Opinion synchronization in group-structured populations: A route to cooperation under noisy indirect reciprocity

Cooperation is fundamental to social and economic systems, yet it is difficult to sustain when individuals have incentives to defect. Reputation-based indirect reciprocity provides an important mechanism for promoting cooperation, but how group structure, private reputations, and gossip-based information exchange jointly shape cooperation remains unclear. Here, we study a group-structured population in which individuals interact within and between groups, maintain private binary reputations, and exchange reputation information through gossip with possible opinion adoption. Action and assessment errors are also considered. Through systematic simulations, we show that cooperation is most effectively promoted when within-group interaction is accompanied by within-group gossip. This effect is driven by reputation synchronization within groups rather than by global reputation consensus. Frequent gossip among group members aligns within-group reputation opinions, strengthening social norms that reward cooperation and punish defection. Although gossip concentrated within groups may increase opinion disagreement between groups, its positive effect on within-group cooperation dominates, leading to higher overall cooperation. Further experiments show that increasing gossip frequency strengthens reputation synchronization, and that the resulting cooperation remains stable over time and robust across different group sizes.