  Previous research has shown how indirect reciprocity can promote cooperation
through evolutionary game theoretic models. Most work in this field assumes a
separation of time-scales: individuals' reputations equilibrate at a fast time
scale for given frequencies of strategies while the strategies change slowly
according to the replicator dynamics. Much of the previous research has focused
on the behaviour and stability of equilibria for the replicator dynamics. Here
we focus on the underlying reputational dynamics that occur on a fast time
scale. We describe reputational dynamics as systems of differential equations
and conduct stability analyses on their equilibria. We prove that reputations
converge to a unique equilibrium for each of the five standard norms whether
assessments are public or private. These results confirm a crucial but
previously unconfirmed assumption underlying the theory of indirect reciprocity
for the most studied set of norms.
