Non-participant externalities reshape the evolution of altruistic punishment

While voluntary participation is a key mechanism that enables altruistic punishment to emerge, its explanatory power typically rests on the common assumption that non-participants have no impact on the public good. Yet, given the decentralized nature of voluntary participation, opting out does not necessarily preclude individuals from influencing the public good. Here, we revisit the role of voluntary participation by allowing non-participants to exert either positive or negative impacts on the public good. Using evolutionary analysis in a well-mixed finite population, we find that positive externalities from non-participants lower the synergy threshold required for altruistic punishment to dominate. In contrast, negative externalities raise this threshold, making altruistic punishment harder to sustain. Notably, when non-participants have positive impacts, altruistic punishment thrives only if non-participation is incentivized, whereas under negative impacts, it can persist even when non-participation is discouraged. Our findings reveal that efforts to promote altruistic punishment must account for the active role of non-participants, whose influence can make or break collective outcomes.