The paper contains the attempt to integration of the classical evolutionary game theory based on replicator dynamics and the state based approach of Houston and Mcnamara. In the new approach, individuals have different heritable strategies, however the individuals carrying the same strategy can differ on the state, role or situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths) can lead to the change of state of the competing individuals. The special cases of the new framework of stage structured models where the state changes describe developmental steps or aging are derived. New approach is illustrated by the example of Owner-Intruder game with explicit dynamics of the role changes. New model is the generalization of the demographic version of the Hawk-Dove game, the difference is that opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Intruders check random nest sites, and play the Hawk-Dove game with the Owner if they are occupied. Interesting feedback mechanism is produced by fluxes of individuals between subpopulations. Owners produce newborns which become Intruders, since they should find a free nest site to reproduce.
This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection -- as a result of competition fashioned by game-theoretic consideration -- into the Wright--Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokke…
Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player-two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games -- the Stag Hunt game, the Snowdrift game, the Prisoner…
We study the multi-strategy stochastic evolutionary game with death-birth updating in expanding spatial populations of size $N\to \infty$. The model is a voter model perturbation. For typical populations, we require perturbation strengths satisfying $1/N\ll w\ll 1$. Under appropriate conditions on the space, the limiting density processes of strategy are proven to obey the replicator equation, and the normalized fluctuations converge to a Gaussian process with the Wright-Fisher covariance function in the limiting densities. As an application, we resolve in the positive a conjecture from the bi…
In this paper, we consider the replicator-mutator dynamics for pairwise social dilemmas where the payoff entries are random variables. The randomness is incorporated to take into account the uncertainty, which is inevitable in practical applications and may arise from different sources such as lack of data for measuring the outcomes, noisy and rapidly changing environments, as well as unavoidable human estimate errors. We analytically and numerically compute the probability that the replicator-mutator dynamics has a given number of equilibria for four classes of pairwise social dilemmas (Priso…
In this paper, we study the number of equilibria of the replicator-mutator dynamics for both deterministic and random multi-player two-strategy evolutionary games. For deterministic games, using Decartes' rule of signs, we provide a formula to compute the number of equilibria in multi-player games via the number of change of signs in the coefficients of a polynomial. For two-player social dilemmas (namely, the Prisoner's Dilemma, Snowdrift, Stag Hunt, and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria wh…
We derive both the finite and infinite population spatial replicator dynamics as the fluid limit of a stochastic cellular automaton. The infinite population spatial replicator is identical to the model used by Vickers and our derivation justifies the addition of a diffusion to the replicator. The finite population form generalizes the results by Durett and Levin on finite spatial replicator games. We study the differences in the two equations as they pertain to a one-dimensional rock-paper-scissors game.
People tend to align their use of language to the linguistic behaviour of their own ingroup and to simultaneously diverge from the language use of outgroups. This paper proposes to model this phenomenon of sociolinguistic identity maintenance as an evolutionary game in which individuals play the field and the dynamics are supplied by a multi-population extension of the replicator-mutator equation. Using linearization, the stabilities of all dynamic equilibria of the game in its fully symmetric two-population special case are found. The model is then applied to an empirical test case from adole…
The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a family of connected networks with communities, where nodes in the same community have the same degree. This replicator equation is applied to the study of different classes of games, exploring the impact of the graph structure on the equilibria of the evolutionary dynamics.
In evolutionary game theory, it is customary to be partial to the dynamical models possessing fixed points so that they may be understood as the attainment of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or chaotic solution is many a time perceived as a shortcoming of the corresponding game dynamic because (Nash) equilibrium play is supposed to be robust and persistent behaviour, and any other behaviour in nature is deemed transient. Consequently, there is a lack of attempt to connect the non-fixed point solutions with the game theoretic concepts. Here we provide a…
Populations composed of a collection of subpopulations (demes) with random migration between them are quite common occurrences. The emergence and sustenance of cooperation in such a population is a highly researched topic in the evolutionary game theory. If the individuals in every deme are considered to be either cooperators or defectors, the migration dilemma can be envisaged: The cooperators would not want to migrate to a defector-rich deme as they fear of facing exploitation; but without migration, cooperation can not be established throughout the network of demes. With a view to studying …
In this paper we present a new modelling framework combining replicator dynamics (which is the standard model of frequency dependent selection) with the model of an age-structured population. The new framework allows for the modelling of populations consisting of competing strategies carried by individuals who change across their life cycle. Firstly the discretization of the McKendrick von Foerster model is derived. It is shown that the Euler--Lotka equation is satisfied when the new model reaches a steady state (i.e. stable frequencies between the age classes). This discretization consists of…
Replicator dynamics, the continuous-time analogue of Multiplicative Weights Updates, is the main dynamic in evolutionary game theory. In simple evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is periodic \cite{zeeman1980population}, however, its behavior in higher dimensions is not well understood. We provide a complete characterization of its behavior in zero-sum evolutionary games. We prove that, if and only if, the system has an interior Nash equilibrium, the dynamics exhibit Poincaré recurrence, i.e., almost all orbits come arbitrary close to their initial cond…
We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner's Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner's Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors …
The now classical replicator equation describes a wide variety of biological phenomena, including those in theoretical genetics, evolutionary game theory, or in the theories of the origin of life. Among other questions, the permanence of the replicator equation is well studied in the local, well-mixed case. Inasmuch as the spatial heterogeneities are key to understanding the species coexistence at least in some cases, it is important to supplement the classical theory of the non-distributed replicator equation with a spatially explicit framework. One possible approach, motivated by the porous …
The tragedy of the commons (TOC) is a ubiquitous social dilemma witnessed in interactions between a population of living entities and shared resources available to them: The individuals in the population tend to selfishly overexploit a common resource as it is arguably the rational choice, or in case of non-human beings, it may be an evolutionarily uninvadable action. How to avert the TOC is a significant problem related to the conservation of resources. It is not hard to envisage situations where the resource could be self-renewing and the size of the population may be dependent on the state …
The sampling of interaction partners depends on often implicit modelling assumptions, yet has marked effects on the dynamics in evolutionary games. One particularly important aspect is whether or not competitors also interact. Population structures naturally affect sampling such that in a microscopic interpretation of the replicator dynamics in well-mixed populations competing individuals do not interact but do interact in structured populations. In social dilemmas interactions with competitors invariably inhibit cooperation while limited local interactions in structured populations support co…
We study effects of strategy-dependent time delays on equilibria of evolving populations. It is well known that time delays may cause oscillations in dynamical systems. Here we report a novel behavior. We show that microscopic models of evolutionary games with strategy-dependent time delays lead to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous models, stationary states of such dynamics depend continuously on time delays. We show that in games with an interior stat…
Here we consider a game theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios w…
The tragedy of the commons (TOC) is an unfortunate situation where a shared resource is exhausted due to uncontrolled exploitation by the selfish individuals of a population. Recently, the paradigmatic replicator equation has been used in conjunction with a phenomenological equation for the state of the shared resource to gain insight into the influence of the games on the TOC. The replicator equation, by construction, models a fixed infinite population undergoing microevolution. Thus, it is unable to capture any effect of the population growth and the carrying capacity of the population altho…
Evolution of distribution of strategies in game theory is an interesting question that has been studied only for specific cases. Here I develop a general method to extend analysis of the evolution of continuous strategy distributions given bi-linear and quadratic payoff functions for any initial distribution to answer the following question: given the initial distribution of strategies in a game, how will it evolve over time? I look at several specific examples, including normal distribution on the entire line, normal truncated distribution, as well as exponential, uniform and Gamma distributi…
We analyze from basic physical considerations the Darwinian competition for reproduction (evolutionary dynamics) of strategists in a Public Goods Game, the archetype for $n$-agent (group) economical and biological interactions. In the proposed setup, the population is organized into groups, being the individual fitness linked to the group performance, while the evolutionary dynamics takes place globally. Taking advantage of (groups) permutation symmetry, the nonlinear analysis of the "mesoscale" Markov phase space for many competing groups is feasible to a large extent, regarding the expected …
In this paper, I model and study the process of natural selection between all possible mixed strategies in classical two-player two-strategy games. I derive and solve an equation that is a natural generalization of the Taylor-Jonker replicator equation that describes dynamics of pure strategy frequencies. I then investigate the evolution of not only frequencies of pure strategies but also of total distribution of mixed strategies. I show that the process of natural selection of strategies for all games obeys the dynamical Principle of minimum of information gain.
Evolutionary games on graphs have been extensively studied. A variety of graph structures, graph dynamics, and behaviours of replicators have been explored. These models have primarily been studied in the framework of facilitation of cooperation, and much previous research has shed light on this field of study. However, there has been little attention devoted to truncation selection as most models employ proportional selection (such as in the replicator equation) or `imitate the best.' Here we systematically explore truncation selection on periodic square lattices, where replicators below a fi…
In this letter, we deal with evolutionary game theoretic learning processes for population games on networks with dynamically evolving communities. Specifically, we propose a novel mathematical framework in which a deterministic, continuous-time replicator equation on a community network is coupled with a closed dynamic flow process between communities that is governed by an environmental feedback mechanism, resulting in co-evolutionary dynamics. Through a rigorous analysis of the system of differential equations obtained, we characterize the equilibria of the coupled dynamical system. Moreove…
We study the evolutionary dynamics of games under environmental feedback using replicator equations for two interacting populations. One key feature is to consider jointly the co-evolution of the dynamic payoff matrices and the state of the environment: the payoff matrix varies with the changing environment and at the same time, the state of the environment is affected indirectly by the changing payoff matrix through the evolving population profiles. For such co-evolutionary dynamics, we investigate whether convergence will take place, and if so, how. In particular, we identify the scenarios w…
In many biological systems, natural selection acts simultaneously on multiple levels of organization. This scenario typically presents an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. Generalizing previous work on multilevel selection in evolutionary game theory, we consider a hyperbolic PDE model of a group-structured population, in which members within a single group compete with each other for individual-level replication; while the group also competes against other groups for group-level replication…
Discrete-time replicator map is a prototype of evolutionary selection game dynamical models that have been very successful across disciplines in rendering insights into the attainment of the equilibrium outcomes, like the Nash equilibrium and the evolutionarily stable strategy. By construction, only the fixed point solutions of the dynamics can possibly be interpreted as the aforementioned game-theoretic solution concepts. Although more complex outcomes like chaos are omnipresent in the nature, it is not known to which game-theoretic solutions they correspond. Here we construct a game-theoreti…
The classical idea of evolutionarily stable strategy (ESS) modeling animal behavior does not involve any spatial dependence. We considered a spatial Hawk-Dove game played by animals in a patchy environment with wrap around boundaries. We posit that each site contains the same number of individuals. An evolution equation for analyzing the stability of the ESS is found as the mean dynamics of the classical frequency dependent Moran process coupled via migration and nonlocal payoff calculation in 1D and 2D habitats. The linear stability analysis of the model is performed and conditions to observe…
Game theory has been widely applied to many areas including economics, biology and social sciences. However, it is still challenging to quantify the global stability and global dynamics of the game theory. We developed a landscape and flux framework to quantify the global stability and global dynamics of the game theory. As an example, we investigated the models of three-strategy games: a special replicator-mutator game, the repeated prison dilemma model. In this model, one stable state, two stable states and limit cycle can emerge under different parameters. The repeated Prisoner's Dilemma sy…
