  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 prisoner's 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.
