Mean Field Approximations Of Fixation Time Distributions Of Evolutionary Game Dynamics On Graphs

Frequency dependent selection processes are traditional models for evolutionary game theory.In a population of N players,we assume that s₁ and S2 are two strategies for each player.Given an initial state of n s₁ players,the quantity of interest of the evolutionary process is the fixation probability that the population ends up in a state of N s₁-players.Deterministic replicator equations are helpful to understand the dynamics of infinite,homogeneous,and well-mixed populations.In this case,ordinary differential equations are used to describe the evolution of the density of the strategy and the mean fixation time often provides good estimates for the speed of fixations.However,we show that the mean fixation time is not sufficient to characterize the speed of fixations to absorbing states when simulating game dynamics in finite sized populations.The numerical results show even large deviations when the population has complex network structure.In order to have a better understanding of these phenomena,we consider the evolutionary dynamics as a Markov process.With the mean field assumption,we obtain the state transition matrix of the population.Then,we show the probability distribution function of fixation times explicitly.Numerical simulations of the dynamics are implemented on complex network topologies.In particular,we choose the standard deviation of the distribution function to discuss the influence of initial densities on the evolutionary times in both prisoner's dilemma and coordination game models.The numerical simulations are in good agreement with the theoretical predictions.