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

Evolutionary stability of cooperation by the leading eight norms in indirect reciprocity under noisy and private assessment