Indirect reciprocity in populations with group structure

Indirect reciprocity, whereby individuals cooperate with others of good standing, is a powerful and parsimonious mechanism to sustain a high level of cooperation. Studies of indirect reciprocity typically assume either that each individual forms their own private assessment of the reputations of others or, conversely, that everyone shares a consensus view of the reputations in the population. Here we develop a mathematical framework for analyzing intermediate cases between strictly private and fully public information. We introduce group structure in information availability: the population is stratified into disjoint groups, and reputation information is shared only within each group. Even when social interactions and strategic imitations are well-mixed across the entire population, the partitioning of reputation information into disjoint groups weakens the fitness advantage of cooperative individuals, because it makes it less likely that interacting players will agree about each other's reputations. In the limit of a single group, or in the limit of a large number of groups, we recover the classical cases of fully public, or completely private, information. We analyze the effect of group structure on the stability of behavioral strategies, and we identify critical values for the number of groups required to support cooperation, under four different social norms of reputation assessment