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

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.