Acceptor control in model ecosystems

The analysis of models of evolutionary games requires explicit consideration of both evolutionary game rules and mutants which infinitesimally break these rules. For example, the Scotch Auction is an evolutionary game which lacks both a rule-obeying evolutionarily stable strategy and an asymptotically stable polymorphism of rule-obeying strategies. However, an infinitesimal rule-breaking, or cheating, mutant can be found which is an evolutionarily stable strategy against rule-obeying strategies. Such cheating strategies can spread through populations initially playing the Scotch Auction, effectively changing the rules of the game. Moreover, the extent of such rule-change will then tend to increase. Thus, the Scotch Auction is a transient evolutionary game, being the initial point of a seemingly orthogenetic game evolutionary process. This sort of transience suggests that the “progressive” nature of evolution may be due in part to those game features of evolutionary processes which make the success of adaptations relative to the level of extant adaptation among competitors, predators, etc.     

Perfect reciprocity is the only evolutionarily stable strategy in the continuous iterated prisoner's dilemma

Theoretical studies have shown that cooperation tends to evolve when interacting individuals have positively correlated phenotypes. In the present article, we explore the situation where this correlation results from information exchange between social partners, and behavioral flexibility. We consider the game 'continuous iterated prisoner's dilemma'. The level of cooperation expressed by individuals in this game, together with their ability to respond to one another, both evolve as two aspects of their behavioral strategy. The conditions for a strategy to be evolutionarily stable in this game are degenerate, and earlier works were thus unable to find a single ESS. However, a detailed invasion analysis, together with the study of evolution in finite populations, reveals that natural selection favors strategies whereby individuals respond to their opponent's actions in a perfectly mirrored (i.e., correlated) fashion. As a corollary, the overall payoff of social interactions (i.e., the amount of cooperation) is maximized because couples of correlated partners effectively become the units of selection.     

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.     

Evolutionary Stability in the Asymmetric Volunteer's Dilemma

It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer''s dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a “strong” player is greater than the “weak” players in the model of Diekmann (1993). This contradicts Selten''s (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.     

Mistakes allow evolutionary stability in the repeated prisoner's dilemma game

The repeated prisoner's dilemma game has been widely used in analyses of the evolution of reciprocal altruism. Recently it was shown that no pure strategy could be evolutionarily stable in the repeated prisoner's dilemma. Here I show that if there is always some probability that individuals will make a mistake, then a pure strategy can be evolutionarily stable provided that it is "strong perfect equilibria" against itself. To be a strong perfect equilibrium against itself, a strategy must be the best response to itself after every possible sequence of behavior. I show that both unconditional defection and a modified version of tit-for-tat have this property.     

Evolutionary stability of ideal free dispersal strategies in patchy environments

A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244-251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt's (Am Nat 140:1010-1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.     

The ideal free distribution as an evolutionarily stable state in density‐dependent population games

In classical games that have been applied to ecology, individual fitness is either density independent or population density is fixed. This article focuses on the habitat selection game where fitness depends on the population density that evolves over time. This model assumes that changes in animal distribution operate on a fast time scale when compared to demographic processes. Of particular interest is whether it is true, as one might expect, that resident phenotypes who use density‐dependent optimal foraging strategies are evolutionarily stable with respect to invasions by mutant strategies. In fact, we show that evolutionary stability does not require that residents use the evolutionarily stable strategy (ESS) at every population density; rather it is the combined resident–mutant system that must be at an evolutionary stable state. That is, the separation of time scales assumption between behavioral and ecological processes does not imply that these processes are independent. When only consumer population dynamics in several habitats are considered (i. e. when resources do not undergo population dynamics), we show that the existence of optimal foragers forces the resident‐mutant system to approach carrying capacity in each habitat even though the mutants do not die out. Thus, the ideal free distribution (IFD) for the single‐species habitat selection game becomes an evolutionarily stable state that describes a mixture of resident and mutant phenotypes rather than a strategy adopted by all individuals in the system. Also discussed is how these results are affected when animal distribution and demographic processes act on the same time scale.     

Group formation games with reappraisal

The formation of a social group, such as the group of individuals sharing a territory, depends on the interaction between choices made by individuals to stay or disperse. The process can be modelled as a multi-player variant of the well-known War of Attrition in evolutionary game theory, as shown by Blackwell (1997; J. Theor. Biol.189, 175-181). In this paper, we extend the set of strategies defined there by allowing reappraisal during the game. We give a formal analysis of the evolutionarily stable strategy, where one exists, and illustrate it with an example based on badger (Meles meles) territoriality. The results predict that group size will be well adapted to, and very sensitive to, the precise conditions under which the game is played, and give an indication of the potential for parent-offspring conflict.     

Involuntary defection and the evolutionary origins of empathy

An occasional involuntary defection (IVD)--being unable, rather than unwilling, to donate help to others--is an intrinsic attribute of reciprocal cooperation (2003. J. Theor. Biol. 225-285). In fact, it is easy to see that--barring special circumstances--individuals that can donate help whenever requested do not need help of the same kind from others. That is, it is by no means clear why such individuals should participate in symmetric reciprocity interactions. In this paper, I analyse the consequences of introducing IVD into direct reciprocity modeling and show that a simple form of empathy--not retaliating after being punished for IVD--is a prerequisite for evolutionarily stable cooperation. Furthermore: we will see that the stability of this, empathic retaliator, strategy increases with the number of opportunities for cooperative exchanges in the life of an average individual.     

Equilibrium Distributions of Populations of Biological Species on Networks of Social Sites

ABSTRACT

We investigate the problem of how a population of biological species would distribute over a given network of social sites so that their social contacts through the connected sites can be maximized (or minimized). This problem has applications in modelling the behaviours of social (or solitary) species such as the development of social groups in human society and the spread of solitary animals in distant habitats. We show that this problem can be formulated as an evolutionary game, with the equilibrium state of the game corresponding to a strategy for choosing the residing sites, each with a certain probability, or equivalently, to a distribution of the population on these sites. The game has a symmetric payoff matrix, and can therefore be analyzed via the solution of a corresponding quadratic programme: An equilibrium strategy of the game is a KKT point of the quadratic programme, which may be a local maximizer, local minimizer, or saddle point, but it is evolutionarily stable if and only if it is a strict local maximizer. In general, with a goal to maximize the social contacts, the species tend to spread on network sites where there are dense connections such as a complete subnetwork or in other words, a network clique. We show that at equilibrium, the population may or may not distribute on a network clique, but the stability of the equilibrium state does depend on the structure of the selected subnetwork. In particular, we show that the distribution of the population on a maximal network clique is evolutionarily stable unless the clique is ‘attached’ to another clique of the same or larger size, when the population may be able to switch or expand to the neighbouring clique to increase or at least maintain its total amount of contacts. However, the distribution of the population on a non-clique subnetwork is always evolutionarily unstable or weakly evolutionarily stable at the very best, for the population can always move away from its current distribution without decreasing its total amount of contacts. We conclude that the strategies to spread on maximal network cliques are not only equilibrium strategies but also evolutionarily more stable than those on non-clique subnetworks, thus theoretically reaffirming the evolutionary advantages of joining social cliques in social networks for social species.     

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.     

The importance of mechanisms for the evolution of cooperation

Studies aimed at explaining the evolution of phenotypic traits have often solely focused on fitness considerations, ignoring underlying mechanisms. In recent years, there has been an increasing call for integrating mechanistic perspectives in evolutionary considerations, but it is not clear whether and how mechanisms affect the course and outcome of evolution. To study this, we compare four mechanistic implementations of two well-studied models for the evolution of cooperation, the Iterated Prisoner''s Dilemma (IPD) game and the Iterated Snowdrift (ISD) game. Behavioural strategies are either implemented by a 1 : 1 genotype–phenotype mapping or by a simple neural network. Moreover, we consider two different scenarios for the effect of mutations. The same set of strategies is feasible in all four implementations, but the probability that a given strategy arises owing to mutation is largely dependent on the behavioural and genetic architecture. Our individual-based simulations show that this has major implications for the evolutionary outcome. In the ISD, different evolutionarily stable strategies are predominant in the four implementations, while in the IPD each implementation creates a characteristic dynamical pattern. As a consequence, the evolved average level of cooperation is also strongly dependent on the underlying mechanism. We argue that our findings are of general relevance for the evolution of social behaviour, pleading for the integration of a mechanistic perspective in models of social evolution.     

Whence tit-for-tat?

Summary In theoretical and empirical studies of the evolution of cooperation, the tit-for-tat strategy (i.e. cooperate unless your partner did not cooperate in the previous interaction) is widely considered to be of central importance. Nevertheless, surprisingly little is known about the conditions in which tit-for-tat appears and disappears across generations in a population of interacting individuals. Here, we apply a newly developed classifier-system model (EvA) in addressing this issue when the key features of interactions are caricatured using the iterated prisoner's dilemma game. Our simple representation of behavioural strategies as algorithms composed of two interacting rules allowed us to determine conditions in which tit-for-tat can replace noncooperative strategies and vice versa. Using direct game-theoretic analysis and simulations with the EvA model, we determined that no strategy is evolutionarily stable, but larger population sizes and longer sequences of interactions between individuals can yield transient dominance by tit-for-tat. Genetic drift among behaviourally equivalent strategies is the key mechanism underlying this dominance. Our analysis suggests that tit-for-tat could be important in nature for cognitively simple organisms of limited memory capacity, in strongly kin-selected or group-selected populations, when interaction sequences between individuals are relatively short, in moderate-sized populations of widely interacting individuals and when defectors appear in the population with moderate frequency.     

Rare but severe concerted punishment that favors cooperation

As one of the mechanisms that are supposed to explain the evolution of cooperation among unrelated individuals, costly punishment, in which altruistic individuals privately bear the cost to punish defection, suffers from such drawbacks as decreasing individuals’ welfare, inducing second-order free riding, the difficulty of catching defection, and the possibility of triggering retaliation. To improve this promising mechanism, here we propose an extended Public Goods game with rare but severe concerted punishment, in which once a defector is caught punishment is triggered and the cost of punishment is equally shared among the remainder of the group. Analytical results show that, when the probability for concerted punishment is above a threshold, cooperating is, while defecting is not, an evolutionarily stable strategy in finite populations, and that this way of punishment can considerably decrease the total cost of inhibiting defection, especially in large populations.     

THE EVOLUTION OF STRATEGY VARIATION: WILL AN ESS EVOLVE?

Evolutionarily stable strategy (ESS) models are widely viewed as predicting the strategy of an individual that when monomorphic or nearly so prevents a mutant with any other strategy from entering the population. In fact, the prediction of some of these models is ambiguous when the predicted strategy is "mixed", as in the case of a sex ratio, which may be regarded as a mixture of the subtraits "produce a daughter" and "produce a son." Some models predict only that such a mixture be manifested by the population as a whole, that is, as an "evolutionarily stable state"; consequently, strategy monomorphism or polymorphism is consistent with the prediction. The hawk-dove game and the sex-ratio game in a panmictic population are models that make such a "degenerate" prediction. We show here that the incorporation of population finiteness into degenerate models has effects for and against the evolution of a monomorphism (an ESS) that are of equal order in the population size, so that no one effect can be said to predominate. Therefore, we used Monte Carlo simulations to determine the probability that a finite population evolves to an ESS as opposed to a polymorphism. We show that the probability that an ESS will evolve is generally much less than has been reported and that this probability depends on the population size, the type of competition among individuals, and the number of and distribution of strategies in the initial population. We also demonstrate how the strength of natural selection on strategies can increase as population size decreases. This inverse dependency underscores the incorrectness of Fisher's and Wright's assumption that there is just one qualitative relationship between population size and the intensity of natural selection.     

Differential games in evolutionary theory: Height growth strategies of trees

Differential game theory is applied to the analysis of evolutionarily stable strategies (ESS) in this article. A general form for the evolutionary differential game is introduced in the case of intra-specific competition, and a connection between the ESS and the mathematical Nash solution concept is indicated. A dynamic ESS is found for the height growth strategies of trees. A hierarchical model is introduced to account for different time constants in simultaneous selection processes. Differential evolutionary games are compared with static evolutionary games utilizing the hierarchical approach.     

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.     

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.     

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.     

Play and the evolution of fairness: a game theory model

Bekoff [J. Consci. Stud. 8 (2001) 81] argued that mammalian social play is a useful behavioral phenotype on which to concentrate in order to learn more about the evolution of fairness. Here, we build a game theoretical model designed to formalize some of the ideas laid out by Bekoff, and to examine whether ‘fair’ strategies can in fact be evolutionarily stable. The models we present examine fairness at two different developmental stages during an individual's ontogeny, and hence we create four strategies—fair at time 1/fair at time 2, not fair at time 1/not fair at time 2, fair at time 1/not fair at time 2, not fair at time 1/fair at time 2. Our results suggest that when considering species where fairness can be expressed during two different developmental stages, acting fairly should be more common than never acting fairly. In addition, when no one strategy was evolutionarily stable, we found that all four strategies we model can coexist at evolutionary equilibrium. Even in the absence of an overwhelming database from which to test our model, the general predictions we make have significant implications for the evolution of fairness.