Repeated Games with Observable Actions in Continuous Time: Costly Transfers in Repeated Cooperation

I propose a way to formulate and solve for sub-game perfect equilibria of continuous-time repeated games with both observable and unobservable actions. The main idea is to study directly self-enforcing agreements corresponding to the interaction, without setting up the full extensive-form game. To discipline observable deviations, I restrict that players are stuck with their deviations for a small amount of time. This restriction simultaneously makes the model tractable and ensures that agreements are well defined. To illustrate, I consider a setting with two players colluding with imperfectly observable productive actions and observable money transfers. Transfers are costly: only a fraction of the amount sent is delivered. I introduce self-enforcing public agreements, which mimic pure-strategy public perfect equilibria from discrete time. For a fixed interest rate, I characterize the set of payoffs attainable in such agreements. In optimal agreements, costly transfers are used rarely, only when the participation constraint of either player binds.