The evolution of cooperation in voluntary public goods game with shared-punishment

A large number of theoretical and experimental studies have shown that punishment and voluntary participation can promote cooperation, which offers an explanation for the emergence of cooperation. In this paper, individuals can freely choose whether to participate in the public goods game, and cooperators perform both cooperation and punishment. On the one hand, cooperators invest in public goods, and on the other hand, they punish the defectors and share the cost of punishment. Furthermore, considering the impact of population structure, we investigate the evolution of cooperation in well-mixed populations with the replicator dynamics and structured populations with the Fermi update rule, respectively. Our results indicate that the evolution of cooperation is obviously different between the two populations. In well-mixed populations, the voluntary participation mechanism prevents defection from becoming an evolutionary stable strategy. When the fine is low, the strategy of non-participation will be the only evolutionary stable strategy. With the increase of the fine, both cooperation and non-participation will be the evolutionary stable strategies. When the fine is high, cooperation will become the only evolutionary stable strategy. In structured populations, cooperators, defectors and loners can coexist under mild punishment, which means that population structure is favorable to the emergence of cooperation. However, cooperators occupy the population with higher fine threshold compared to well-mixed populations, which indicates that population structure is not always favorable to the evolution of cooperation.