  Indirect reciprocity is one of the main mechanisms to explain the emergence
and sustainment of altruism in societies. The standard approach to indirect
reciprocity are reputation models. These are games in which players base their
decisions on their opponent's reputation gained in past interactions with other
players (moral assessment). The combination of actions and moral assessment
leads to a large diversity of strategies, thus determining the stability of any
of them against invasions by all the others is a difficult task. We use a
variant of a previously introduced reputation-based model that let us
systematically analyze all these invasions and determine which ones are
successful. Accordingly we are able to identify the third-order strategies
(those which, apart from the action, judge considering both the reputation of
the donor and that of the recipient) that are evolutionarily stable. Our
results reveal that if a strategy resists the invasion of any other one sharing
its same moral assessment, it can resist the invasion of any other strategy.
However, if actions are not always witnessed, cheaters (i.e., individuals with
a probability of defecting regardless of the opponent's reputation) have a
chance to defeat the stable strategies for some choices of the probabilities of
cheating and of being witnessed. Remarkably, by analyzing this issue with
adaptive dynamics we find that whether a honest population resists the invasion
of cheaters is determined by a Hamilton-like rule---with the probability that
the cheat is discovered playing the role of the relatedness parameter.
