Indirect reciprocity is a mechanism that explains large-scale cooperation in humans. In indirect reciprocity, individuals use reputations to choose whether or not to cooperate with a partner and update others' reputations. A major question is how the rules to choose their actions and the rules to update reputations evolve. In the public reputation case, where all individuals share the evaluation of others, social norms called Simple Standing (SS) and Stern Judging (SJ) have been known to maintain cooperation. However, in the case of private assessment where individuals independently evaluate others, the mechanism of maintenance of cooperation is still largely unknown. This study theoretically shows for the first time that cooperation by indirect reciprocity can be evolutionarily stable under private assessment. Specifically, we find that SS can be stable, but SJ can never be. This is intuitive because SS can correct interpersonal discrepancies in reputations through its simplicity. On the other hand, SJ is too complicated to avoid an accumulation of errors, which leads to the collapse of cooperation. We conclude that moderate simplicity is a key to success in maintaining cooperation under the private assessment. Our result provides a theoretical basis for evolution of human cooperation.

翻译：间接互惠是一种解释人类大规模合作的机制。在间接互惠中，个体利用声誉决定是否与伙伴合作，并更新他人的声誉。一个核心问题在于：个体选择行动的策略以及更新声誉的规则如何演化。在公共声誉情形（所有个体共享对他人的评价）下，已知“简单立场”（Simple Standing, SS）和“严厉评判”（Stern Judging, SJ）等社会规范能够维持合作。然而，在私人评价情形（个体独立评估他人）下，合作维持的机制在很大程度上仍是未知。本研究首次从理论上证明，在私人评价下，间接互惠式的合作能够达到演化稳定。具体而言，我们发现SS可以保持稳定，而SJ则永不能。这一结论具有直观性：SS因其简单性能够纠正个体间声誉的差异；反之，SJ则过于复杂，难以避免错误的累积，从而导致合作崩溃。我们得出结论：适度简单性是私人评价下维持合作取得成功的关键。本研究为人类合作的演化提供了理论基础。