An open problem in evolutionary game dynamics is to understand the effect of peer pressure on cooperation in a quantitative manner. Peer pressure can be modeled by punishment, which has been proved to be an effective mechanism to sustain cooperation among selfish individuals. We investigate a symmetric punishment strategy, in which an individual will punish each neighbor if their strategies are different, and vice versa. Because of the symmetry in imposing the punishment, one might expect intuitively the strategy to have little effect on cooperation. Utilizing the prisoners dilemma game as a prototypical model of interactions at the individual level, we find, through simulation and theoretical analysis, that proper punishment, when even symmetrically imposed on individuals, can enhance cooperation. Besides, we find that the initial density of cooperators plays an important role in the evolution of cooperation driven by mutual punishment.
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