Cooperation evolution of public goods games under probabilistic punishment rule for reputational altruism

This study investigates the influence of individual retaliatory sentiment on punishment behavior, and explores how reputation-based indirect altruistic punishment promotes cooperation evolution in structured group public goods games. We define individual retaliatory sentiment as the probability of implementing punitive actions, and propose a reputation-based probabilistic altruistic punishment rule: punishers can adjust benefit distribution rules to penalize defectors while rewarding high-reputation individuals. The results demonstrate that under this mechanism, punishers form latent alliance relationships transcending social structures through reputation-based indirect reciprocity to collectively resist defectors, with strong retaliatory sentiment towards neighboring defectors significantly suppressing the survival of defection. Further analysis reveals that cooperation evolution achieves optimal promotion when the co-beneficiary capacity for indirect reciprocity maintains a moderate level. Specifically, smaller collective punishment co-beneficiary capacity combined with strong historical reputation memory allows some defectors to evade punishment by leveraging past good behaviors, inadvertently fostering defection through reciprocal benefits. Conversely, weaker memory capacity prevents punishers from establishing effective indirect reciprocal relationships through reputation, thereby weakening punitive efficacy. Moderate co-beneficiary capacity ensures cooperation evolution across varying memory intensities, while excessive capacity leads to non-punishers’ parasitic benefits triggering defection resurgence. Additionally, higher synergy factors enhance the cooperation-promoting effect of this punishment mechanism. These findings deepen our understanding of individual punitive behaviors in evolutionary game theory.