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PubMed1997

Evolutionarily stable strategy distributions for the Repeated Prisoner's Dilemma.

This paper introduces the idea of an evolutionarily stable strategy distribution, which generalizes the idea of an evolutionarily stable strategy; roughly speaking, an evolutionarily stable strategy distribution is a finite set of symbiotic strategies which is unaffected by low levels of mutation. This idea is then applied to the n-person Repeated Prisoner's Dilemma, of which the usual Repeated Prisoner's Dilemma is the special case n=2. Given some standard assumptions on what mutations are possible, it is shown that if the probability of future interactions is sufficiently large, there are no evolutionarily stable strategy distributions. (And hence no evolutionarily stable strategies.) An example is given of an evolutionarily stable strategy distribution in the case when the set of possible mutant strategies is restricted.

Animals↗

PubMed1997

Multilocus evolutionarily stable strategy models: additive effects.

A common reservation about Evolutionarily Stable Strategy analyses is that they fail to account explicitly for Mendelian genetics, and therefore may produce biologically unfeasible predictions. This note shows that Evolutionarily Stable Strategy and multilocus genetic analyses agree for one simple model of frequency-dependent selection under the assumption that the effects of alleles and loci are completely additive. A numeric example illustrated the concepts and results involved.

Animals↗

PubMed2004

Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay.

We construct two models of discrete-time replicator dynamics with time delay. In the social-type model, players imitate opponents taking into account average payoffs of games played some units of time ago. In the biological-type model, new players are born from parents who played in the past. We consider two-player games with two strategies and a unique mixed evolutionarily stable strategy. We show that in the first type of dynamics, it is asymptotically stable for small time delays and becomes unstable for big ones when the population oscillates around its stationary state. In the second type of dynamics, however, evolutionarily stable strategy is asymptotically stable for any size of a time delay.

Animals↗

PubMed1990

Group selection among alternative evolutionarily stable strategies.

Many important models of the evolution of social behavior have more than one evolutionarily stable strategy (ESS). Examples include co-ordination games, contests, mutualism, reciprocity, and sexual selection. Here we show that when there are multiple evolutionarily stable strategies, selection among groups can cause the spread of the strategy that has the lowest extinction rate or highest probability of contributing to the colonization of empty habitats, and that this may occur even when groups are usually very large, migration rates are substantial, and "extinction" entails only the disruption of the group and the dispersal of its members. The main requirements are: (1) individuals drawn from a single surviving group make up a sufficiently large fraction newly formed groups, and (2) the processes increasing the frequency of successful strategies within groups are strong compared to rate of migration among groups. The latter condition suggests that this form of group selection will be particularly important when behavioral variation is culturally acquired.

Animals↗

PubMed1988

Evolutionarily stable strategies of mutual help between relatives having unequal fertilities.

The evolutionarily stable strategy of mutual help between relatives having unequal fertilities is studied in a kin selection model, which also takes into account competition between kins and the possibility of reciprocation. It turns out that competition and reciprocation can establish ESSs which are completely different from those expected by Hamilton's basic theory.

Altruism↗

PubMed2005

Evolutionarily stable strategies and short-term selection in Mendelian populations re-visited.

This note concerns a one locus, two allele, random mating diploid population, subject to frequency-dependent viability selection. It is already known that in such a population, any evolutionarily stable strategies (ESS), if only accessible by the genotype-to-phenotype mapping, is the phenotypic image of a stable genetic equilibrium (Eshel, I. 1982. Evolutionarily stable strategies and viability selection in Mendelian populations. Theor. Popul. Biol. 22(2), 204-217; Cressman et al. 1996. Evolutionary stability in strategic models of single-locus frequency-dependent viability selection. J. Math. Biol. 34, 707-733). The opposite is not true. We find necessary and sufficient parametric conditions for global convergence to the ESS, but we also demonstrate conditions under which, although a unique, genetically accessible ESS exists, there is another, "non-phenotypic" genetically stable equilibrium.

Animals↗

PubMed1997

Effect of time delay and evolutionarily stable strategy.

In this paper, a simple two-phenotype model with time delay is investigated. The main results are that: (i) the stability of the interior equilibrium point of the pure strategy model not only depends on the property of the payoff matrix but also the effect of time delay; (ii) the conditions of the evolutionarily stable strategy in the two-phenotype model with time delay are completely identical with the conditions in the two-phenotype model with no time delay; and (iii) a mixed evolutionarily stable strategy can be an unstable equilibrium state of the population in the two-phenotype model with time delay.

Animals↗

PubMed2000

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.

Animals↗

PubMed1988

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.

Behavior↗

PubMed2000

Evolutionarily Stable Strategies for Consuming a Structured Resource.

A general consumer-resource model assuming discrete consumers and a continuously structured resource is examined. We study two foraging behaviors, which lead to fixed and flexible patch residence times, in conjunction with a simple consumer energetics model linking resource consumption, foraging behavior, and metabolic costs. Results indicate a single, evolutionarily stable foraging strategy for fixed and flexible foraging in a nonspatial environment, but flexible foraging in a spatial environment leads to consumer grouping, which affects the resource distribution such that no single foraging strategy can exclude all other strategies. This evolutionarily stable coexistence of multiple foraging strategies may help explain a dichotomous pattern observed in a wide variety of natural systems.

competitive coexistence↗

PubMed1997

A general technique for computing evolutionarily stable strategies based on errors in decision-making.

Realistic models of contests between animals will often involve a series of state-dependent decisions by the contestants. Computation of evolutionarily stable strategies for such state-dependent dynamic games are usually based on damped iterations of the best response map. Typically this map is discontinuous so that iterations may not converge and even if they do converge it may not be clear if the limiting strategy is a Nash equilibrium. We present a general computational technique based on errors in decision making that removes these computational difficulties. We show that the computational technique works for a simple example (the Hawk-Dove game) where an analytic solution is known, and prove general results about the technique for more complex games. It is also argued that there is biological justification for inclusion of the types of errors we have introduced.

Animals↗

PubMed2004

Determining the optimal developmental route of Strongyloides ratti: an evolutionarily stable strategy approach.

Understanding the processes that drive parasite evolution is crucial to the development of management programs that sustain long-term, effective control of infectious disease in the face of parasite adaptation. Here we present a novel evolutionarily stable strategy (ESS) model of the developmental decisions of a nematode parasite, Strongyloides ratti. The genus Strongyloides exhibits an unusual developmental plasticity such that progeny from an individual may either develop via a direct (homogonic) route, where the developing larvae are infective to new hosts, or an indirect (heterogonic) route, where the larvae develop into free-living, dioecious adults that undergo at least one bout of sexual reproduction outside the host, before producing offspring that develop into infective larvae. The model correctly predicts a number of observed features of the parasite's behavior and shows that this plasticity may be adaptive such that pure homogonic development, pure heterogonic development, or a mixed strategy may be optimal depending on the prevailing environmental conditions, both within and outside the host. Importantly, our results depend only on the benefits of an extra round of reproduction in the environment external to the host and not on benefits to sexual reproduction through the purging of deleterious mutation or the generation of novel, favorable genotypes. The ESS framework presented here provides a powerful, general approach to predict how macroparasites, the agents of many of the world's most important infectious diseases, will evolve in response to the various selection pressures imposed by different control regimes in the future.

Adaptation, Biological↗

PubMed2003

Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics.

An integrodifferential equations model for the distribution of individuals with respect to the age at maturity is considered. Mutation is modeled by an integral operator. Results concerning the behaviour of the steady states and their relation to evolutionarily stable strategies when the mutation rate is small are given. The same results are obtained for a (rather) general class of models that include the one mentioned before.

Age Distribution↗

PubMed1993

Environmental effects on fitness-sets shape and evolutionarily stable strategies.

Most models on the evolution of sex allocation and life-history traits are based on the existence of compensations between these traits, and often consider them as linear. With a simple model of physiological response to richness of the environment, we show that not only can these compensations take many different shapes, but also that this shape varies as a function of the resource level. Consequently, evolutionarily stable strategies (ESSs) calculation can give different results for the same two functions according to resource level. Thus, selection can act in different directions depending on the "quality" of the environment. Moreover, genetic variability in the resource allocation strategies is likely to be shown better in intermediate environments, where the proportions of allocation have the most crucial effect on the phenotype.

Animals↗

PubMed1984

Evolutionarily stable strategies in food selection models with fitness sets.

Most current models for optimal food selection apply to ecological and behavioural optimization. In this paper optimal food selection theory is extended to apply to evolutionary optimization. A general evolutionary model for optimal food selection must incorporate the concept of fitness sets--or that variables, changing as a result of natural selection in evolutionary time, cannot, in general, vary independently of each other. A "Charnov type" optimal food selection model with a fitness set is investigated, and evolutionarily stable strategy (ESS) solutions of the evolutionary variables (i.e., the efficiencies of using available food types) are found. From this analysis it follows that the relative frequency of various food types in the environment may, under specified conditions, influence the evolutionarily optimal diet. Secondly, the analysis demonstrates that a food type not in the optimal diet may, in evolutionary time, be added to this by becoming more abundant. Thirdly, it follows from the analysis that the ecological result of MacArthur and Pianka, that food types are worth eating even if there is competition for them, is not generally applicable when referring to an evolutionary time scale. Finally, it is pointed out that for the diet to be an ESS, it is necessary that the consumer's density is stable and that the consumer's population dynamics are subjected to some density-dependent factor.

Animals↗

PubMed2002

Are there really no evolutionarily stable strategies in the iterated prisoner's dilemma?

The evolutionary form of the iterated prisoner's dilemma (IPD) is a repeated game where players strategically choose whether to cooperate with or exploit opponents and reproduce in proportion to game success. It has been widely used to study the evolution of cooperation among selfish agents. In the past 15 years, researchers proved over a series of papers that there is no evolutionarily stable strategy (ESS) in the IPD when players maintain long-term relationships. This makes it difficult to make predictions about what strategies can actually persist as prevalent in a population over time. Here, we show that this no ESS finding may be a mathematical technicality, relying on implausible players who are "too perfect" in that their probability of cooperating on any move is arbitrarily close to either 0 or 1. Specifically, in the no ESS proof, all strategies were allowed, meaning that after a strategy X experiences any history H, X cooperates with an unrestricted probability p (X, H) where 0< or =p (X, H)< or =1. Here, we restrict strategies to the set S in which X is a member of S [corrected] if after any H, X cooperates with a restricted probability p (X, H) where e< or =p (X, H)< or =1-e and 0<e<1/2. The variables e and 1-e may be thought of as the biological limits to how perfect (pure) a strategy can be. In S, we first prove that an ESS must be a Nash equilibrium and boundary strategy of S where a boundary strategy has allowable probabilities of cooperating of e or 1-e after any history. We then prove that X is an ESS in S if and only if X is a Nash equilibrium in the set consisting of only S's boundary strategies. Thus, in searching for ESSs in S, we only need to consider what happens when boundary strategies interact with one another. This greatly simplifies the search for ESSs. Using these results, we finally show that when e is sufficiently small, exactly three one-move memory ESSs exist in S: [1] Pavlov-e which generally cooperates (i.e. cooperates with maximum probability 1-e) after both players either simultaneously cooperated or simultaneously defected; otherwise, Pavlov-e generally defects (i.e. cooperates with minimum probability e). [2] Grudge-e which generally cooperates after both players simultaneously cooperated; otherwise, Grudge-e generally defects. [3] ALLD-e which generally defects after all histories.

Animals↗

PubMed1979

An evolutionarily stable strategy approach to indiscriminate spite.

AN individual behaves spitefully when it harms itself in order to harm another individual more(1). Hamilton(1,2) predicted that spite may evolve if it is expressed only in those encounters that occur between individuals of less than average relatedness. More recently Verner(3) suggested that territory size may become super-optimal because of a selective advantage arising from the spiteful exclusion of others from limited resources. His model is essentially different from Hamilton's in that spite is directed at individuals indiscriminately with respect to relatedness. Recently Rothstein(4) has shown analytically that the initial spread of spiteful traits will be very slow in all but the smallest populations. He also argued verbally that indiscriminate spite can never be evolutionarily stable even if it should spread (see also Davies(5)). The question of evolutionary stability is clearly important, but its resolution requires an analytical approach. We report here an approach based on Maynard Smith's(6) concept of the evolutionarily stable strategy (ESS), a strategy which, when common, does better than any alternative strategy played by a rare mutant. We show that spite can be an ESS, but that the magnitude of spite will be small in large populations.

Journal Article↗

PubMed2003

Honesty and cheating in cleaning symbioses: evolutionarily stable strategies defined by variable pay-offs.

Game-theory models have indicated that the evolution of mixed strategies of cheating and honesty in many mutualisms is unlikely. Moreover, the mutualistic nature of interspecific interactions has often been difficult to demonstrate empirically. We present a game-theory analysis that addresses these issues using cleaning symbioses among fishes as a model system. We show that the assumption of constant pay-offs in existing models prevents the evolution of evolutionarily stable mixed strategies of cheating and honesty. However, when interaction pay-offs are assumed to be density dependent, mixed strategies of cheating and honesty become possible. In nature, cheating by clients often takes the form of retaliation by clients against cheating cleaners, and we show that mixed strategies of cheating and honesty evolve within the cleaner population when clients retaliate. The dynamics of strategies include both negative and positive effects of interactions, as well as density-dependent interactions. Consequently, the effects of perturbations to the model are nonlinear. In particular, we show that under certain conditions the removal of cleaners may have little impact on client populations. This indicates that the underlying mutualistic nature of some interspecific interactions may be difficult to demonstrate using simple manipulation experiments.

Animals↗