Consider the following situation involving two agents who belong to a large society. One of the agents needs help to avoid a big loss. The other agent may either incur a low cost to help him or do nothing. If agents do not recognize each other, providing incentives for socially optimal behavior (helping) is, in general, very difficult. We use a repeated anonymous random matching setting in a large society to understand how, in the previous
situation, help may take place in equilibrium. We find explicit equilibria that, unlike other models proposed in the literature, feature smooth aggregate behavior over time and robustness to many perturbations, such as the presence of behavioral types or trembles. We consider the joint limit of increasing the size of the society and making it more interactive (or patient.) Under this limit, our equilibria resemble the tit-for-tat strategy for the
prisoner’s dilemma, introducing some small probability of forgiveness. The model is also applied to bilateral trade, where the mechanism used to spread deviations is transmissive instead of contagious. The smooth evolution of the aggregate variables over time makes the model suitable for empirical work.