Synergy and discount of punishment in the public goods game

Abstract

Punishment has been verified as an important mechanism for promoting cooperation. Ethnographic evidence shows that punishment is coordinated within the group. Thus, we take into account the nonlinear group interaction and propose the synergetic and discounted punishment depicting how fines imposed on defectors nonlinearly accumulate within the group. We further model the replicator dynamics on regular networks and investigate the equilibria of evolutionary dynamics in structured populations. Our results show that cooperation can not be the unique stable equilibrium in well-mixed populations. However, for structured populations, the results indicate that cooperation can be the only stable equilibrium. The interior equilibrium is always an unstable equilibrium in well-mixed and structured populations, which means that cooperation and defection cannot coexist. For low values of β, the high network degree k is favorable to promoting cooperation. On the contrary, the low network degree k becomes beneficial to promoting cooperation with β increases.