  Reputation is one of key mechanisms to maintain human cooperation, but its
analysis gets complicated if we consider the possibility that reputation does
not reach consensus because of erroneous assessment. The difficulty is
alleviated if we assume that reputation and cooperation do not take binary
values but have continuous spectra so that disagreement over reputation can be
analysed in a perturbative way. In this work, we carry out the analysis by
expanding the dynamics of reputation to the second order of perturbation under
the assumption that everyone initially cooperates with good reputation. The
second-order theory clarifies the difference between Image Scoring and Simple
Standing in that punishment for defection against a well-reputed player should
be regarded as good for maintaining cooperation. Moreover, comparison among the
leading eight shows that the stabilizing effect of justified punishment weakens
if cooperation between two ill-reputed players is regarded as bad. Our analysis
thus explains how Simple Standing achieves a high level of stability by
permitting justified punishment and also by disregarding irrelevant information
in assessing cooperation. This observation suggests which factors affect the
stability of a social norm when reputation can be perturbed by noise.
