  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.
