Asymmetric strategy setup solve the Prisoner's Dilemma of the evolution of mutualism

Explaining the evolution of cooperation represents one of the greatest challenges in both evolutionary biology and social science. However, asymmetrical interaction, as one typical characteristic of cooperative system, has not been sufficiently considered in the existing literature for the evolution of cooperation. Incorporating the asymmetry in the strategy sets, we here construct an asymmetric game model with the so-called carrot-stick strategy, which is a mixed strategy of reward and punishment. Based on mathematical analyses, it is unveiled that this asymmetric interaction can lower the dilemma of cooperation: the dominant players and recipient players might coexist through cycle frequency. Further analysis shows that the multi-equilibria are possible, which depend on the payoff parameters and the initial conditions. That is to say, by setting up different values of the cost-to-benefit, the punishment-to-benefit, and the reward-to-benefit, we can recover three basic evolutionary dynamics of systems: dominance; bistability and coexistence. These theoretical observations are consistent with existing empirical outcomes that asymmetric sanction or reward of host species might solve the conflicts between the actors in the fig-fig wasp mutualism or in the cleaner fish-client mutualism. It is thus suggested that this asymmetric strategy setup may shed new light into the solution of social dilemmas.