Dynamics In Finite Populations Research Articles

Frequency-dependent returns in nonlinear public goods games.

When individuals interact in groups, the evolution of cooperation is traditionally modelled using the framework of public goods games. These models often assume that the return of the public goods depends linearly on the fraction of contributors. In contrast, in real-life public goods interactions, the return can depend on the size of the investor pool as well. Here, we consider a model in which the multiplication factor (marginal per capita return) for the public goods depends linearly on how many contribute, which results in a nonlinear model of public goods. This simple model breaks the curse of dominant defection found in linear public goods interactions and gives rise to richer dynamical outcomes in evolutionary settings. We provide an in-depth analysis of the more varied decisions by the classical rational player in nonlinear public goods interactions as well as a mechanistic, microscopic derivation of the evolutionary outcomes for the stochastic dynamics in finite populations and in the deterministic limit of infinite populations. This kind of nonlinearity provides a natural way to model public goods with diminishing returns as well as economies of scale.

Fixation of cooperation in evolutionary games with environmental feedbacks

The interaction between strategy and environment widely exists in nature and society. Traditionally, evolutionary dynamics in finite populations are described by the Moran process, where the environment is constant. Therefore, we model the Moran process with environmental feedbacks. Our results show that the selection intensity, which is closely related to the population size, exerts varying influences on evolutionary dynamics. In the case of the specific payoff matrix, cooperation cannot be favored by selection in extremely small-sized populations. The medium-sized populations are beneficial for the evolution of cooperation under intermediate selection intensities. For weak or strong selection intensities, the larger the population size, the more favorable it is for the evolution of cooperation. In the case of the generalized payoff matrix, the low incentives for the defector to cooperate in the degraded state cannot promote the emergence of cooperation. As the incentive for the defector to cooperate in the degraded state increases, selection favors cooperation or defection depending on the population size and selection intensity. For large values of the incentive for the defector facing the cooperative opponent to cooperate in the degraded state, selection always favors cooperation. We further investigate the impact of the time-scale on the fixation probability of cooperation.

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.

The stability of three-strategy cyclic dominance dynamics in finite populations

In finite populations, the stability of three-strategy cyclic dominance dynamics is investigated. A model for three-strategy cyclic dominance games is proposed. The stability of dynamical system corresponding to the model is compared with that of replicator dynamics, it verifies that the results of finite populations are quite different from that of replicator dynamics.

Quasi-neutral evolution in populations under small demographic fluctuations

We study an eco-evolutionary dynamics in finite populations of two haploid asexually reproducing allelic types. We focus on the quasi-neutral case when individual types differ only in their intrinsic birth and death rates but have the same expected lifetime reproductive output. We assume that the population size can fluctuate stochastically. We solve the Kolmogorov forward equation in the population whose size fluctuates only minimally and show that the fixation probability is decreasing with the increasing turnover rate. We also show that when the mutant’s turnover is small enough, selection favors the mutant replacing residents. Similarly, when the turnover is high enough, selection opposes the replacement. This basic result has previously been demonstrated numerically for the contact process and shown analytically for the Moran process; the current paper extends this analysis to provide an analytical proof for the contact process. We also demonstrate numerically that our results extend for general fluctuating populations and beyond the quasi-neutral case.

The Evolution of Cooperation and Reward in a Corrupt Environment

The maintenance of cooperative behaviors in many complex social systems has always been a great challenge. It has been suggested that costly punishment and reward can both facilitate the evolution of cooperation. Recent theoretical work, however, reveals that the positive role of centralized punishment in promoting cooperation has been challenged when violators can bribe centralized authorities to escape from sanctions. Naturally, the question arises as to how cooperation evolves when defectors can bribe rewarders for getting the reward. Here, we propose an evolutionary game theoretical model in which defectors can choose to bribe the rewarders probabilistically, and meanwhile rewarders will stochastically receive bribes from defectors in the public goods game. We theoretically study deterministic dynamics in infinite populations, and find that cooperators, defectors, and rewarders can coexist and the fraction of each strategist in the population remains unchanged in the coexistence state. Furthermore, we numerically investigate stochastic dynamics in finite populations, and reveal that cooperative behaviors can be maintained since the population can spend most of the time in the region where cooperators, defectors, and rewarders coexist.

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.

Stationary Stability for Evolutionary Dynamics in Finite Populations

We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the Moran process with mutation and generalizations, as well as a generalized notion of evolutionary stability that includes mutation called an incentive stable state (ISS) candidate. For sufficiently large populations, extrema of the stationary distribution are ISS candidates and we give a family of Lyapunov quantities that are locally minimized at the stationary extrema and at ISS candidates. In various examples, including for the Moran and Wright–Fisher processes, we show that the local maxima of the stationary distribution capture the traditionally-defined evolutionarily stable states. The classical stability theory of the replicator dynamic is recovered in the large population limit. Finally we include descriptions of possible extensions to populations of variable size and populations evolving on graphs.

Intrinsic noise in systems with switching environments.

We study individual-based dynamics in finite populations, subject to randomly switching environmental conditions. These are inspired by models in which genes transition between on and off states, regulating underlying protein dynamics. Similarly, switches between environmental states are relevant in bacterial populations and in models of epidemic spread. Existing piecewise-deterministic Markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching. Here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear-noise approximation. Specifically, we derive the stationary distributions of a number of model systems, in good agreement with simulations. This improves existing approaches which are limited to the regimes of fast and slow switching.

In and out of equilibrium I: Evolution of strategies in repeated games with discounting

In the repeated prisoner's dilemma there is no strategy that is evolutionarily stable, and a profusion of neutrally stable ones. But how stable is neutrally stable? We show that in repeated games with large enough continuation probabilities, where the stage game is characterized by a conflict between individual and collective interests, there is always a neutral mutant that can drift into a population that is playing an equilibrium, and create a selective advantage for a second mutant. The existence of stepping stone paths out of any equilibrium determines the dynamics in finite populations playing the repeated prisoner's dilemma.

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.

Fixation in finite populations evolving in fluctuating environments.

The environment in which a population evolves can have a crucial impact on selection. We study evolutionary dynamics in finite populations of fixed size in a changing environment. The population dynamics are driven by birth and death events. The rates of these events may vary in time depending on the state of the environment, which follows an independent Markov process. We develop a general theory for the fixation probability of a mutant in a population of wild-types, and for mean unconditional and conditional fixation times. We apply our theory to evolutionary games for which the payoff structure varies in time. The mutant can exploit the environmental noise; a dynamic environment that switches between two states can lead to a probability of fixation that is higher than in any of the individual environmental states. We provide an intuitive interpretation of this surprising effect. We also investigate stationary distributions when mutations are present in the dynamics. In this regime, we find two approximations of the stationary measure. One works well for rapid switching, the other for slowly fluctuating environments.

Evolutionary dynamics in finite populations with zealots

We investigate evolutionary dynamics of two-strategy matrix games with zealots in finite populations. Zealots are assumed to take either strategy regardless of the fitness. When the strategy selected by the zealots is the same, the fixation of the strategy selected by the zealots is a trivial outcome. We study fixation time in this scenario. We show that the fixation time is divided into three main regimes, in one of which the fixation time is short, and in the other two the fixation time is exponentially long in terms of the population size. Different from the case without zealots, there is a threshold selection intensity below which the fixation is fast for an arbitrary payoff matrix. We illustrate our results with examples of various social dilemma games.

Heterogeneity in background fitness acts as a suppressor of selection

We introduce the concept of heterogeneity in background fitness to evolutionary dynamics in finite populations. Background fitness is specific to an individual but not linked to its strategy. It can be thought of as a property that is related to the physical or societal position of an individual, but is not dependent on the strategy that is adopted in the evolutionary process under consideration. In our model, an individual's total fitness is the sum of its background fitness and the fitness derived from using a specific strategy. This approach has important implications for the imitation of behavioural strategies: if we imitate others for their success, but can only adopt their behaviour and not their social and economic ties, we may imitate in vain. We study the effect of heterogeneity in background fitness on the fixation of a mutant strategy with constant fitness. We find that heterogeneity suppresses selection, but also decreases the time until a novel strategy either takes over the population or is lost again. We derive analytical solutions of the fixation probability in small populations. In the case of large total background fitness in a population with maximum inequality, we find a particularly simple approximation of the fixation probability. Numerical simulations suggest that this simple approximation also holds for larger population sizes.

A tale of two contribution mechanisms for nonlinear public goods

Amounts of empirical evidence, ranging from microbial cooperation to collective hunting, suggests public goods produced often nonlinearly depend on the total amount of contribution. The implication of such nonlinear public goods for the evolution of cooperation is not well understood. There is also little attention paid to the divisibility nature of individual contribution amount, divisible vs. non-divisible ones. The corresponding strategy space in the former is described by a continuous investment while in the latter by a continuous probability to contribute all or nothing. Here, we use adaptive dynamics in finite populations to quantify and compare the roles nonlinearity of public-goods production plays in cooperation between these two contribution mechanisms. Although under both contribution mechanisms the population can converge into a coexistence equilibrium with an intermediate cooperation level, the branching phenomenon only occurs in the divisible contribution mechanism. The results shed insight into understanding observed individual difference in cooperative behavior.

Fixation and escape times in stochastic game learning

Evolutionary dynamics in finite populations is known to fixate eventually in the absence of mutation. We here show that a similar phenomenon can be found in stochastic game dynamical batch learning, and investigate fixation in learning processes in a simple 2×2 game, for two-player games with cyclic interaction, and in the context of the best-shot network game. The analogues of finite populations in evolution are here finite batches of observations between strategy updates. We study when and how such fixation can occur, and present results on the average time-to-fixation from numerical simulations. Simple cases are also amenable to analytical approaches and we provide estimates of the behaviour of so-called escape times as a function of the batch size. The differences and similarities with escape and fixation in evolutionary dynamics are discussed.

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.