Repeated Games with General Discounting

We introduce a general class of time discounting, which includes time-inconsistent ones, into repeated games with perfect monitoring. A strategy pro le is called an agent subgame perfect equilibrium if there is no pro table one-shot deviation at any history. We characterize strongly symmetric agent subgame perfect equilibria for repeated games with symmetric stage game. We nd that the harshest punishment takes different forms given different biases. When players are future biased, the harshest punishment is supported by a version of stick-and-carrot strategy. When players are present biased, the harshest punishment may take a more complex form. In particular, the worst punishment path may need to be cyclical. We also nd that the worst punishment payoff is different from the stage game minmax payoff even when players are patient. For some class of discounting, we show that the worst punishment payoff is larger than the stage game minmax payoff with present bias and smaller than the stage game minmax payo with future bias. We also characterize the set of limit equilibrium payoffs as the length of periods converges to 0, without changing the intertemporal structure of biases.