  Reputation is a powerful mechanism to enforce cooperation among unrelated
individuals through indirect reciprocity, but it suffers from disagreement
originating from private assessment, noise, and incomplete information. In this
work, we investigate stability of cooperation in the donation game by regarding
each player's reputation and behaviour as continuous variables. Through
perturbative calculation, we derive a condition that a social norm should
satisfy to give penalties to its close variants, provided that everyone
initially cooperates with a good reputation, and this result is supported by
numerical simulation. A crucial factor of the condition is whether a
well-reputed player's donation to an ill-reputed co-player is appreciated by
other members of the society, and the condition can be reduced to a threshold
for the benefit-cost ratio of cooperation which depends on the reputational
sensitivity to a donor's behaviour as well as on the behavioural sensitivity to
a recipient's reputation. Our continuum formulation suggests how indirect
reciprocity can work beyond the dichotomy between good and bad even in the
presence of inhomogeneity, noise, and incomplete information.
