Effect of private information on indirect reciprocity

Indirect reciprocity is one of the key mechanisms for the evolution of cooperation. It relies on mutual monitoring and assessments, i.e., individuals collect information about the past behavior of others and judge whether that behavior is ``good'' or ``bad.'' A player will not be helped if labeled with a bad image. There are many ways for assessing others, each of which can be interpreted as an elementary form of a moral sense (i.e., a view on what is good or bad). The information can be either public or private: private information can lead to mismatches between the opinions of individuals even when they share the same moral sense. In this paper, the effect of private information on the best-known assessment rules is investigated. In order to calculate payoffs, the concept of an image matrix is introduced. It describes who is good in the eyes of whom, and its time evolution is given by a probabilistic Boolean automaton. In contrast to the public information case, private information leads to the collapse of the sterner assessment rule. Alternatively, stable polymorphisms may subsist, with the milder rule and a more simple-minded rule coexisting together with unconditional cooperators; thus, cooperation can be sustained by indirect reciprocity even in the absence of public information.