Cooperation Under Stochastic Punishment in Social Dilemma Situations

Dear Editor, This letter is concerned with the evolutionary dynamics of cooperative strategies in social dilemma situations. Stochastic punishment has been proposed, in which whether an individual acts as a punisher or not is stochastic and depends on its preference for punishment. Meanwhile, both the cost of punishment and whether a defector would be punished are also stochastic. In previous models, the cost of punishment is considered to be either constant or proportional to the number of individuals to be punished. Furthermore, the hypothesis that all defectors should be penalized is frequently adopted. Actually, some defectors may refrain from being punished due to the presence of noise or the limitation of the punishment cost, and the cost of punishment is also dependent on the number of punishers. Thus, we establish an analytic model of stochastic punishment for infinite and well-mixed populations, investigate the effects of stochastic punishment on the evolution of cooperation, and analyze how to support the evolution of cooperation effectively when the stochastic punishment is possible. The objective of this letter is to design a cooperation-promoting stochastic punishment that will allow the system to evolve to full cooperation. The replicator equations have been used to explore the evolutionary dynamics of cooperation under stochastic punishment, and the conditions under which cooperation is favored by natural selection have been specified.