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.     

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.     

Evolutionary game theory as a framework for studying biological invasions

Although biological invasions pose serious threats to biodiversity, they also provide the opportunity to better understand interactions between the ecological and evolutionary processes structuring populations and communities. However, ecoevolutionary frameworks for studying species invasions are lacking. We propose using game theory and the concept of an evolutionarily stable strategy (ESS) as a conceptual framework for integrating the ecological and evolutionary dynamics of invasions. We suggest that the pathways by which a recipient community may have no ESS provide mechanistic hypotheses for how such communities may be vulnerable to invasion and how invaders can exploit these vulnerabilities. We distinguish among these pathways by formalizing the evolutionary contexts of the invader relative to the recipient community. We model both the ecological and the adaptive dynamics of the interacting species. We show how the ESS concept provides new mechanistic hypotheses for when invasions result in long- or short-term increases in biodiversity, species replacement, and subsequent evolutionary changes.     

Revisiting matrix games: the concept of neighborhood invader strategies

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.     

Evolutionarily stable strategy carbon allocation to foliage, wood, and fine roots in trees competing for light and nitrogen: an analytically tractable, individual-based model and quantitative comparisons to data

We present a model that scales from the physiological and structural traits of individual trees competing for light and nitrogen across a gradient of soil nitrogen to their community-level consequences. The model predicts the most competitive (i.e., the evolutionarily stable strategy [ESS]) allocations to foliage, wood, and fine roots for canopy and understory stages of trees growing in old-growth forests. The ESS allocations, revealed as analytical functions of commonly measured physiological parameters, depend not on simple root-shoot relations but rather on diminishing returns of carbon investment that ensure any alternate strategy will underperform an ESS in monoculture because of the competitive environment that the ESS creates. As such, ESS allocations do not maximize nitrogen-limited growth rates in monoculture, highlighting the underappreciated idea that the most competitive strategy is not necessarily the "best," but rather that which creates conditions in which all others are "worse." Data from 152 stands support the model's surprising prediction that the dominant structural trade-off is between fine roots and wood, not foliage, suggesting the "root-shoot" trade-off is more precisely a "root-stem" trade-off for long-lived trees. Assuming other resources are abundant, the model predicts that forests are limited by both nitrogen and light, or nearly so.     

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.     

Fighting for food: a dynamic version of the Hawk-Dove game

Summary The Hawk-Dove game (Maynard Smith, 1982) has been used to analyse conflicts over resources such as food. At the evolutionarily stable strategy (ESS), either a proportionp* of animals always play Hawk, or each animal has a probabilityp* of playing Hawk. We modify the standard Hawk-Dove game to include a state variable,x, that represents the animal's level of energy reserves. A strategy is now a rule for choosing an action as a function ofx and time of day. We consider a non-reproductive period and adopt the criterion of minimizing mortality over this period. We find the ESS, which has the form play Hawk if reserves are belowc* (t) at timet, otherwise play Dove. This ESS is very different from the ESS in the standard Hawk-Dove game. It is a pure ESS that depends on the animal's state and on time. Furthermore, it is characterized by the strong condition that any single mutant that does not adopt the ESS suffers a reduction in fitness. The standard Hawk-Dove game assumes pay-offs that are related to fitness; our approach starts from a definition of fitness and derives the pay-offs in the process of finding the ESS. When the environment becomes worse (e.g. food becomes less reliable or energy expenditure increases) the ESS changes in such a way as to increase the proportion of animals that will play Hawk.     

Learning to Choose Among Social Foraging Strategies in Adult House Sparrows (Passer domesticus)

Social foragers may be regarded as being engaged in a producer–scrounger game in which they can search for food independently or join others who have discovered food. Research on the producer–scrounger game has focused mainly on the different factors influencing its evolutionarily stable strategy (ESS) solution, but very little is known about the actual mechanisms that shape players' decisions. Recent work has shown that early experience can affect producer–scrounger foraging tendencies in young house sparrows and that in nutmeg mannikins learning is involved in reaching the ESS. Here, we show that direct manipulation of the success rate experienced by adult sparrows when following others can change their strategy choice on the following day. We presented to live sparrows an experimental regime, where stuffed adult house sparrows in a feeding position were positioned on a foraging grid that included two reward regimes: a positive one, in which the stuffed models were placed near food, and a negative one, in which the models were placed away from food. There was a significant increase in joining behavior after the positive treatment (exhibited by 84% of the birds), but no change after the negative treatment. Further analysis demonstrated that sparrows more frequently used the strategy with which they were more successful (usually joining) and that differences in strategy use were correlated with differences in success. These results suggest that adult birds can monitor their success and learn to choose among social foraging strategies in the producer–scrounger game.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.     

Threat displays are not handicaps

Within a general framework of handicap signalling it was proposed that threat displays are handicaps, they can work only if they put the signaller at a disadvantage, which is only acceptable to honest signallers. The aim of the present article is to investigate this proposal with the help of a simple game-theoretical model. It was found: (1) that the use of cost-free signals is an ESS against the invasion of handicapped signals even if cheating is played as part of a mixed strategy in the population; (2) that the use of handicaps may be an ESS against cost-free signals but only if we assume that the invading cost-free signal is not accepted by weak individuals as a signal of strength; (3) that the establishment of a handicapped signal in the first place is an unresolved problem, because both cost free signals and negative-handicaps are evolutionarily stable against the invasion of handicaps; (4) that in contrast to handicaps the use of negative-handicaps can invade a population using cost-free signals (a negative-handicap is a signal which may serve other functions as well); (5) that negative-handicaps are ESS against cost-free signals as well as against handicaps; and (6) thus, the most likely evolutionary end point is that the biggest negative-handicap would be used as a threat display. This is a posture, which prepares the animal most efficiently to fight; hence, most probably it is the initial position of the fighting technique of the given species. (7) Finally, the investigation of the threat displays of well-studied taxa (great tit, cats, dogs, and hoofed mammals) confirms that threat displays are indeed negative-handicaps. They do not put the user into a disadvantaged position, instead the initial position of the species specific fighting technique is used as a threat display as predicted by the present model.     

Individual-based simulations of the War of Attrition

The War of Attrition model of John Maynard Smith predicts a single, mixed evolutionarily stable strategy (ESS) for animal contests which are settled by conventional displays with no assessment of the opponent's fighting ability. We test the predictions of the model by simulating the evolution of strategies in a finite population of animals under various assumptions on how possible strategies are coded and mutated. While our simulations for the most part confirm the predictions of the model, we also discovered some significant deviations from the theoretically predicted ESS. Specifically, we found that if inheritance of strategies is somewhat imprecise, then a population can evolve that achieves on average a higher payoff than a population at the theoretically predicted ESS. Moreover, if the ESS is realized as a polymorphism of fixed persistence times, then for small populations, sufficiently stringent statistical tests will reject the hypothesis that these times are distributed as theoretically predicted.     

Invasions cause biodiversity loss and community simplification in vertebrate food webs

Global change is increasing the occurrence of perturbation events on natural communities, with biological invasions posing a major threat to ecosystem integrity and functioning worldwide. Most studies addressing biological invasions have focused on individual species or taxonomic groups to understand both, the factors determining invasion success and their effects on native species. A more holistic approach that considers multispecies communities and species’ interactions can contribute to a better understanding of invasion effects on complex communities. Here we address biological invasions on species‐rich food webs. We performed in silico experiments on empirical vertebrate food webs by introducing virtual species characterised by different ecological roles and belonging to different trophic groups. We varied a number of invasive species traits, including their diet breadth, the number of predators attacking them, and the bioenergetic thresholds below which invader and native species become extinct. We found that simpler food webs were more vulnerable to invasions, and that relatively less connected mammals were the most successful invaders. Invasions altered food web structure by decreasing species richness and the number of links per species, with most extinctions affecting poorly connected birds. Our food web approach allows identifying the combinations of trophic factors that facilitate or prevent biological invasions, and it provides testable predictions on the effects of invasions on the structure and dynamics of multitrophic communities.     

The unified framework for biological invasions: a forest fungal pathogen perspective

Biological invasions in forests are growing in number and importance globally. The best studied examples are those caused by plants and animals, including insects. In contrast, forest invasions caused by microbes, including fungi, have received much lower levels of attention, particularly in the invasion biology literature. This can at least to some extent be due to the large number of these organisms involved and the fact that the majority of these have yet to be discovered and described. This is equally true for tree-infecting fungi, many of which are devastating pathogens responsible for dramatic invasions in natural and planted forests. This situation is changing through the application of molecular genetic tools that make it possible to accurately identify fungal tree pathogens, to determine their origins, pathways of movement, their modes of reproduction and change; all of which can influence invasions. The role and relevance of symbioses between tree pathogens and insects in forest invasions is also gaining increased attention. So too is our understanding that trees live in close association with large numbers of microbes that make up their holobiome. This has substantial relevance to invasion biology (Zenni et al. 2017). This commentary highlights four emerging issues that need to be considered regarding the invasions by fungal pathogens of trees and it emphasizes opportunities to better understand their relevance and impacts on natural and planted forests. A call is also made for plant pathologists to work more closely with ecologists such that fungal pathogens become more commonly integrated into invasion biology programmes.     

THE EVOLUTION OF VIRULENCE IN PATHOGENS WITH VERTICAL AND HORIZONTAL TRANSMISSION

The idea that vertical transmission of parasites selects for lower virulence is widely accepted. However, little theoretical work has considered the evolution of virulence for parasites with mixed horizontal plus vertical transmission. Many human, animal, and plant parasites are transmitted both vertically and horizontally, and some horizontal transmission is generally necessary to maintain parasites at all. We present a population-dynamical model for the evolution of virulence when both vertical and horizontal transmission are present. In the simplest such model, up to two infectious strains can coexist within one host population. Virulent, vertically transmitted pathogens can persist in a population when they provide protection against more virulent, horizontally transmitted strains. When virulence is maintained by a correlation with horizontal transmission rates, increased levels of vertical transmission always lower the evolutionarily stable (ESS) level of virulence. Contrary to existing theory, however, increases in opportunities for horizontal transmission also lower the ESS level of virulence. We explain these findings in light of earlier work and confirm them in simulations including imperfect vertical transmission. We describe further simulations, in which both vertical and horizontal transmission rates are allowed to evolve. The outcome of these simulations depends on whether high levels of vertical transmission are possible with low virulence. Finally, we argue against the notion of a virulence-avirulence continuum between horizontal and vertical transmission, and discuss our results in relation to empirical studies of transmission and virulence.     

Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models

This paper studies the correspondence between Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models. Whenever the fitness function is sufficiently continuous, any large-population ESS corresponds to a symmetric Nash equilibrium in the game that describes the simultaneous interaction of the individuals in the population, and any strict, symmetric Nash equilibrium in that game corresponds to a large-population ESS. This correspondence continues to hold, approximately, in finite populations; and it holds exactly for strict pure-strategy equilibria in sufficiently large finite populations. By contrast, a sequence of (mixed-strategy) finite-population ESSs can converge, as the population grows, to a limit that is not a large-population ESS, and a large-population ESS need not be the limit of any sequence of finite-population ESSs.     

A theory for the evolutionary game

We present an evolutionary game theory. This theory differs in several respects from current theories related to Maynard Smith's pioneering work on evolutionary stable strategies (ESS). Most current work deals with two person matrix games. For these games the strategy set is finite. We consider evolutionary games which are defined over a continuous strategy set and which permit any number of players. Matrix games are included as a bilinear continuous game. However, under our definition, such games will not posses an ESS on the interior of the strategy set. We extend previous work on continuous games by developing an ESS definition which permits the ESS to be composed of a coalition of several strategies. This definition requires that the coalition must not only be stable with respect to perturbations in strategy frequencies which comprise the coalition, but the coalition must also satisfy the requirement that no mutant strategies can invade. Ecological processes are included in the model by explicitly considering population size and density dependent selection.     

Is Food Worth Fighting for? ESS’s in Mixed Populations of Kleptoparasites and Foragers

We extend the game theoretic model of kleptoparasitism discussed in Broom et al. (2004), by considering a population of foragers consisting of two groups with different behaviours—those who forage and steal from other feeders, and those who only forage. We a sume that those who do not steal have a better foraging rate than those who are also looking out for opportunities to steal. We also allow either type to resist an attack or not resist. We look for Evolutionary Stable States, of either a mixture of the two behaviours, or where the whole population has just one of these behaviours. We find nine such ESS’s, dependent on the environmental parameters, although in fact only five of these are distinguishable. In general, we find that if the overall population density is low, food-stealing becomes less viable, and there is an ESS consisting of only foragers. Conversely, when there are many animals looking for, and finding, food, there is an ESS consisting of just kleptoparasites (which are also foraging). In between, an ESS will contain both pure-foragers and stealers. There is some empirical evidence of such behaviours. We find that when there is a mixture of the two types, they must both have the same resistive behaviour. We can thus have some individuals challenging for food but not resisting challenges, and others not challenging and not resisting. This shows how aggressive behaviour may be context-dependent, as seen in practice.     

The patch distributed producer-scrounger game

Grouping in animals is ubiquitous and thought to provide group members antipredatory advantages and foraging efficiency. However, parasitic foraging strategy often emerges in a group. The optimal parasitic policy has given rise to the producer-scrounger (PS) game model, in which producers search for food patches, and scroungers parasitize the discovered patches. The N-persons PS game model constructed by Vickery et al. (1991. Producers, scroungers, and group foraging. American Naturalist 137, 847-863) predicts the evolutionarily stable strategy (ESS) of frequency of producers that depends on the advantage of producers and the number of foragers in a group. However, the model assumes that the number of discovered patches in one time unit never exceeds one. In reality, multiple patches could be found in one time unit. In the present study, we relax this assumption and assumed that the number of discovered patches depends on the producers’ variable encounter rate with patches (λ). We show that strongly depends on λ within a feasible range, although it still depends on the advantage of producer and the number of foragers in a group. The basic idea of PS game is the same as the information sharing (parasitism), because scroungers are also thought to parasitize informations of locations of food patches. Horn (1968) indicated the role of information-parasitism in animal aggregation (Horn, H.S., 1968. The adaptive significance of colonial nesting in the Brewer's blackbird (euphagus cyanocephalus). Ecology 49, 682-646). Our modified PS game model shows the same prediction as the Horn's graphical animal aggregation model; the proportion of scroungers will increase or animals should adopt colonial foraging when resource is spatiotemporally clumped, but scroungers will decrease or animals should adopt territorial foraging if the resource is evenly distributed.     

The ESS under spatial variation with applications to sex allocation

I derive a new approximation which uses the backward Kolmogorov equation to describe evolution when individuals have variable numbers of offspring. This approximation is based on an explicit fixed population size assumption and therefore differs from previous models. I show that for individuals to accept an increase in the variance of offspring number, they must be compensated by an increase in mean offspring number. Based on this model and any given set of feasible alleles, an evolutionary stable strategy (ESS) can be found. Four types of ESS are possible and can be discriminated by graphical methods. These ESS values depend on population size, but population size can be reinterpreted as deme size in a structured population. I adapt this theory to the problem of sex allocation under variable returns to male and female function and derive the ESS sex allocation strategy. I show that allocation to the more variable sexual function should be reduced, but that this effect decreases as population size increases and as variability decreases. These results are compared with results from exact matrix models and computer simulations, all of which show strong congruence.     

The asymmetric war of attrition

The paper, which has an informal discussion at the end, provides a game theoretical analysis of the asymmetric “war of attrition” with incomplete information. This is a contest where animals adopt different roles like “owner” and “intruder” in a territorial conflict, and where the winner is the individual prepared to persist longer. The term incomplete information refers to mistakes in the identification of roles. The idea by Parker & Rubenstein (1981) is mathematically worked out and confirmed that there exists only a single evolutionarily stable strategy (ESS) for the model with a continuum of possible levels of persistence and no discontinuities in the increase of cost during attrition. The ESS prescribes to settle the conflict according to “who has more to gain or less to pay for persistence”. The only evolutionarily stable convention is thus to give the player access to the resource who has the role which is favoured with respect to payoffs. By contrast, it was shown earlier (Hammerstein, 1981) for various asymmetric versions of the “Hawks-Doves” model that an ESS can exist which appears paradoxical with respect to payoffs. The nature of this contrast is further analyzed by introducing elements of discreteness in the asymmetric war of attrition. It turns out that some conditions must be satisfied in order to have the possibility of an alternative ESS which is not of the above simple commonsense type. First, a decision to persist (or escalate) further in a contest must typically commit a contestant to go on fighting for a full “round”, before he can give up without danger. Second, such a “discontinuity” must occur at a level of persistence where the contest is still cheap, and, finally, errors in the identification of roles must be rare.