A simple rule for robust stabilization of evolutionary dynamics

The evolution of cooperation opens a prominent window to investigate the organizing properties in complex systems. The boom of complex networks paves the way to describe structured population and it has been introduced into evolutionary game theory recently. Yet, for multi-player games, theoretical works so far have been only on cycle and well-mixed population. With the metaphor of public goods game, we have considered the stochastic dynamics in finite structured population where the average degree of the population is k. A simple rule, i.e., the benefit to cost ratio should exceed (k + 1)²/(k + 3), is obtained analytically. This rule not only guarantees the emergence of cooperation, but also ensures the stabilization of cooperation under weak selection. Besides, inspired by pair approximation, we find some evidence supporting the thesis of spatial reciprocity in multiple players games to some extend. Our results may give a criteria suggesting when cooperation can be favored in finite structured population.