Role of mutual punishment in the snowdrift game

The effects of punishment on cooperation have drawn increasing attention. In this paper, we propose a new mechanism of punishment, in which an individual will punish each neighbor if their strategies are different, and vice versa. We incorporate the mutual punishment into the snowdrift game. Results for well-mixed and structured populations have shown that, for no punishment or small values of punishment fine, the fraction of cooperators continuously decreases with the temptation to defect. However, for large values of punishment fine, there exists an abrupt transition point, at which the fraction of cooperators suddenly drops from 1 to 0. Compared to no punishment, mutual punishment promotes cooperation when the temptation to defect is small but inhibits cooperation when the temptation to defect is large. For weak (strong) temptation to defect, the cooperation level increases (decreases) with the punishment fine. For moderate temptation to defect, there exists an optimal value of the punishment fine that leads to the highest cooperation level.