Reply to "Comment on 'Stochastic dynamics of the prisoner's dilemma with cooperation facilitators' ".

The fixation properties of a simple prisoner's dilemma game in the presence of "cooperation facilitators" have recently been investigated in finite and well-mixed populations for various dynamics [Mobilia, Phys. Rev. E 86, 011134 (2012)]. In a Comment, Miękisz claims that, for cooperation to be favored by selection in the standard prisoner's dilemma games with facilitators, it suffices that f(C)>f(D) (where f(C/D) are the respective fitnesses of cooperators and defectors). In this Reply, we show that, in generic prisoner's dilemma games with ℓ cooperation facilitators, it is generally not sufficient that a single cooperator has a higher fitness than defectors to ensure that selection favors cooperation. In fact, it is also necessary that selection promotes the replacement of defection by cooperation in a population of size N, which requires that the fixation probability of a single cooperator exceeds (N-ℓ)(-1). This replacement condition is independent of f(C)>f(D) and, when the payoff for mutual defection is negative, it is shown to be more stringent than the invasion condition. Our results, illustrated by a series of examples, considerably generalize those reported in the paper [Phys. Rev. E 86, 011134 (2012)] and in the aforementioned Comment whose claims are demonstrated to be relevant only for a special subclass of prisoner's dilemma games.