Exact conditions for evolutionary stability in indirect reciprocity under noise

Indirect reciprocity is a key mechanism for large-scale cooperation. This mechanism captures the insight that in part, people help others to build and maintain a good reputation. To enable such cooperation, appropriate social norms are essential. They specify how individuals should act based on each others’ reputations, and how reputations are updated in response to individual actions. Although previous work has identified several norms that sustain cooperation, a complete analytical characterization of all evolutionarily stable norms remains lacking, especially when assessments or actions are noisy. In this study, we provide such a characterization for the public assessment regime. This characterization reproduces known results, such as the leading eight norms, but it extends to more general cases, allowing for various types of errors and additional actions including costly punishment. We also identify norms that impose a fixed payoff on any mutant strategy, analogous to the zero-determinant strategies in direct reciprocity. These results offer a rigorous foundation for understanding the evolution of cooperation through indirect reciprocity and the critical role of social norms.Author summary: Understanding how cooperation can evolve and be sustained is a central question in evolutionary biology and social science. One prominent explanation is indirect reciprocity, where individuals help others to build a good reputation and receive help in future. For this mechanism to work, societies rely on social norms — shared rules that specify how actions are judged and thereby how reputations are updated. Previous studies have proposed specific norms that support cooperation. However, it has remained unclear what general conditions make a norm evolutionarily stable. In this study, we develop a mathematical framework to analytically derive such conditions. Our theory reproduces well-known results, and it extends to more complex scenarios involving non-negligible errors and costly punishment. These findings deepen our understanding of the evolution of cooperation and offer insights into how robust social norms can emerge and persist, even in noisy environments.