Cooperation is essential for both human and artificial life societies, yet understanding how to promote it remains a complex challenge. Indirect reciprocity, where individuals cooperate to maintain a good reputation, is one mechanism to encourage cooperation. To promote stable cooperation, society needs social norms that stipulate how individuals should behave and how they should evaluate others. Previous research has identified a set of effective social norms, called the “leading eight”, for achieving evolutionarily stable cooperation. In this study, we expand on a classical framework in two significant ways. First, we include norms that update the reputations of passive receivers. Second, we introduce stochasticity to social norms. We theoretically derived the necessary and sufficient conditions for evolutionarily stable norms that result in full cooperation within this generalized model. Our findings offer a new perspective on prior research and provide a foundation for future studies in this field.

Direct reciprocity is one of the key mechanisms accounting for cooperation in our social life. According to recent understanding, most of the classical strategies for direct reciprocity fall into one of two classes, ‘partners’ or ‘rivals.’ A ‘partner’ is a generous strategy achieving mutual cooperation, and a ‘rival’ never lets the co-player become better off. They have different working conditions: For example, partners show good performance in a large population, whereas rivals do in head-to-head matches. Using exhaustive enumeration using a super-computer, we demonstrate the existence of strategies that are partners as well as rivals, called ‘friendly rivals.’ Among them, we focus on a human-interpretable strategy, named ‘CAPRI’ after its five characteristic ingredients, i.e., cooperate, accept, punish, recover, and defect otherwise. Our evolutionary simulation shows excellent performance of CAPRI regardless of environmental conditions.