Interaction stochasticity may hinder cooperation in the spatial public goods game

We propose an analytic model to explore the effect of interaction stochasticity on the evolution of cooperation in spatial public goods game. The results show that whether cooperation can dominate in populations crucially depends on the player's probability of opting-out. Stochastic opting-out enhances cooperation as long as the probability of opting-out is less than a threshold depending on the graph's degree. Otherwise the promoting effect of spatial structures on cooperation is hindered even neutralized by stochastic opting-out. Moreover, there exists an intermediate optimal probability with which the advantage of cooperation over defection is maximized in the evolutionary race. Interestingly, the optimal probability is related to the percolation threshold of the underlying graph. Our findings illustrate that spatial structures may not facilitate cooperation when stochastic opting-out is allowed, and provide a link between physics and social sciences.