Stochastic stability analysis of evolutionary two-player games on regular graphs

We study evolutionary two-player games and identify stochastically stable equilibria of the network games restricted to infinite populations on regular graphs. The players update their strategies according to four different rules: birth–death, death–birth, imitation and pairwise comparison for prisoner’s dilemma and snowdrift games, respectively. For two-player games on regular graphs, we show that there is a unique stochastically stable equilibrium for infinite populations. For the prisoner’s dilemma game, if the benefit-to-cost ratio is larger than k+2 (k is the degree of a regular graph), the networked game has a higher fraction of cooperators than that for a well-mixed population. For the snowdrift game, the fraction of cooperators in a regular graph would be higher than that of the well-mixed population, if the benefit-to-cost ratio is larger than 1.5. Under certain conditions, the lower graph connectivity can lead to the emergence of more cooperators. Finally, some numerical simulation examples are given to demonstrate the theoretical results.