  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.
