Evolutionary dynamics of cooperation in optional public goods games with loners' punishment

Volunteering is a common mechanism for promoting cooperation, characterized by the self-organized cyclic dynamics among cooperators, defectors, and loners. Although this mechanism protects cooperators from being overwhelmed by defectors, it does not guarantee their ultimate success. Defectors are motivated to exploit cooperators due to their contributions to public goods. This raises a question: Can punishing cooperators disrupt the balance of cyclic invasion among the three strategies, ultimately leading to their dominance? Based on experimental evidence of economic sanctions imposed by loners on cooperators, this paper introduces loners who punish cooperators into optional public goods games. We develop an evolutionary model of a four-strategy game within a structured population. By exploring the spatial dynamics of the system and the evolutionary patterns of specific strategy distributions, we study the properties of phase transitions and the underlying mechanisms. Our analysis aims to elucidate how cooperation evolves under such punishment. The results indicate that these punishers can create a three-strategy cyclic state with cooperators and defectors. Under specific conditions, the system can enter from this cyclic phase to a pure cooperator phase. Additionally, when loners have a high fixed income, the system may enter a coexistence phase, including cooperators and two types of loners. In this phase, traditional loners form protective enclaves to safeguard cooperators from severe punishment.