This paper investigates how cooperation can be sustained in large societies even in the presence of agents who never cooperate. In order to do this, we consider a large but finite society where in each period agents are randomly matched in pairs. Nature decides, within each match, which agent needs help in order to avoid some loss, and which agent can help him incurring a cost smaller than the loss. Each agent observes only his own history, and we assume that agents do not recognize each other. We introduce and characterize a class of equilibria, named linear equilibria, in which cooperation takes place. Within this class, efficiency can be achieved with simple one-period strategies, which are close to a tit-for-tat strategy when the society is large, and which generate smooth dynamics of the expected average level of cooperation. Unlike previously suggested equilibria in similar environments, our equilibria are robust to the presence of behavioral agents and other perturbations of the base model. We also apply our model to bilateral trade with many traders, where we find that the mechanism of transmitting defections is transmissive and not contagious as in our base model.
