Frequency-dependent Moran Process Research Articles

Fixation of strategies with the Moran and Fermi processes in evolutionary games

A model of stochastic evolutionary game dynamics with finite population was built. It combines the standard Moran and Fermi rules with two strategies cooperation and defection. We obtain the expressions of fixation probabilities and fixation times. The one-third rule which has been found in the frequency dependent Moran process also holds for our model. We obtain the conditions of strategy being an evolutionarily stable strategy in our model, and then make a comparison with the standard Moran process. Besides, the analytical results show that compared with the standard Moran process, fixation occurs with higher probabilities under a prisoner’s dilemma game and coordination game, but with lower probabilities under a coexistence game. The simulation result shows that the fixation time in our mixed process is lower than that in the standard Fermi process. In comparison with the standard Moran process, fixation always takes more time on average in spatial populations, regardless of the game. In addition, the fixation time decreases with the growth of the number of neighbors.

Extreme Selection Unifies Evolutionary Game Dynamics in Finite and Infinite Populations

We show that when selection is extreme-the fittest strategy always reproduces or is imitated-the unequivalence between the possible evolutionary game scenarios in finite and infinite populations resolves, in the sense that the three generic outcomes-dominance, coexistence, and mutual exclusion-emerge in well-mixed populations of any size. We consider the simplest setting of a 2-player-2-strategy symmetric game and the two most common microscopic definitions of strategy spreading-the frequency-dependent Moran process and the imitation process by pairwise comparison-both in the case allowing any intensity of selection. We show that of the seven different invasion and fixation scenarios that are generically possible in finite populations-fixation being more or less likely to occur and rapid compared to the neutral game-the three that are possible in large populations are the same three that occur for sufficiently strong selection: (1) invasion and fast fixation of one strategy; (2) mutual invasion and slow fixation of one strategy; (3) no invasion and no fixation. Moreover (and interestingly), in the limit of extreme selection 2 becomes mutual invasion and no fixation, a case not possible for finite intensity of selection that better corresponds to the deterministic case of coexistence. In the extreme selection limit, we also derive the large population deterministic limit of the two considered stochastic processes.

Evolution of fairness and coalition formation in three-person ultimatum games

We consider the evolution of fairness and coalition formation in a three-person ultimatum game in which the coalition value depends on its size. Traditional game theory, which assumes selfish and rational players, predicts the largest and efficient coalition with a proposer exploiting most of the total value. In a stochastic evolutionary model (the frequency-dependent Moran process with mutations) where players make errors in estimating the payoffs and strategies of others, evolutionary selection favors the formation of a two-person subcoalition under weak selection and in the low mutation limit if and only if its coalition value exceeds a high proportion (0.7) of that of the largest coalition. Proposers offer 30–35% of the subcoalition value to a coalition member, excluding a non-member. Multilateral bargaining is critically different from the bilateral one. Coalition-forming behavior may cause economic inefficiency and social exclusion. Stochastic evolutionary game theory thus provides theoretical support to explain the behavior of human subjects in economic experiments of a three-person ultimatum game.

Fixation Probabilities of Strategies for Bimatrix Games in Finite Populations

Recent developments in stochastic evolutionary game theory in finite populations yield insights that complement the conventional deterministic evolutionary game theory in infinite populations. However, most studies of stochastic evolutionary game theory have investigated dynamics of symmetric games, although not all social and biological phenomena are described by symmetric games, e.g., social interactions between individuals having conflicting preferences or different roles. In this paper, we describe the stochastic evolutionary dynamics of two-player \(2 \times 2\) bimatrix games in finite populations. The stochastic process is modeled by a frequency-dependent Moran process without mutation. We obtained the fixation probability that the evolutionary dynamics starting from a given initial state converges to a specific absorbing state. Applying the formula to the ultimatum game, we show that evolutionary dynamics favors fairness. Furthermore, we present two novel concepts of stability for bimatrix games, based on our formula for the fixation probability, and demonstrate that one of the two serves as a criterion for equilibrium selection.

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth-death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

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 alpha and beta. alpha-players and beta-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.

Deterministic evolutionary game dynamics in finite populations

Evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals. Typically, the microscopic definition of strategy spreading is stochastic such that the dynamics becomes deterministic only in infinitely large populations. Here, we present a microscopic birth-death process that has a fully deterministic strong selection limit in well-mixed populations of any size. Additionally, under weak selection, from this process the frequency-dependent Moran process is recovered. This makes it a natural extension of the usual evolutionary dynamics under weak selection. We find simple expressions for the fixation probabilities and average fixation times of the process in evolutionary games with two players and two strategies. For cyclic games with two players and three strategies, we show that the resulting deterministic dynamics crucially depends on the initial condition in a nontrivial way.

Drift reversal in asymmetric coevolutionary conflicts: influence of microscopic processes and population size

The coevolutionary dynamics in finite populations currently is investigated in a wide range of disciplines, as chemical catalysis, biological evolution, social and economic systems. The dynamics of those systems can be formulated within the unifying framework of evolutionary game theory. However it is not a priori clear which mathematical description is appropriate when populations are not infinitely large. Whereas the replicator equation approach describes the infinite population size limit by deterministic differential equations, in finite populations the dynamics is inherently stochastic which can lead to new effects. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection in finite populations based on microscopic processes. In asymmetric conflicts between two populations with a cyclic dominance, a finite-size dependent drift reversal was demonstrated, depending on the underlying microscopic process of the evolutionary update. Cyclic dynamics appears widely in biological coevolution, be it within a homogeneous population, or be it between disjunct populations as female and male. Here explicit analytic address is given and the average drift is calculated for the frequency-dependent Moran process and for different pairwise comparison processes. It is explicitely shown that the drift reversal cannot occur if the process relies on payoff differences between pairs of individuals. Further, also a linear comparison with the average payoff does not lead to a drift towards the internal fixed point. Hence the nonlinear comparison function of the frequency-dependent Moran process, together with its usage of nonlocal information via the average payoff, is the essential part of the mechanism.

Monotone Imitation Dynamics in Large Populations

We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [3] and the related processes proposed by Benaim and Weibull [2] and Traulsen et al. [24], as well as the frequency-dependent Moran process studied by Fudenberg et al. [15]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.

Monotone imitation dynamics in large populations

We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benaı¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.

Finite populations choose an optimal language

This paper studies the evolution of a proto-language in a finite population under the frequency-dependent Moran process. A proto-language can be seen as a collection of concept-to-sign mappings. An efficient proto-language is a bijective mapping from objects of communication to used signs and vice versa. Based on the comparison of fixation probabilities, a method for deriving conditions of evolutionary stability in a finite population [Nowak et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 246–650], it is shown that efficient proto-languages are the only strategies that are protected by selection, which means that no mutant strategy can have a fixation probability that is greater than the inverse population size. In passing, the paper provides interesting results about the comparison of fixation probabilities as well as Maynard Smith's notion of evolutionary stability for finite populations [Maynard Smith, 1988. Can a mixed strategy be stable in a finite population? J. Theor. Biol. 130, 247–251] that are generally true for games with a symmetric payoff function.

Pairwise comparison and selection temperature in evolutionary game dynamics

Recently, the frequency-dependent Moran process has been introduced in order to describe evolutionary game dynamics in finite populations. Here, an alternative to this process is investigated that is based on pairwise comparison between two individuals. We follow a long tradition in the physics community and introduce a temperature (of selection) to account for stochastic effects. We calculate the fixation probabilities and fixation times for any symmetric 2 × 2 game, for any intensity of selection and any initial number of mutants. The temperature can be used to gauge continuously from neutral drift to the extreme selection intensity known as imitation dynamics. For some payoff matrices the distribution of fixation times can become so broad that the average value is no longer very meaningful.

Stochasticity and evolutionary stability

In stochastic dynamical systems, different concepts of stability can be obtained in different limits. A particularly interesting example is evolutionary game theory, which is traditionally based on infinite populations, where strict Nash equilibria correspond to stable fixed points that are always evolutionarily stable. However, in finite populations stochastic effects can drive the system away from strict Nash equilibria, which gives rise to a new concept for evolutionary stability. The conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations. We show that the two concepts can be derived from the frequency dependent Moran process in different limits. Our results help to determine the appropriate stability concept in large finite populations. The general validity of our findings is demonstrated showing that the same results are valid employing vastly different co-evolutionary processes.

Coevolutionary dynamics in large, but finite populations

Coevolving and competing species or game-theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection with two strategies in finite populations based on microscopic processes [A. Traulsen, J. C. Claussen, and C. Hauert, Phys. Rev. Lett. 95, 238701 (2005)]. Here we generalize this approach in a twofold way: First, we extend the framework to an arbitrary number of strategies and second, we allow for mutations in the evolutionary process. The deterministic limit of infinite population size of the frequency-dependent Moran process yields the adjusted replicator-mutator equation, which describes the combined effect of selection and mutation. For finite populations, we provide an extension taking random drift into account. In the limit of neutral selection, i.e., whenever the process is determined by random drift and mutations, the stationary strategy distribution is derived. This distribution forms the background for the coevolutionary process. In particular, a critical mutation rate uc is obtained separating two scenarios: above uc the population predominantly consists of a mixture of strategies whereas below uc the population tends to be in homogeneous states. For one of the fundamental problems in evolutionary biology, the evolution of cooperation under Darwinian selection, we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes. This approach complements simulation results and provides a deeper, systematic understanding of coevolutionary dynamics.