Punishment in the form of shared cost promotes altruism in the cooperative dilemma games

One phenomenon or social institution often observed in multi-agent interactions is the altruistic punishment, i.e. the punishment of unfair behavior by others at a personal cost. Inspired by the works focusing on punishment and the intricate mechanism behind it, we theoretically study the strategy evolution in the framework of two-strategy game models with the punishment on defectors, moreover, the cost of punishing will be evenly shared among the cooperators. Theoretical computations suggest that larger punishment on defectors or smaller punishment cost incurred by cooperators will enhance the fixation of altruistic cooperation in the population. Through the replicate dynamics, the group size of the randomly selected individuals from the sufficiently large population will notably affect the strategy evolution in populations nested within a dilemma. By theoretical modeling the concept of shared cost for punishment from one point of view, our findings underscore the importance of punishment with shared cost as a factor in real-life decisions in an evolutionary game context.