Proportional cost for punishment enhances spatial reciprocity in evolutionary games

Punishment is an effective method to promote cooperation. Previously, all the defectors will be punished in each generation. However, this exists great discrepancy with the real system, which becomes necessary to seek out a new approach. In this paper, the individuals are checked with the proportion (p) in the entire system and they will be punished if they choose defection. In addition, we consider the strength of punishment (β) based on the payoff of the punished individual. The fine of punishment is a percentage of payoff of each individual. The effect of punishment is quantitative investigated in the prisoner’s dilemma game and the snowdrift game. Interestingly, we find that punishment successfully reduces the number of individuals who have lower or higher payoffs. In other word, punishment makes more and more individuals’ payoffs gather in a certain range. Last, we find that distribution of payoffs also can form cluster.