Game Dynamics In Finite Populations Research Articles

The path integral formula for the stochastic evolutionary game dynamics

Although the long-term behavior of stochastic evolutionary game dynamics in finite populations has been fully investigated, its evolutionary characteristics in a limited period of time is still unclear. In order to answer this question, we introduce the formulation of the path integral approach for evolutionary game theory. In this framework, the transition probability is the sum of all the evolutionary paths. The path integral formula of the transition probability is expected to be a new mathematical tool to explore the stochastic game evolutionary dynamics. As an example, we derive the transition probability for stochastic evolutionary game dynamics by the path integral in a limited period of time with the updating rule of the Wright-Fisher process.

An introduction to ABED: Agent-based simulation of evolutionary game dynamics

ABED is free and open-source software for simulating evolutionary game dynamics in finite populations. We explain how ABED can be used to simulate a wide range of dynamics considered in the literature and many novel dynamics. In doing so, we introduce a general model of revisions for dynamic evolutionary models, one that decomposes strategy updates into selection of candidate strategies, payoff determination, and choice among candidates. Using examples, we explore ways in which simulations can complement theory in increasing our understanding of strategic interactions in finite populations.

Evolutionary game dynamics of combining the imitation and aspiration-driven update rules

So far, most studies on evolutionary game dynamics in finite populations have concentrated on a single update rule. However, given the impacts of the environment and individual cognition, individuals may use different update rules to change their current strategies. In light of this, the current paper reports on a study that constructed a mixed stochastic evolutionary game dynamic by combining the imitation and aspiration-driven update processes. The target was to clarify the influences of the aspiration-driven process on the evolution of the level of cooperation by considering the behavior of a population in which individuals have two strategies available: cooperation and defection. Through a numerical analysis of unstructured populations and simulation analyses of structured populations and of the random-matching model, the following results were found. First, the mean fraction of cooperators varied alongside the probability with which the individual adopted the aspiration-driven update rule. In the Prisoner's Dilemma and coexistence games, the aspiration-driven update process promoted cooperation in the well-mixed population but inhibited it in structured ones and the random-matching model; however, in the coordination game, the aspiration-driven update process was seen to exert the opposite effect on cooperation by inhibiting the latter in a homogeneously mixed population but promoting it in structured ones and in the random-matching model. Second, the mean fraction of cooperators changed with the aspiration level in the differently structured populations and random-matching model, and there appeared a phase transition point. Third, the evolutionary characteristics of the mean fraction of cooperators maintained robustness in the differently structured populations and random-matching model. These results extend evolutionary game theory.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

Evolutionary game dynamics of the Wright-Fisher process with different selection intensities

Evolutionary game dynamics in finite populations can be described by a frequency-dependent, stochastic Wright-Fisher process. The fitness of individuals in a population is not only linked to environmental conditions but also tightly coupled to the types and frequencies of competitors, leading to different types of individuals with different selection intensities. We studied a 2 × 2 symmetric game in a finite population and established a dynamic model of the Wright-Fisher process by introducing different selection intensities for different strategies. Thus, we provided another effective way to study the evolutionary dynamics of a finite population and obtained the analytical expressions of fixation probabilities under weak selection. The fixation probability of a strategy is not only related to a game matrix but also to different selection intensities. The conditions required for natural selection to favor one strategy and for that strategy to be an evolutionary stable strategy (ESSN) are specified in our model. We compared our results with those of a Moran dynamic process with different selection intensities to explore these two processes better. In the two processes, the conditions conducive to the strategy's taking fixation are the same. By simulation analysis, the dynamic relationships between the fixation probabilities and selection intensities were intuitively observed in the prisoner's dilemma, coordination, and coexistence games. The fixation probability of the cooperative strategy in the prisoner's dilemma decreases with the increase of its own selection intensity. In the coexistence and coordination games, the fixation probability of the cooperative strategy increases with its own selection intensity. For the three types of games, the fixation probability of the cooperative strategy decreases with the increase of the selection intensity of the defection strategy.

Fitness-based models and pairwise comparison models of evolutionary games are typically different—even in unstructured populations

The modeling of evolutionary game dynamics in finite populations requires microscopic processes that determine how strategies spread. The exact details of these processes are often chosen without much further consideration. Different types of microscopic models, including in particular fitness-based selection rules and pairwise comparison dynamics, are often used as if they were interchangeable. We challenge this view and investigate how robust these choices on the micro-level really are. We focus on a key macroscopic quantity, the probability for a single mutant to take over a population of wild-type individuals. We show that even in unstructured populations there is only one pair of a fitness-based process and a pairwise comparison process leading to identical outcomes for arbitrary games and for all intensities of selection. This strong restriction is not relaxed even when the class of pairwise comparison processes is broadened. This highlights the perils of making arbitrary choices at the micro-level without regard of the consequences at the macro-level.

The structure of mutations and the evolution of cooperation.

Evolutionary game dynamics in finite populations assumes that all mutations are equally likely, i.e., if there are strategies a single mutation can result in any strategy with probability . However, in biological systems it seems natural that not all mutations can arise from a given state. Certain mutations may be far away, or even be unreachable given the current composition of an evolving population. These distances between strategies (or genotypes) define a topology of mutations that so far has been neglected in evolutionary game theory. In this paper we re-evaluate classic results in the evolution of cooperation departing from the assumption of uniform mutations. We examine two cases: the evolution of reciprocal strategies in a repeated prisoner's dilemma, and the evolution of altruistic punishment in a public goods game. In both cases, alternative but reasonable mutation kernels shift known results in the direction of less cooperation. We therefore show that assuming uniform mutations has a substantial impact on the fate of an evolving population. Our results call for a reassessment of the “model-less” approach to mutations in evolutionary dynamics.

The [formula omitted] law of evolutionary dynamics in community-structured population

Evolutionary game dynamics in finite populations provide a new framework to understand the selection of traits with frequency-dependent fitness. Recently, a simple but fundamental law of evolutionary dynamics, which we call σ law, describes how to determine the selection between two competing strategies: in most evolutionary processes with two strategies, A and B, strategy A is favored over B in weak selection if and only if σR+S>T+σP. This relationship holds for a wide variety of structured populations with mutation rate and weak selection under certain assumptions. In this paper, we propose a model of games based on a community-structured population and revisit this law under the Moran process. By calculating the average payoffs of A and B individuals with the method of effective sojourn time, we find that σ features not only the structured population characteristics, but also the reaction rate between individuals. That is to say, an interaction between two individuals are not uniform, and we can take σ as a reaction rate between any two individuals with the same strategy. We verify this viewpoint by the modified replicator equation with non-uniform interaction rates in a simplified version of the prisoner's dilemma game (PDG).

Imitation, internal absorption and the reversal of local drift in stochastic evolutionary games

AbstractEvolutionary game dynamics in finite populations is typically subject to noise, inducing effects which are not present in deterministic systems, including fixation and extinction. In the first part of this paper we investigate the phenomenon of drift reversal in finite populations, taking into account that drift is a local quantity in strategy space. Secondly, we study a simple imitation dynamics, and show that it can lead to fixation at internal mixed-strategy fixed points even in finite populations. Imitation in infinite populations is adequately described by conventional replicator dynamics, and these equations are known to have internal fixed points. Internal absorption in finite populations on the other hand is a novel dynamic phenomenon. Due to an outward drift in finite populations this type of dynamic arrest is not found in other commonly studied microscopic dynamics, not even in those with the same deterministic replicator limit as imitation.

Effects of migration on the evolutionary game dynamics in finite populations with community structures

We investigate the impacts of migration on the evolutionary game dynamics in finite populations with community structures in the framework of evolutionary game theory. In contrast to deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Based on the analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that it is most likely to find the population in the community where all individuals have the lower migration rate. Furthermore, we show that reducing the difference between the migration rates of distinct communities can increase the first hitting time to the homogeneous absorbing state and can prolong the coexistence time of different species, promoting the conservation of biodiversity.

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.

Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations

Evolutionary game dynamics of two strategies in finite population is studied by continuous probabilistic approach. Besides frequency dependent selection, mutation was also included in this study. The equilibrium probability density functions of abundance, expected time to extinction or fixation were derived and their numerical solutions are calculated as illustrations. Meanwhile, individual-based computer simulations are also done. A comparison reveals the consistency between theoretical analysis and simulations.

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.

TRANSFORMING THE DILEMMA

How does natural selection lead to cooperation between competing individuals? The Prisoner's Dilemma captures the essence of this problem. Two players can either cooperate or defect. The payoff for mutual cooperation, R, is greater than the payoff for mutual defection, P. But a defector versus a cooperator receives the highest payoff, T, where as the cooperator obtains the lowest payoff, S. Hence, the Prisoner's Dilemma is defined by the payoff ranking T > R > P > S. In a well-mixed population, defectors always have a higher expected payoff than cooperators, and therefore natural selection favors defectors. The evolution of cooperation requires specific mechanisms. Here we discuss five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity (or graph selection). Each mechanism leads to a transformation of the Prisoner's Dilemma payoff matrix. From the transformed matrices, we derive the fundamental conditions for the evolution of cooperation. The transformed matrices can be used in standard frameworks of evolutionary dynamics such as the replicator equation or stochastic processes of game dynamics in finite populations.

The one-third law of evolutionary dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a “one-third law” of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency 1 3 . This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the “number” 1 3 is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.

Tit-for-tat or win-stay, lose-shift?

The repeated Prisoner's Dilemma is usually known as a story of tit-for-tat (TFT). This remarkable strategy has won both of Robert Axelrod's tournaments. TFT does whatever the opponent has done in the previous round. It will cooperate if the opponent has cooperated, and it will defect if the opponent has defected. But TFT has two weaknesses: (i) it cannot correct mistakes (erroneous moves) and (ii) a population of TFT players is undermined by random drift when mutant strategies appear which play always-cooperate (ALLC). Another equally simple strategy called ‘win-stay, lose-shift’ (WSLS) has neither of these two disadvantages. WSLS repeats the previous move if the resulting payoff has met its aspiration level and changes otherwise. Here, we use a novel approach of stochastic evolutionary game dynamics in finite populations to study mutation–selection dynamics in the presence of erroneous moves. We compare four strategies: always-defect (ALLD), ALLC, TFT and WSLS. There are two possible outcomes: if the benefit of cooperation is below a critical value then ALLD is selected; if the benefit of cooperation is above this critical value then WSLS is selected. TFT is never selected in this evolutionary process, but lowers the selection threshold for WSLS.

The different limits of weak selection and the evolutionary dynamics of finite populations

Evolutionary theory often resorts to weak selection, where different individuals have very similar fitness. Here, we relate two ways to introduce weak selection. The first considers evolutionary games described by payoff matrices with similar entries. This approach has recently attracted a lot of interest in the context of evolutionary game dynamics in finite populations. The second way to introduce weak selection is based on small distances in phenotype space and is a standard approach in kin-selection theory. Whereas both frameworks are interchangeable for constant fitness, frequency-dependent selection shows significant differences between them. We point out the difference between both limits of weak selection and discuss the condition under which the differences vanish. It turns out that this condition is fulfilled by the popular parametrization of the prisoner's dilemma in benefits and costs. However, for general payoff matrices differences between the two frameworks prevail.

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.

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 × 2 games, we give a complete analysis of the long-run behavior when mutation rates are small. For 3 × 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.

Stochastic dynamics of invasion and fixation

We study evolutionary game dynamics in finite populations. We analyze an evolutionary process, which we call pairwise comparison, for which we adopt the ubiquitous Fermi distribution function from statistical mechanics. The inverse temperature in this process controls the intensity of selection, leading to a unified framework for evolutionary dynamics at all intensities of selection, from random drift to imitation dynamics. We derive a simple closed formula that determines the feasibility of cooperation in finite populations, whenever cooperation is modeled in terms of any symmetric two-person game. In contrast with previous results, the present formula is valid at all intensities of selection and for any initial condition. We investigate the evolutionary dynamics of cooperators in finite populations, and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations, as a function of a given initial number of cooperators, showing how this interplay strongly affects the approach to fixation of a given trait in finite populations, leading to counterintuitive results at different intensities of selection.