Stable polymorphism of cooperators and punishers in a public goods game

The role of punishment in the maintenance of cooperation has been emphasized recently. However, the maintenance of punishment is not an obvious consequence because punishment itself is also a public good; it is costly to perform and hence vulnerable to exploitation. For example, cooperative punishers, who help others and punish free riders, are disadvantageous in competition against pure cooperators, who cooperate but do not punish free riders. In addition, pure punishers, who do not help others but punish free riders, have been considered to be selfish in conventional models, because they do not perform cooperation. Instead, here we assume that performing either cooperation or punishment is sufficient to avoid accusation from others because not only cooperators but also pure punishers contribute to their society by reducing a threat of free riders. Under such an assumption, we analyzed the evolutionary dynamics of pure cooperators, pure punishers and free riders. We first showed that cooperation is never maintained in a well-mixed population. When the population is spatially structured, however, oscillatory rock-paper-scissors dynamics among those strategies appear. We further find that a stable polymorphism of the three strategies is attained when mutation is introduced. Our results indicate that cooperation and punishment can be stably maintained even when one cannot perform cooperation and punishment at the same time, and provide insights about the coexistence of cooperation and punishment in real societies.