Self-Enforcing Cooperation with Graduated Punishments∗

Case studies of self-enforcing cooperation in repeated interactions usually find that the punishments inflicted on deviators are mild to start with, and increase only if there is evidence of persistent deviation. We model this using a combination of imperfect monitoring and asymmetric information in a one-sided prisoner’s dilemma. Call the player with the temptation to cheat player B, and the other player A. Player B can be one of several types, distinguished by the size of this temptation; B’s type is his private information. A’s payoff may be high or low, and the probability of a low payoff is higher if B has cheated. If player A receives the low payoff, he can inflict an immediate punishment on B, at a cost to himself. We find that A’s optimal choice of punishment is generally a non-concave problem, and the comparative statics can be discontinuous and asymmetric. If the temptation of each type of B is non-stochastic, A’s optimal choice of punishment is not lowered after a high payoff, and may be raised or lowered after a bad payoff depending on the prior distribution of B’s types. With stochastic temptations, as the prior probability of B being a bad type increases, the optimal level of punishment chosen by A is initially low and decreasing, but then jumps to a higher level and increases further thereafter, and for very high levels of this probability the relationship may be terminated. Non-extreme levels of punishment are most informative about B’s type, and the value of this information to A is highest when the prior probabilities are also non-extreme. ∗This is an extremely preliminary statement of work still in progress. It was prepared by Dixit for an informal seminar presentation to elicit feedback, and has not yet been read or approved by the coauthors. Comments very welcome, addressed to [email protected] .