Evolutionary prisoner's dilemma games with voluntary participation.

Voluntary participation in public good games has recently been demonstrated to be a simple yet effective mechanism to avoid deadlocks in states of mutual defection and to promote persistent cooperative behavior. Apart from cooperators and defectors a third strategical type is considered: the risk averse loners who are unwilling to participate in the social enterprise and rather rely on small but fixed earnings. This results in a rock-scissors-paper type of cyclic dominance of the three strategies. In the prisoner's dilemma, the effects of voluntary participation crucially depend on the underlying population structure. While leading to homogeneous states of all loners in well-mixed populations, we demonstrate that cyclic dominance produces self-organizing patterns on square lattices but leads to different types of oscillatory behavior on random regular graphs: the temptation to defect determines whether damped, periodic, or increasing oscillations occur. These Monte Carlo simulations are complemented by predictions from pair approximation reproducing the results for random regular graphs particularly well.