  Indirect reciprocity is a key mechanism that promotes cooperation in social
dilemmas by means of reputation. Although it has been a common practice to
represent reputations by binary values, either `good' or `bad', such a
dichotomy is a crude approximation considering the complexity of reality. In
this work, we studied norms with three different reputations, i.e., `good',
`neutral', and `bad'. Through massive supercomputing for handling more than
thirty billion possibilities, we fully identified which norms achieve
cooperation and possess evolutionary stability against behavioural mutants. By
systematically categorizing all these norms according to their behaviours, we
found similarities and dissimilarities to their binary-reputation counterpart,
the leading eight. We obtained four rules that should be satisfied by the
successful norms, and the behaviour of the leading eight can be understood as a
special case of these rules. A couple of norms that show counter-intuitive
behaviours are also presented. We believe the findings are also useful for
designing successful norms with more general reputation systems.
