On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.     

Mutation-selection equilibrium in games with multiple strategies

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of n×n games in the limit of weak selection.     

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n×n payoff matrix. Players can use mixed strategies, which are given by the vector (p₁,…,p_n). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_n. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.     

Active linking in evolutionary games

In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at which they seek new interactions. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In a limiting case, the linking dynamics introduces a simple transformation of the payoff matrix. We outline conditions for evolutionary stability. As a specific example, we study the interaction between cooperators and defectors. We find a simple relationship that characterizes those linking dynamics which allow natural selection to favour cooperation over defection.     

Mutation in evolutionary games can increase average fitness at equilibrium

We study game dynamical interactions between two strategies, A and B, and analyse whether the average fitness of the population at equilibrium can be increased by adding mutation from A to B. Classifying all two by two games with payoff matrix [(a,b),(c,d)], we show that mutation from A to B enhances the average fitness of the whole population (i) if both a and d are less than (b + c)/2 and (ii) if c is less than b. Furthermore, we study conditions for maximizing the productivity of strategy A, and we analyse the effect of mutations in both directions. Depending on the biological system, a mutation in an evolutionary game can be interpreted as a genetic alteration, a cellular differentiation, a change in gene expression, an accidental or deliberate modification in cultural transmission, or a learning error. In a cultural context, our results indicate that the equilibrium payoff of the population can be increased if players sometimes choose the strategy with lower payoff. In a genetic context, we have shown that for frequency-dependent selection mutation can enhance the average fitness of the population at equilibrium.     

Effects of asynchronism on evolutionary games

We analyze the influence of the update dynamics on symmetric 2-player evolutionary games, which are among the most used tools to study the emergence of cooperation in populations of interacting agents. A synchronous dynamics means that, at each time step, all the agents of the population update their strategies simultaneously. An extreme case of asynchronism is sequential dynamics, in which only one agent is updated each time. We first show that these two opposite update dynamics can lead to very different outcomes and that sequential dynamics is detrimental to the emergence of cooperation only when the probability of imitating the most successful neighbors is high. In this sense, we can say that, when the update dynamics has some influence, in general asynchronism is beneficial to the emergence of cooperation. We then explore the consequences of using intermediate levels of asynchronism, where only a fraction of the agents update their behavior each time. In general, the level of cooperation changes smoothly and monotonically as we gradually go from synchronous to sequential dynamics. However, there are some exceptions that should be taken into account. In addition, the results show that the possibility of agents taking irrational decisions has a key role in the sensitivity of these models to changes in the update dynamics. Explanations for the observed behaviors are advanced.     

How can we model selectively neutral density dependence in evolutionary games

The problem of density dependence appears in all approaches to the modelling of population dynamics. It is pertinent to classic models (i.e., Lotka-Volterra's), and also population genetics and game theoretical models related to the replicator dynamics. There is no density dependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the question arises: How is unlimited population growth suppressed in frequency-dependent models? Two categories of solutions can be found in the literature. In the first, replicator dynamics is independent of background fitness. In the second type of solution, a multiplicative suppression coefficient is used, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory and basic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reaches the maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitly consider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium value of population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectories is the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and then remains constant and dependent on the rate of turn-over of individuals.     

Low fertility in humans as the evolutionary outcome of snowballing resource games

For decades, evolutionary biologists and anthropologists havepuzzled over the negative relationship that exists between wealthand fertility in humans. Particularly mystifying have been that(1) humans do not appear to translate their reproductive resourcesinto additional offspring, and (2) attempts to model naturalselection resulting in a negative relationship between amountof economic resources and fertility have all predicted the oppositerelationship. In this article, we use game theory to derivethe evolutionarily stable ratio of offspring investment versusresource generation when the continuing survival of offspringlineages is strongly affected by long-term resource accumulation.The model generates the prediction that fertility should belower when there are more resources available and when moreintensive investment in resource generation has the potentialto acutely increase the survival probability of descendant offspringlineages. This prediction provides a simple and general evolutionaryexplanation for why fertility negatively correlates with wealthboth within and between human populations. Indeed, this mayprovide a new understanding of low fertility in contemporaryhuman groups in addition to furthering our understanding thedemographic transition in general.     

Mutualism and evolutionary multiplayer games: revisiting the Red King

Coevolution of two species is typically thought to favour the evolution of faster evolutionary rates helping a species keep ahead in the Red Queen race, where ‘it takes all the running you can do to stay where you are’. In contrast, if species are in a mutualistic relationship, it was proposed that the Red King effect may act, where it can be beneficial to evolve slower than the mutualistic species. The Red King hypothesis proposes that the species which evolves slower can gain a larger share of the benefits. However, the interactions between the two species may involve multiple individuals. To analyse such a situation, we resort to evolutionary multiplayer games. Even in situations where evolving slower is beneficial in a two-player setting, faster evolution may be favoured in a multiplayer setting. The underlying features of multiplayer games can be crucial for the distribution of benefits. They also suggest a link between the evolution of the rate of evolution and group size.     

Strategic analysis in evolutionary genetics and the theory of games

This paper is written in memory of John Maynard Smith. In a brief survey it discusses essential aspects of how game theory in biology relates to its counterpart in economics, the major transition in game theory initiated by Maynard Smith, the discrepancies between genetic and phenotypic models in evolutionary biology, and a balanced way of reconciling these models. In addition, the paper discusses modern problems in understanding games at the genetic level using the examples of conflict between endosymbionts and their hosts, and the molecular interactions between parasites and the mammalian immune system.     

Evolutionary stable strategies and game dynamics for n-person games

This note contains a generalization of the definition of an evolutionary stable strategy and of the corresponding game dynamics from 2-person to n-person games. This broader framework also allows modelling of several interacting populations or of populations containing different types of individuals, for example males and females.     

The dynamical attainability of ESS in evolutionary games

In this paper, the attainability of ESS of the evolutionary game among n players under the frequency-independent selection is studied by means of a mathematical model describing the dynamical development and a concept of stability (strongly determined stability). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of fitness increasing. It is shown that (1) the ESS solution is not always evolutionarily attainable in the evolutionary dynamics, (2) in the game where the interaction between two species is completely competitive, the Nash solution is always attainable, and (3) one of two species may attain the state of minimum fitness as a result of evolution. The attainability of ESS is also examined in two game models on the sex ratio of wasps and aphids in light of our criterion of the attainability of ESS.     

The evolutionary stability of automimicry

Internal defences such as toxins cannot be detected from a distance by a predator, and are likely to be costly to produce and maintain. Populations of well-defended prey may therefore be vulnerable to invasion from rare 'cheater' mutants that do not produce the toxin themselves but obtain some protection from their resemblance to their better defended conspecifics (automimicry). Although it is well established that well-defended and weakly defended morphs may coexist stably in protected dimorphisms, recent theoretical work suggests that such dimorphisms would not be resistant to invasion by novel mutants with defence levels intermediate to those present. Given that most defences (including toxins) are likely to be continuous traits, this implies that automimicry may tend to be a transitory phenomenon, and thus less likely to explain variation in defence levels in nature. In contrast to this, we show that automimicry can also be evolutionarily stable for continuous traits, and that it may evolve under a wide range of conditions. A recently developed geometric method allows us to determine directly from a trade-off curve whether an evolutionarily stable defence dimorphism is at all possible, and to make some qualitative inferences about the ecological conditions that may favour it.     

Mating games: the evolution of human mating transactions

We propose a new, evolutionary, game-theoretic model of conditionalhuman mating strategies that integrates currently disconnectedbodies of data into a single mathematically-explicit theoryof human mating transactions. The model focuses on the problemof how much resource a male must provide to a female to secureand retain her as a mate. By using bidding-game models, we showhow the male's minimally required resource incentive variesas a function of his own mate value, the value of the female,and the distribution of the mate values of their available alternativemates. The resulting theory parsimoniously accounts for strategicpluralism within the sexes, mate choice differences betweenthe sexes, and assortative mating, while generating a rich setof testable new predictions about human mating behavior.     

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.     

Role of plant abundance in determining the abundance of herbivorous insects

Summary The proposal (Gaston and Lawton 1988a, b) that small species of insects are more abundant because they have lower per capita resource requirements than large species does not hold for aphids (Dixon 1990a). There are good theoretical grounds, supported by empirical data, for the suggestion that host specific aphids that live on uncommon plants incur great losses in finding their host plants and as a consequence have a lower realized r _m and are rarer than aphids living on common plants. This is also likely to apply to other organisms that are host specific and time-limited dispersers.     

Biological auctions with multiple rewards

The competition for resources among cells, individuals or species is a fundamental characteristic of evolution. Biological all-pay auctions have been used to model situations where multiple individuals compete for a single resource. However, in many situations multiple resources with various values exist and single reward auctions are not applicable. We generalize the model to multiple rewards and study the evolution of strategies. In biological all-pay auctions the bid of an individual corresponds to its strategy and is equivalent to its payment in the auction. The decreasingly ordered rewards are distributed according to the decreasingly ordered bids of the participating individuals. The reproductive success of an individual is proportional to its fitness given by the sum of the rewards won minus its payments. Hence, successful bidding strategies spread in the population. We find that the results for the multiple reward case are very different from the single reward case. While the mixed strategy equilibrium in the single reward case with more than two players consists of mostly low-bidding individuals, we show that the equilibrium can convert to many high-bidding individuals and a few low-bidding individuals in the multiple reward case. Some reward values lead to a specialization among the individuals where one subpopulation competes for the rewards and the other subpopulation largely avoids costly competitions. Whether the mixed strategy equilibrium is an evolutionarily stable strategy (ESS) depends on the specific values of the rewards.     

The unstable dynamics of multiple alternative reproductive tactics

Although negative frequency-dependent fitness is argued to allow the stable coexistence of two alternative reproductive types (such as resource defenders and reproductive parasites), no existing theory has considered a third strategy where resource defenders invest differentially in defence against reproductive parasites. Here, we present the results of a three-strategy game, where reproductive parasites interact with two resource defenders: 'Susceptibles' defend more resources but lose more reproductive success to parasites. 'Immunes' lose less to parasites, but immunity carries a reproductive cost. We show that the inclusion of a third strategy dramatically changes the evolutionary dynamics, such that for a wide range of parameter values, our model predicts the continuous sequential invasion of the three strategies instead of stable coexistence. Our results therefore limit the generality of the prediction that frequency-dependent fitness necessarily allows alternative reproductive tactics to coexist at equilibrium and may also explain the observed dynamics of some multiple-strategy systems.     

Oscillatory dynamics in rock-paper-scissors games with mutations

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude.