Evolutionary games in a generalized Moran process with arbitrary selection strength and mutation

By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2×2 games in a well-mixed population with a finite size is investigated. In the model, the individuals' payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength \beta and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when \varepsilon is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the 'Hawk–Dove' game and the 'Coordinate' game are used to illustrate our model. We give the evolutionary stable strategy (ESS) of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.