Indirect reciprocity with optional games and monitoring of interactions between defectors

We study evolution of cooperation by indirect reciprocity with optional interactions. There are repeated interactions between two types of players, cooperators and defectors, in a population of finite size. Previously, we considered the scenario where an encounter between a cooperator and a defector results in the defector's identity being revealed with some probability, while an encounter between two defectors does not reveal their identities. Here, we study a generalization of this model: an encounter between a cooperator and a defector results in the defector's identity being revealed with probability Q_C; an encounter between two defectors results in each of those defectors' identities being revealed independently with probability Q_D. We find that larger values of Q_D can significantly increase both the average payoffs for cooperators and, in a dynamical setting, the basin of attraction for cooperation. Moreover, if Q_C is sufficiently small and Q_D is sufficiently large, then we find a new behavior over the previous model in which cooperators and defectors can stably coexist. We also study hesitation to cooperate with unknown individuals and defectors refusing interactions with known defectors to preserve their own unknown status.