  Cooperation is a crucial aspect of social life, yet understanding the nature
of cooperation and how it can be promoted is an ongoing challenge. One
mechanism for cooperation is indirect reciprocity. According to this mechanism,
individuals cooperate to maintain a good reputation. This idea is embodied in a
set of social norms called the ``leading eight''. When all information is
publicly available, these norms have two major properties. Populations that
employ these norms are fully cooperative, and they are stable against invasion
by alternative norms. In this paper, we extend the framework of the leading
eight in two directions. First, we allow social norms to be stochastic. Such
norms allow individuals to evaluate others with certain probabilities. Second,
we consider norms in which also the reputations of passive recipients can be
updated. Using this framework, we characterize all evolutionarily stable norms
that lead to full cooperation in the public information regime. When only the
donor's reputation is updated, and all updates are deterministic, we recover
the conventional model. In that case, we find two classes of stable norms: the
leading eight and the `secondary sixteen'. Stochasticity can further help to
stabilize cooperation when the benefit of cooperation is comparably small.
Moreover, updating the recipients' reputations can help populations to recover
more quickly from errors. Overall, our study highlights a remarkable trade-off
between the evolutionary stability of a norm and its robustness with respect to
errors. Norms that correct errors quickly require higher benefits of
cooperation to be stable.
