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.     

Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game

Understanding the mechanisms that can lead to the evolution of cooperation through natural selection is a core problem in biology. Among the various attempts at constructing a theory of cooperation, game theory has played a central role. Here, we review models of cooperation that are based on two simple games: the Prisoner's Dilemma, and the Snowdrift game. Both games are two‐person games with two strategies, to cooperate and to defect, and both games are social dilemmas. In social dilemmas, cooperation is prone to exploitation by defectors, and the average payoff in populations at evolutionary equilibrium is lower than it would be in populations consisting of only cooperators. The difference between the games is that cooperation is not maintained in the Prisoner's Dilemma, but persists in the Snowdrift game at an intermediate frequency. As a consequence, insights gained from studying extensions of the two games differ substantially. We review the most salient results obtained from extensions such as iteration, spatial structure, continuously variable cooperative investments, and multi‐person interactions. Bridging the gap between theoretical and empirical research is one of the main challenges for future studies of cooperation, and we conclude by pointing out a number of promising natural systems in which the theory can be tested experimentally.     

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.     

Stability in negotiation games and the emergence of cooperation

Consider a two-player game in which each player contributes a costly resource to the common good of the pair. For such contests, the Nash equilibrium contribution, x*, is one for which neither player can increase its pay-off by unilaterally altering its contribution from x*. We study an elaboration of this game, which allows the players to exchange x-offers back and forth in a negotiation phase until they converge to a final pair of contributions, x1 and x2. A significant feature of such negotiation games, hitherto unrecognized, is the existence of a set of neutrally stable equilibrium points in negotiation phase space. To explore the long-term evolutionary outcome of such games, we simulate populations containing various mixtures of negotiation strategies and, contrary to previous results, we often find convergence to a contribution that is more cooperative than the Nash equilibrium. Mathematical analysis suggests why this might be happening, and provides a novel and robust explanation for cooperation, that negotiation can facilitate the evolution of cooperative behaviour.     

Extrapolating Weak Selection in Evolutionary Games

In evolutionary games, reproductive success is determined by payoffs. Weak selection means that even large differences in game outcomes translate into small fitness differences. Many results have been derived using weak selection approximations, in which perturbation analysis facilitates the derivation of analytical results. Here, we ask whether results derived under weak selection are also qualitatively valid for intermediate and strong selection. By “qualitatively valid” we mean that the ranking of strategies induced by an evolutionary process does not change when the intensity of selection increases. For two-strategy games, we show that the ranking obtained under weak selection cannot be carried over to higher selection intensity if the number of players exceeds two. For games with three (or more) strategies, previous examples for multiplayer games have shown that the ranking of strategies can change with the intensity of selection. In particular, rank changes imply that the most abundant strategy at one intensity of selection can become the least abundant for another. We show that this applies already to pairwise interactions for a broad class of evolutionary processes. Even when both weak and strong selection limits lead to consistent predictions, rank changes can occur for intermediate intensities of selection. To analyze how common such games are, we show numerically that for randomly drawn two-player games with three or more strategies, rank changes frequently occur and their likelihood increases rapidly with the number of strategies . In particular, rank changes are almost certain for , which jeopardizes the predictive power of results derived for weak selection.     

Stochastic evolutionary dynamics of bimatrix games

Evolutionary game dynamics of two-player asymmetric games in finite populations is studied. We consider two roles in the game, roles α and β. α-players and β-players interact and gain payoffs. The game is described by a pair of matrices, which is called bimatrix. One's payoff in the game is interpreted as its fecundity, thus strategies are subject to natural selection. In addition, strategies can randomly mutate to others. We formulate a stochastic evolutionary game dynamics of bimatrix games as a frequency-dependent Moran process with mutation. We analytically derive the stationary distribution of strategies under weak selection. Our result provides a criterion for equilibrium selection in general bimatrix games.     

For whom is it more beneficial to stop interactions with defectors: Cooperators or defectors?

Cooperation is a mysterious evolutionary phenomenon and its mechanisms require elucidation. When cooperators can stop interactions with defectors, the evolution of cooperation becomes possible; this is one mechanism that facilitates the evolution of cooperation. Here, stopping interactions with defectors is beneficial not only for cooperators but also for defectors. The question then arises, for whom is stopping interactions with defectors more beneficial: cooperators or defectors? By utilizing evolutionary game theory, I addressed this question using a two-player game involving four strategies: (1) cooperators who stop the interaction if the current partner is a defector, (2) cooperators who attempt to maintain a relationship with anyone, (3) defectors who stop the interaction if the current partner is a defector, and (4) defectors who attempt to maintain a relationship with anyone. Our results show that, at equilibrium, the ratio of cooperators who stop the interaction if the current partner is a defector to cooperators who attempt to maintain a relationship with anyone is larger than the ratio of defectors who stop the interaction if the current partner is a defector to defectors who attempt to maintain a relationship with anyone. Thus, cooperators rather than defectors are more likely to stop interactions with defectors at equilibrium. This result is consistent with a previous experimental study in which a positive correlation was detected between the degree of individuals’ cooperativeness and how accurately the individuals recognize whether other individuals are cooperators or defectors. Thus, the theoretical work presented in this study provides relevant insights into the natural phenomena of cooperation and recognition.     

Nash Equilibria in Multi-Agent Motor Interactions

Social interactions in classic cognitive games like the ultimatum game or the prisoner''s dilemma typically lead to Nash equilibria when multiple competitive decision makers with perfect knowledge select optimal strategies. However, in evolutionary game theory it has been shown that Nash equilibria can also arise as attractors in dynamical systems that can describe, for example, the population dynamics of microorganisms. Similar to such evolutionary dynamics, we find that Nash equilibria arise naturally in motor interactions in which players vie for control and try to minimize effort. When confronted with sensorimotor interaction tasks that correspond to the classical prisoner''s dilemma and the rope-pulling game, two-player motor interactions led predominantly to Nash solutions. In contrast, when a single player took both roles, playing the sensorimotor game bimanually, cooperative solutions were found. Our methodology opens up a new avenue for the study of human motor interactions within a game theoretic framework, suggesting that the coupling of motor systems can lead to game theoretic solutions.     

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.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

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.     

Comparative statics of games between relatives

According to Hamilton's theory of kin selection, species tend to evolve behavior such that each organism appears to be attempting to maximize its inclusive fitness. In particular, two neighbors are likely to help each other if the cost of doing so is less than the benefit multiplied by r, their coefficient of relatedness. Since the latter is less than unity, mutual altruism benefits both neighbors. However, is it theoretically possible that acting so as to maximize the inclusive, rather than personal, fitness may harm both parties. This may occur in strategic symmetric pairwise interactions (more specifically, nxn games), in which the outcome depends on both sides' actions. In this case, the equilibrium outcome may be less favorable to the interactants' personal fitness than if each of them acted so as to maximize the latter. This paper shows, however, that such negative effect of relatedness on fitness is incompatible with evolutionary stability. If the symmetric equilibrium strategies are evolutionarily stable, a higher coefficient of relatedness can only entail higher personal fitness for the two neighbors. This suggests that negative comparative statics as above are not likely to occur in nature.     

Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

Stochastic stability in three-player games

Animal behavior and evolution can often be described by game-theoretic models. Although in many situations the number of players is very large, their strategic interactions are usually decomposed into a sum of two-player games. Only recently were evolutionarily stable strategies defined for multi-player games and their properties analyzed [Broom, M., Cannings, C., Vickers, G.T., 1997. Multi-player matrix games. Bull. Math. Biol. 59, 931–952]. Here we study the long-run behavior of stochastic dynamics of populations of randomly matched individuals playing symmetric three-player games. We analyze the stochastic stability of equilibria in games with multiple evolutionarily stable strategies. We also show that, in some games, a population may not evolve in the long run to an evolutionarily stable equilibrium.     

On Nash Equilibrium and Evolutionarily Stable States That Are Not Characterised by the Folk Theorem

In evolutionary game theory, evolutionarily stable states are characterised by the folk theorem because exact solutions to the replicator equation are difficult to obtain. It is generally assumed that the folk theorem, which is the fundamental theory for non-cooperative games, defines all Nash equilibria in infinitely repeated games. Here, we prove that Nash equilibria that are not characterised by the folk theorem do exist. By adopting specific reactive strategies, a group of players can be better off by coordinating their actions in repeated games. We call it a type-k equilibrium when a group of k players coordinate their actions and they have no incentive to deviate from their strategies simultaneously. The existence and stability of the type-k equilibrium in general games is discussed. This study shows that the sets of Nash equilibria and evolutionarily stable states have greater cardinality than classic game theory has predicted in many repeated games.     

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.     

Evolutionarily stable strategy and invader strategy in matrix games

In this paper, we consider the concepts of evolutionarily stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in single species with frequency-dependent interactions. We find some general relationships among the three concepts in matrix games. The main conclusion is that ESS and NIS are equivalent to each other and are both equivalent to local superiority; a strategy with global superiority must be a GIS; a GIS may not be equivalent to its global superiority in games with more than two players; and in any two-player matrix game a GIS is just equivalent to its global superiority. In two-player games, globally asymptotic stability in the replicator dynamics has also been shown. Equivalent conditions for the three concepts stated by payoff comparisons are given and are applied to examples involved.     

Evolutionary games and population dynamics: maintenance of cooperation in public goods games

The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behaviour seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the Prisoner's Dilemma and public goods games. Traditional approaches to studying the problem of cooperation assume constant population sizes and thus neglect the ecology of the interacting individuals. Here, we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation. In public goods games, cooperation can gain a foothold if the population density depends on the average population payoff. Decreasing population densities, due to defection leading to small payoffs, results in smaller interaction group sizes in which cooperation can be favoured. This feedback between ecological dynamics and game dynamics can generate stable coexistence of cooperators and defectors in public goods games. However, this mechanism fails for pairwise Prisoner's Dilemma interactions and the population is driven to extinction. Our model represents natural extension of replicator dynamics to populations of varying densities.     

Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept

This paper contains the basic extensions of classical evolutionary games (multipopulation and density dependent models). It is shown that classical bimatrix approach is inconsistent with other approaches because it does not depend on proportion between populations. The main conclusion is that interspecific proportion parameter is important and must be considered in multipopulation models. The paper provides a synthesis of both extensions (a metasimplex concept) which solves the problem intrinsic in the bimatrix model. It allows us to model interactions among any number of subpopulations including density dependence effects. We prove that all modern approaches to evolutionary games are closely related. All evolutionary models (except classical bimatrix approaches) can be reduced to a single population general model by a simple change of variables. Differences between classic bimatrix evolutionary games and a new model which is dependent on interspecific proportion are shown by examples.     

Spectacular phenomena and limits to rationality in genetic and cultural evolution

In studies of both animal and human behaviour, game theory is used as a tool for understanding strategies that appear in interactions between individuals. Game theory focuses on adaptive behaviour, which can be attained only at evolutionary equilibrium. We suggest that behaviour appearing during interactions is often outside the scope of such analysis. In many types of interaction, conflicts of interest exist between players, fuelling the evolution of manipulative strategies. Such strategies evolve out of equilibrium, commonly appearing as spectacular morphology or behaviour with obscure meaning, to which other players may react in non-adaptive, irrational ways. We present a simple model to show some limitations of the game-theory approach, and outline the conditions in which evolutionary equilibria cannot be maintained. Evidence from studies of biological interactions seems to support the view that behaviour is often not at equilibrium. This also appears to be the case for many human cultural traits, which have spread rapidly despite the fact that they have a negative influence on reproduction.