resource exceeds the cost of losing a fight, the cost of displaying is sufficiently small, and assessment of resource holding power is reasonably accurate but not perfect, the evolutionarily stable strategy (ESS) prompts those contestants who perceive themselves as the likely losers to initiate fights, while it prompts those contestants who perceive themselves as the likely winners to wait for the adversary to attack or retreat.     

Genetical ESS-models. I. Concepts and basic model

Evolutionarily Stable Strategies (ESS) in phenotypic models are used to explain the evolution of animal interactive behaviour. As the behavioural features under consideration are assumed to be genetically determined, the question arises how underlying a genetical system might affect the results of phenotypic ESS-models. This question can be fully treated in terms of ESS-theory. A method of designing Genetical ESS-Models is proposed, which transfers the question of evolutionary stability to a "lower" level, the genetical basis. Genetical ESS-models - although nonlinear even in the simplest cases - can be analysed in a way that is familiar to ESS-theorists and yield immediate results on gene pool ESSs, which then may or may not maintain ESSs on the phenotypic level. Moreover, general results can be obtained to characterize evolutionarily stable gene pool states and their interrelation with commonsense, phenotypic ESSs. This part of the article presents the basic concepts and an outline of the method of genetical ESS-models. It gives, as a demonstration, a complete analysis for phenotypic two-strategy models (linear or nonlinear) based on a diploid, diallelic single-locus system under random mating. The results in this case suggest that a phenotypic ESS should indeed be expected to evolve but, maybe, only after passing through a succession of temporarily stable states.     

The evolution of payoff matrices: providing incentives to cooperate

Most of the work in evolutionary game theory starts with a model of a social situation that gives rise to a particular payoff matrix and analyses how behaviour evolves through natural selection. Here, we invert this approach and ask, given a model of how individuals behave, how the payoff matrix will evolve through natural selection. In particular, we ask whether a prisoner's dilemma game is stable against invasions by mutant genotypes that alter the payoffs. To answer this question, we develop a two-tiered framework with goal-oriented dynamics at the behavioural time scale and a diploid population genetic model at the evolutionary time scale. Our results are two-fold: first, we show that the prisoner's dilemma is subject to invasions by mutants that provide incentives for cooperation to their partners, and that the resulting game is a coordination game similar to the hawk-dove game. Second, we find that for a large class of mutants and symmetric games, a stable genetic polymorphism will exist in the locus determining the payoff matrix, resulting in a complex pattern of behavioural diversity in the population. Our results highlight the importance of considering the evolution of payoff matrices to understand the evolution of animal social systems.     

The cost of dishonesty

The handicap principle states that stable biological signals must be honest and costly to produce. The cost of the signal should reflect the true quality of the signaller. Here, it is argued that honest signalling may be maintained although the used signals are not handicaps. A game theoretic model in the form of a game of signalling is presented: all the existing evolutionarily stable strategies (ESSs) are found. Honest and cheap signalling of male quality is shown to be evolutionarily stable if females divorce the mate if it turns out that he has cheated about his quality. However, for this ESS to apply, the cost of lost time must not be too great. The stability of the honest signalling is based on deceivers being prevented from spreading in the population because they suffer from a cost of divorce. Under some fairly strict conditions, a mixed polymorphism of dishonesty and honesty represents another possible ESS.     

Evolutionary matrix games and optimization theory

An evolutionarily stable strategy (ESS) is only required to be capable of resisting invasion by rare mutant strategies. In contrast, an absolute invader strategy (AIS) is a rare mutant strategy that can invade any established strategy. We show that the predictions of the outcome of evolution made by optimization models are compatible with those made by the classical expected payoff comparisons in matrix games. We also show that if a matrix game has an AIS that AIS is unique and is also an ESS. But an ESS need not be an AIS. In pure-strategy submodels, an AIS need not be unique. An AIS of a matrix game has global asymptotic stability property in the game dynamics which involve only pure strategies including the AIS.     

Multiple patterns of parental care

Many animals show multiple patterns of parental care, where more than one of the four basic patterns (biparental care, uniparental care by males or females, or no care) is present within a single population during a single breeding season. We consider three reasons for the existence of multiple patterns of parental care: (1) mixed-strategy behaviours; (2) time-dependent behaviour with parents changing their care decision during the breeding season; and (3) quality differences between individuals leading to different care decisions being made depending on the qualities of both parents. The basic framework we use to investigate these is a two-stage game-theoretical model, and we highlight the importance of including feedback between the parental care decisions made by population members and the probability that a deserting individual will find a new mate. Including this feedback may introduce a nonlinear dependence of the fitness payoffs on the frequencies with which the pure strategies ('care' and 'desert') are played by each of the sexes. This can have important consequences for the existence of evolutionarily stable strategies (ESSs). For example, mixed-strategy ESSs may exist (an outcome forbidden if the feedback is not included) and, in one model, the feedback also prevents uniparental care by either sex from being evolutionarily stable. We also point out that decisions made by animals without dependent offspring can have important consequences for observed parental care behaviour. Copyright 1999 The Association for the Study of Animal Behaviour.     

Kleptoparasitic Melees��Modelling Food Stealing Featuring Contests with Multiple Individuals

Kleptoparasitism is the stealing of food by one animal from another. This has been modelled in various ways before, but all previous models have only allowed contests between two individuals. We investigate a model of kleptoparasitism where individuals are allowed to fight in groups of more than two, as often occurs in real populations. We find the equilibrium distribution of the population amongst various behavioural states, conditional upon the strategies played and environmental parameters, and then find evolutionarily stable challenging strategies. We find that there is always at least one ESS, but sometimes there are two or more, and discuss the circumstances when particular ESSs occur, and when there are likely to be multiple ESSs.     

On the war of attrition and other games among kin

Evolutionarily stable strategies or ESSs of games among kin have been calculated in the literature by both personal-fitness and inclusive-fitness methods. These methods were compared by Hines and Maynard Smith (1979) for games with bilinear payoffs. Although Hines and Maynard Smith regarded the first method as correct, they regarded the second method as useful because the inclusive-fitness conditions for an ESS gave necessary conditions for a personal-fitness ESS in the class of games they considered. In general, however, satisfying the inclusive-fitness conditions is neither necessary nor sufficient for satisfying the personal-fitness conditions, although the two methods may often yield identical ESSs. This result is established by reformulating the classic war-of-attrition model to allow variation in energy reserves, assumed to have a Gamma distribution. For this game, the two methods may disagree for intermediate values of relatedness. By the correct method, if the coefficient of variation in energy reserves is sufficiently high, then the game has a unique ESS in pure strategies at which populations with higher coefficients of variation or relatedness display for shorter times. Unrelated contestants are prepared to expend at least half of their reserves. For populations with lower variation coefficients, the ESS exists only if the cost of displaying per unit time is low compared to the rate at which remaining reserves translate into expected future reproductive success for the victor. The critical variation coefficient, below which the ESS exists regardless of cost, decreases from 0.52 to 0 as the coefficient of relatedness increases from 0 to 1. Although there is no assessment, contests are always won by the animal with greater energy reserves in a population at the ESS.     

Patterns of evolutionarily stable strategies: the maximal pattern conjecture revisited

A conflict is defined by a set of pure strategies [1,..., n] and a payoff matrix, and may have many evolutionarily stable strategies (ESSs). A collection of subsets of the set of pure strategies is called a pattern. If there is an n x n matrix which has ESSs whose supports match those of the pattern, then that pattern is said to be attainable. Much of the work on patterns of ESSs relied upon an unproved conjecture. Subject to some relaxation of the definition of attainability, this conjecture is proved.     

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.     

Runaway social games,genetic cycles driven by alternative male and female strategies,and the origin of morphs

Analysis of evolutionarily stable strategies (ESS) and decade-long field studies indicate that two color morphs of female side-blotched lizards exhibit density- and frequency-dependent strategies. Orange females are r-strategists: they lay large clutches of small progeny that are favored at low density. Conversely, yellow females are K-strategists: they lay small clutches of large progeny that are favored when carrying capacity is exceeded and the population crashes to low density. Interactions among three male morphs resembles a rock-paper-scissors (RPS) game. Fertilization success of males depends on frequency of neighboring morphs. Orange males usurp territory from blue neighbors and thereby mate with many females. However, orange males are vulnerable to cuckoldry by sneaky yellow males that mimic females. The yellow strategy is thwarted in turn by the mate-guarding strategy of blue. Sinervo and Lively (1996) developed a simple asexual model of the RPS game. Here, we model the dynamics of male and female morphs with one- and two-locus genetic models. Male and female games were considered in isolation and modeled as games that were genetically coupled by the same locus. Parameters for payoff matrices, which describe the force of frequency-dependent selection in ESS games, were estimated from free-ranging animals. Period of cycles in nature was 5 years for males and 2 years for females. Only the one locus model with three alleles (o, b, y) was capable of driving rapid cycles in male and female games. Furthermore, the o allele must be dominant to the y allele in females. Finally, the amplitude of male cycles was only reproduced in genetic models which allowed for irreversible plasticity of by genotypes, which is consistent with hormonally-induced changes that transform some males with yellow to dark blue. We also critique experimental designs that are necessary to detect density- and frequency-dependent selection in nature. Finally, runaway ESS games are discussed in the context of self-reinforcing genetic correlations that build and promote the formation of morphotypic variation.     

Genetical ESS-models. II. Multi-strategy models and multiple alleles

The problem of evolutionarily stable strategies (ESS) in sexual populations can be investigated by means of genetical ESS-models which link common sense, phenotypic ESS-models to an underlying genetical system. Thorough results are obtained for multi-strategy models in diploid, panmictic populations on the basis of multi-allelic, one-locus systems. A sexual population will be maintained at a phenotypic ESS if this can possibly be produced by the genotypes currently existing. If there is enough allelic variation, the corresponding gene pool may either be an ESS itself, or belong to an attracting, continuous set of states, which all determine the same evolutionarily stable population. The latter case allows new alleles to enter and spread in the gene pool without disturbing the phenotypic ESS. If a phenotypic ESS cannot be established, ESSs of the genetical model may be found which give rise to stable populations alternatively. Since these depend on the phenotypes determined by the currently existing genotypes, they may be destabilized by the occurrence of new mutations. In this sense, they are less durable than populations maintained at a phenotypic ESS and can be expected to evolve, in the long run, towards a phenotypic ESS.     

Ecological stability,evolutionary stability and the ESS maximum principle

Summary Since the fitness of each individual organism in a biological community may be affected by the strategies of all other individuals in the community, the essential element of a game exists. This game is an evolutionary game where the individual organisms (players) inherit their strategies from continuous play of the game through time. Here, the strategies are assumed to be constants associated with certain adaptive parameters (such as sunlight conversion efficiency for plants or body length in animals) in a set of differential equations which describe the population dynamics of the community. By means of natural selection, these parameters will evolve to a set of strategy values that natural selection, by itself, can no longer modify, i.e. an evolutionarily stable strategy (ESS). For a given class of models, it is possible to predict the outcome of this evolutionary process by determining ESSs using an ESS maximum principle. However, heretofore, the proof of this principle has been based on a limited set of conditions. Herein, we generalize the proof by removing certain restrictions and use instead the concept of an ecological stable equilibrium (ESE). Individuals in a biological community will be at an ESE if fixing the strategies used by the individuals results in stable population densities subject to perturbations in those densities. We present both necessary and sufficient conditions for an ESE to exist and then use the ESE concept to provide a very simple proof of the ESS maximum principle (which is a necessary condition for an ESS). A simple example is used to illustrate the difference between a strategy that maximizes fitness and one that satisfies the ESS maximum principle. In general they are different. We also look for ESEs in Lotka—Volterra competition and use the maximum principle to determine when an ESE will be an ESS. Finally, we examine the applicability of these ideas to matrix games.