Pairwise assessment correlations predict cooperation of the Leading Eight under private assessment

Abstract

Indirect reciprocity can sustain cooperation when individuals use reputations to decide whom to help. The Leading Eight norms sustain such cooperation under public assessment, where all individuals share the same assessments. Under private assessment, however, observers may disagree, yet their assessments remain correlated. Two quantities, the average fraction of good assessments $h$ and the conditional probability $h_{\mathrm G}$ that one observer assesses a player as good given that another does, are known to determine the stability of cooperation, but how $h_{\mathrm G}$ itself is determined has remained open. Within a pairwise approximation, we derive a closed equation for $h_{\mathrm G}$ that, together with that for $h$, determines both quantities for all eight norms, including third-order ones. This description substantially improves agreement with simulations over a mean-field approximation that neglects correlations, and recovers the known classification of the Leading Eight into three classes. The remaining discrepancies are largest for L2 and L5 at low error rates and decrease as assessment error increases. Relating $h_{\mathrm G}$ to the variance of individual goodness suggests that these discrepancies reflect information beyond the first two moments of the goodness distribution. Our framework provides a concise and analytically tractable tool for evaluating social norms under private assessment.