Evolutionary Dynamics Of Coordination Games On Finite Complex Networks

Evolutionary game theory is the game model considering players of limited ratio-nality on networks with heterogeneous interaction. Compared with the classical game theory, evolutionary games on networks reflect the decision-making behavior of players in reality. When an infinite well-mixed system with coordinate games having only two alternative strategies s1 and s2, the replicator equation describing the group’s dynamics predicts that the system has two absorbing states and one unstable fixed point. More specifically, all individuals with s1 strategy is one absorbing state, and all individuals with s2 strategy represents another absorbing state. These two absorbing states are sep-arated by an unstable fixed point. The system is attracted to one of the absorbing states. The probability of which absorbing state the population finally arrives at (fixation prob-ability) depends on the initial density of strategy s1. If the initial density of s1 players is less than the unstable fixed point, the population evolves to the absorbing state of full s2; otherwise, the population evolves into the absorbing state of full s1. Therefore, the fixation probability shows a discontinuous jump when crossing the unstable fixed point. The value of the unstable fixed point can be obtained directly from the payoff matrix.But when the population size is finite, there is a big difference between the numer-ical simulations and the predictions by the replicator equation:the fixation probability around the fixed point of the initial density of strategy is continuous. This difference is particularly prominent when considering the heterogeneous interactions between indi-viduals in the population. The mean time of the system evolving to any absorbing states (the mean fixation time) is much longer than that has been predicted by the well-mixed model. There is no general methods to analyze the evolutionary games on complex networks for now.In this regard, I propose a stochastic model which considers the evolution dy- namics of the strategy’s density as a Markov process. When the degree distribution and the conditional degree distribution are given, the mean field theory yields the tran-sition probability matrix of the strategy’s density, namely the Markov matrix, which completely determines the statistical behavior of the system’s evolutionary dynamics. Afterwards, we show both the fixation probability and the mean fixation time. When the number of individuals in the population is large enough, the master equation of the density of strategy is simplified as the Fokker-Planck equation. However, in the case of finite network as I consider in this work, I will show the conditions under which the replicator equation maybe helpful by comparing the relative contributions of the diffu-sion and drift terms. We convincingly explain the reasons why replicator equation does not give the precise predictions when the initial density is close the unstable fixed point.Our theoretical results have been confirmed by numerical simulations. When the average degree of network is relative small, we note that the coupling between strategies make the mean field approximations not be suitable, which will be a subject of our future work.