Analytical classification of indirect reciprocity with public and private interactions.

Parameter regions for the most cooperative, evolutionarily stable action rules are shown. p denotes the frequency of public interactions; q denotes the probability that a private interaction is observed. Perfect cooperation (using CDCD) is realized above the solid line labeled as ‘3’, whereas only partial cooperation (using CDDD) is achieved below. There are also parameter regions where the lenient strategies, CCCD and CDCC, are ESS. For b/c < 2 there exists a parameter region where no ESS other than DDDD is found (shown in gray). The curves are given by 1: q = p/[(r − 1)(1 − p)], 2: q = p/[(r + 1)p − 1], 3: q = p/(p + r − 1), and 4: q = (r − 1)p/(1 − p), where r ≡ b/c.