Indirect reciprocity with optional interactions and private information

We consider indirect reciprocity with optional interactions and private information. A game is offered between two players and accepted unless it is known that the other person is a defector. Whenever a defector manages to exploit a cooperator, his or her reputation is revealed to others in the population with some probability. Therefore, people have different private information about the reputation of others, which is a setting that is difficult to analyze in the theory of indirect reciprocity. Since a defector loses a fraction of his social ties each time he exploits a cooperator, he is less efficient at exploiting cooperators in subsequent rounds. We analytically calculate the critical benefit-to-cost ratio above which cooperation is successful in various settings. We demonstrate quantitative agreement with simulation results of a corresponding Wright-Fisher process with optional interactions and private information. We also deduce a simple necessary condition for the critical benefit-to-cost ratio.