Indirect reciprocity is a mechanism that explains large-scale cooperation in human societies. In indirect reciprocity, an individual chooses whether or not to cooperate with another based on reputation information, and others evaluate the action as good or bad. Under what evaluation rule (called ``social norm'') cooperation evolves has long been of central interest in the literature. It has been reported that if individuals can share their evaluations (i.e., public reputation), social norms called ``leading eight'' can be evolutionarily stable. On the other hand, when they cannot share their evaluations (i.e., private assessment), the evolutionary stability of cooperation is still in question. To tackle this problem, we create a novel method to analyze the reputation structure in the population under private assessment. Specifically, we characterize each individual by two variables, ``goodness'' (what proportion of the population considers the individual as good) and ``self-reputation'' (whether an individual thinks of him/herself as good or bad), and analyze the stochastic process of how these two variables change over time. We discuss evolutionary stability of each of the leading eight social norms by studying the robustness against invasions of unconditional cooperators and defectors. We identify key pivots in those social norms for establishing a high level of cooperation or stable cooperation against mutants. Our finding gives an insight into how human cooperation is established in a real-world society.

翻译：间接互惠是解释人类社会大规模合作的一种机制。在间接互惠中，个体根据声誉信息选择是否与他人合作，而其他人则将这一行为评价为“好”或“坏”。在何种评价规则（称为“社会规范”）下合作得以演化，一直是文献中的核心议题。已有研究表明，如果个体能够共享评价（即公共声誉），被称为“前八种”的社会规范可以在演化上保持稳定。然而，当个体无法共享评价（即私人评估）时，合作的演化稳定性仍存在疑问。为解决这一问题，我们提出了一种新方法，用于分析私人评估下群体中的声誉结构。具体而言，我们用两个变量来刻画每个个体：“善良度”（群体中认为该个体为“好”的比例）和“自我声誉”（个体自身认为自己是“好”还是“坏”），并分析这两个变量随时间变化的随机过程。通过研究这些规范对无条件合作者和背叛者入侵的稳健性，我们讨论了前八种社会规范各自的演化稳定性。我们识别了这些规范中建立高水平合作或针对突变的稳定合作的关键支点。这一发现为理解现实世界中人类合作的形成提供了洞见。