Cooperation under endogenous punishment in the spatial public goods game

Punishment and network reciprocity have profound implications for the evolution of cooperation. However, existing research on the consequences of cooperation under punishment in social networks has largely relied on agent-based models and laboratory experiments. Moreover, different from the majority of existing studies where punishment is always believed to be deterministic, the individuals' preferences for certain behaviors are always stochastic and vary with the environment. There is an urgent need to explore how cooperation evolves when punishment is stochastic and endogenous in social networks. In this paper, we propose a theoretical model of endogenous punishment in spatial public goods games. Cooperators each can stochastically choose whether to participate in the punishment for defectors. The choice to penalize defectors comes with a price. Whether and how defectors are punished is endogenously determined by the cooperators' preferences for executing the costly punishment. We analyze how cooperation evolves under endogenous punishment based on a regular network in the mean-field limit and outline the conditions under which endogenous punishment can support cooperation. When network reciprocity is unfavorable for cooperation, endogenous punishment can be effective in supporting cooperation. On the contrary, endogenous punishment no longer supports or even hinders the promoting effect of network reciprocity on cooperation. These findings illustrate that the effectiveness of endogenous punishment in fostering cooperation is dependent on the cooperators' willingness to pay for punishment as well as the topology of social networks.