A solution for private assessment in indirect reciprocity using solitary observation

Although indirect reciprocity is a fundamental mechanism in the evolution of human cooperation, most studies assume public assessment in which individuals are not permitted to obtain private assessments of others. Existing studies on private assessment have used individual-based simulations because of the analytical difficulty involved. Here, we develop an analytical method using solitary observation to solve private assessment in indirect reciprocity problem without any approximation. In this study, we formulate a model of solitary observation and calculate the replicator dynamics systems of five leading norms of indirect reciprocity. Indirect reciprocity in private assessment provides a different result to that in public assessment. According to the existence proofs of cooperative evolutionarily stable (CES) points in the system, strict norms (stern judging and shunning) have no CES point in private assessment, while they do in public assessment. Image scoring does not change the system regardless of the assessment types because it does not use second-order information. In tolerant norms (simple standing and staying), the CES points move to co-existence of norms and unconditional cooperators. Despite the fact that there is no central coercive assessment system in private assessment, the average cooperation rate at the CES points in private assessment is greater than that in public assessment. This is because private assessment gives unconditional cooperators a role. Our results also show the superiority of the staying norm. Compared with simple standing, staying has three advantages in private assessment: a higher cooperation rate, easiness of invasion into defectors, and robustness to maintain cooperative evolutionarily stable situations. Our results are applicable to general social dilemmas in relation to private information. Under some dilemmas, norms or assessment rules should be carefully chosen to enable cooperation to evolve.