Evolutionary game dynamics in finite populations with strong selection and weak mutation

We study stochastic game dynamics in finite populations. To this end we extend the classical Moran process to incorporate frequency-dependent selection and mutation. For 2 x 2 games, we give a complete analysis of the long-run behavior when mutation rates are small. For 3 x 3 coordination games, we provide a simple rule to determine which strategy will be selected in large populations. The expected motion in our model resembles the standard replicator dynamics when the population is large, but is qualitatively different when the population is small. Our analysis shows that even in large finite populations the behavior of a replicator-like system can be different from that of the standard replicator dynamics. As an application, we consider selective language dynamics. We determine which language will be spoken in finite large populations. The results have an intuitive interpretation but would not be expected from an analysis of the replicator dynamics.     

Unpredictability induced by unfocused games in evolutionary game dynamics

Evolutionary game theory is a basis of replicator systems and has applications ranging from animal behavior and human language to ecosystems and other hierarchical network systems. Most studies in evolutionary game dynamics have focused on a single game, but, in many situations, we see that many games are played simultaneously. We construct a replicator equation with plural games by assuming that a reward of a player is a simple summation of the reward of each game. Even if the numbers of the strategies of the games are different, its dynamics can be described in one replicator equation. We here show that when players play several games at the same time, the fate of a single game cannot be determined without knowing the structures of the whole other games. The most absorbing fact is that even if a single game has a ESS (evolutionary stable strategy), the relative frequencies of strategies in the game does not always converge to the ESS point when other games are played simultaneously.     

An Application of Evolutionary Game Theory to Social Dilemmas: The Traveler's Dilemma and the Minimum Effort Coordination Game

The Traveler''s Dilemma game and the Minimum Effort Coordination game are two social dilemmas that have attracted considerable attention due to the fact that the predictions of classical game theory are at odds with the results found when the games are studied experimentally. Moreover, a direct application of deterministic evolutionary game theory, as embodied in the replicator dynamics, to these games does not explain the observed behavior. In this work, we formulate natural variants of these two games as smoothed continuous-strategy games. We study the evolutionary dynamics of these continuous-strategy games, both analytically and through agent-based simulations, and show that the behavior predicted theoretically is in accord with that observed experimentally. Thus, these variants of the Traveler''s Dilemma and the Minimum Effort Coordination games provide a simple resolution of the paradoxical behavior associated with the original games.     

Local Replicator Dynamics: A Simple Link Between Deterministic and Stochastic Models of Evolutionary Game Theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner’s dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.     

Equilibrium selection in evolutionary games with random matching of players

We discuss stochastic dynamics of populations of individuals playing games. Our models possess two evolutionarily stable strategies: an efficient one, where a population is in a state with the maximal payoff (fitness) and a risk-dominant one, where players are averse to risks. We assume that individuals play with randomly chosen opponents (they do not play against average strategies as in the standard replicator dynamics). We show that the long-run behavior of a population depends on its size and the mutation level.     

The replicator dynamics with n players and population structure

The well-known replicator dynamics is usually applied to 2-player games and random matching. Here we allow for games with n players, and for population structures other than random matching. This more general application leads to a version of the replicator dynamics of which the standard 2-player, well-mixed version is a special case, and which allows us to explore the dynamic implications of population structure. The replicator dynamics also allows for a reformulation of the central theorem in Van Veelen (2009), which claims that inclusive fitness gives the correct prediction for games with generalized equal gains from switching (or, in other words, when fitness effects are additive). If we furthermore also assume that relatedness is constant during selection - which is a reasonable assumption in a setting with kin recognition - then inclusive fitness even becomes a parameter that determines the speed as well as the direction of selection. For games with unequal gains from switching, inclusive fitness can give the wrong prediction. With equal gains however, not only the sign, but also even the value of inclusive fitness becomes meaningful.     

Evolutionary stability conditions for signaling games with costly signals

The paper investigates the class of signaling games with the following properties: (a) the interests of sender and receiver coincide, (b) different signals incur differential costs, and (c) different events (meanings/types) have different probabilities. Necessary and sufficient conditions are presented for a profile to be evolutionarily stable and neutrally stable, and for a set of profiles to be an evolutionarily stable set.The main finding is that a profile belongs to some evolutionarily stable set if and only if a maximal number of events can be reliably communicated. Furthermore, it is shown that under the replicator dynamics, a set of states with a positive measure is attracted to “sub-optimal” equilibria that do not belong to any asymptotically stable set.     

Evolutionary dynamics of finite populations in games with polymorphic fitness equilibria

The hawk-dove (HD) game, as defined by Maynard Smith [1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge], allows for a polymorphic fitness equilibrium (PFE) to exist between its two pure strategies; this polymorphism is the attractor of the standard replicator dynamics [Taylor, P.D., Jonker, L., 1978. Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145-156; Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge] operating on an infinite population of pure-strategists. Here, we consider stochastic replicator dynamics, operating on a finite population of pure-strategists playing games similar to HD; in particular, we examine the transient behavior of the system, before it enters an absorbing state due to sampling error. Though stochastic replication prevents the population from fixing onto the PFE, selection always favors the under-represented strategy. Thus, we may naively expect that the mean population state (of the pre-absorption transient) will correspond to the PFE. The empirical results of Fogel et al. [1997. On the instability of evolutionary stable states. BioSystems 44, 135-152] show that the mean population state, in fact, deviates from the PFE with statistical significance. We provide theoretical results that explain their observations. We show that such deviation away from the PFE occurs when the selection pressures that surround the fitness-equilibrium point are asymmetric. Further, we analyze a Markov model to prove that a finite population will generate a distribution over population states that equilibrates selection-pressure asymmetry; the mean of this distribution is generally not the fitness-equilibrium state.     

Evolutionary games on networks and payoff invariance under replicator dynamics

The commonly used accumulated payoff scheme is not invariant with respect to shifts of payoff values when applied locally in degree-inhomogeneous population structures. We propose a suitably modified payoff scheme and we show both formally and by numerical simulation, that it leaves the replicator dynamics invariant with respect to affine transformations of the game payoff matrix. We then show empirically that, using the modified payoff scheme, an interesting amount of cooperation can be reached in three paradigmatic non-cooperative two-person games in populations that are structured according to graphs that have a marked degree inhomogeneity, similar to actual graphs found in society. The three games are the Prisoner’s Dilemma, the Hawks-Doves and the Stag-Hunt. This confirms previous important observations that, under certain conditions, cooperation may emerge in such network-structured populations, even though standard replicator dynamics for mixing populations prescribes equilibria in which cooperation is totally absent in the Prisoner’s Dilemma, and it is less widespread in the other two games.     

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

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

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.     

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.     

Autocatalytic networks with translation

We consider the kinetics of an autocatalytic reaction network in which replication and catalytic actions are separated by a translation step. We find that the behaviour of such a system is closely related to second-order replicator equations, which describe the kinetics of autocatalytic reaction networks in which the replicators act also as catalysts. In fact, the qualitative dynamics seems to be described almost entirely be the second-order reaction rates of the replication step. For two species we recover the qualitative dynamics of the replicator equations. Larger networks show some deviations, however. A hypercyclic system consisting of three interacting species can converge toward a stable limit cycle in contrast to the replicator equation case. A singular perturbation analysis shows that the replication-translation system reduces to a second-order replicator equation if translation is fast. The influence of mutations on replication-translation networks is also very similar to the behavior of selection-mutation equations.     

Thalamocortical dynamics of sleep: roles of purinergic neuromodulation

Thalamocortical dynamics, the millisecond to second changes in activity of thalamocortical circuits, are central to perception, action and cognition. Generated by local circuitry and sculpted by neuromodulatory systems, these dynamics reflect the expression of vigilance states. In sleep, thalamocortical dynamics are thought to mediate "offline" functions including memory consolidation and synaptic scaling. Here, I discuss thalamocortical sleep dynamics and their modulation by the ascending arousal system and locally released neurochemicals. I focus on modulation of these dynamics by electrically silent astrocytes, highlighting the role of purinergic signaling in this glial form of communication. Astrocytes modulate cortical slow oscillations, sleep behavior, and sleep-dependent cognitive function. The discovery that astrocytes can modulate sleep dynamics and sleep-related behaviors suggests a new way of thinking about the brain, in which integrated circuits of neurons and glia control information processing and behavioral output.     

Coevolution of trustful buyers and cooperative sellers in the trust game

Many online marketplaces enjoy great success. Buyers and sellers in successful markets carry out cooperative transactions even if they do not know each other in advance and a moral hazard exists. An indispensable component that enables cooperation in such social dilemma situations is the reputation system. Under the reputation system, a buyer can avoid transacting with a seller with a bad reputation. A transaction in online marketplaces is better modeled by the trust game than other social dilemma games, including the donation game and the prisoner's dilemma. In addition, most individuals participate mostly as buyers or sellers; each individual does not play the two roles with equal probability. Although the reputation mechanism is known to be able to remove the moral hazard in games with asymmetric roles, competition between different strategies and population dynamics of such a game are not sufficiently understood. On the other hand, existing models of reputation-based cooperation, also known as indirect reciprocity, are based on the symmetric donation game. We analyze the trust game with two fixed roles, where trustees (i.e., sellers) but not investors (i.e., buyers) possess reputation scores. We study the equilibria and the replicator dynamics of the game. We show that the reputation mechanism enables cooperation between unacquainted buyers and sellers under fairly generous conditions, even when such a cooperative equilibrium coexists with an asocial equilibrium in which buyers do not buy and sellers cheat. In addition, we show that not many buyers may care about the seller's reputation under cooperative equilibrium. Buyers' trusting behavior and sellers' reputation-driven cooperative behavior coevolve to alleviate the social dilemma.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Natural selection and social preferences

A large number of individuals are randomly matched into groups, where each group plays a finite symmetric game. Individuals breed true. The expected number of surviving offspring depends on own material payoff, but may also, due to cooperative breeding and/or reproductive competition, depend on the material payoffs to other group members. The induced population dynamic is equivalent with the replicator dynamic for a game with payoffs derived from those in the original game. We apply this selection dynamic to a number of examples, including prisoners' dilemma games with and without a punishment option, coordination games, and hawk-dove games. For each of these, we compare the outcomes with those obtained under the standard replicator dynamic. By way of a revealed-preference argument, our selection dynamic can explain certain "altruistic" and "spiteful" behaviors that are consistent with individuals having social preferences.     

Differences in dynamics between discrete strategies and continuous strategies in a multi-player game with a linear payoff structure

A deductive analysis concerning replicator dynamics proved that a continuous strategy game (in which a player chooses an arbitrary real number between [0, 1] as a cooperative fraction) has the same equilibrium as a discrete strategy game (in which a player chooses only C or D), which has the same linear payoff structure as a continuous strategy game. The deduction is shown for two-player and multi-player games.