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Stochastic dynamics of the prisoner's dilemma with cooperation facilitators

In the framework of the paradigmatic prisoner's dilemma, we investigate the evolutionary dynamics of social dilemmas in the presence of "cooperation facilitators". In our model, cooperators and defectors interact as in the classical prisoner's dilemma game, where selection favors defection. However, here the presence of a small number of cooperation facilitators enhances the fitness (reproductive potential) of cooperators, while it does not alter that of defectors. In a finite population of size N, the dynamics of the prisoner's dilemma with facilitators is characterized by the probability that cooperation takes over (fixation probability) and by the mean times to reach the absorbing states. These quantities are computed exactly and using Fokker-Planck equations. Our findings, corroborated by stochastic simulations, demonstrate that the influence of facilitators crucially depends on the difference between their density z and the game's cost-to-benefit ratio r. When z>r, the fixation of cooperators is likely in a large population and, under weak selection pressure, invasion and replacement of defection by cooperation is favored by selection if b(z-r)(1-z)>1/N, where 0<b<= 1 is the cooperation payoff benefit. When z<r, the fixation probability of cooperators is exponentially enhanced by the presence of facilitators but defection is the dominating strategy.