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

In a recent paper [Phys. Rev. E 86, 011134 (2012)], Mobilia introduces cooperation facilitators in the standard prisoner's dilemma game. He claims that natural selection favors the replacement of defection by cooperation in the weak-selection case if and only if their frequency satisfies a certain inequality. We show that this is not true, and we point out an error in the author's proof which follows from the improper handling of the large-population limit. In addition, we prove a stronger result that cooperation is favored for any selection strength if and only if the average payoff of cooperation is bigger than the average payoff of defection (which is a weaker condition than the author's inequality). We also show that, if we include self-interaction, then the presence of a fixed number of facilitators causes a rescaling of the payoff matrix, and for their certain frequency, cooperation becomes a dominant strategy, and the prisoner's dilemma simply disappears.