Indirect reciprocity with optional interactions.

Indirect reciprocity is a mechanism for the evolution of cooperation that is relevant for prosocial behavior among humans. Indirect reciprocity means that my behavior towards you also depends on what you have done to others. Indirect reciprocity is associated with the evolution of social intelligence and human language. Most approaches to indirect reciprocity assume obligatory interactions, but here we explore optional interactions. In any one round a game between two players is offered. A cooperator accepts a game unless the reputation of the other player indicates a defector. For a game to take place, both players must accept. In a game between a cooperator and a defector, the reputation of the defector is revealed to all players with probability Q. After a sufficiently large number of rounds the identity of all defectors is known and cooperators are no longer exploited. The crucial condition for evolution of cooperation can be written as hQB>1, where h is the average number of rounds per person and B=(b/c)-1 specifies the benefit-to-cost ratio. We analyze both stochastic and deterministic evolutionary game dynamics. We study two extensions that deal with uncertainty: hesitation and malicious gossip.