The Regularity of the Value Function of Repeated Games with Switching Costs

We study repeated zero-sum games where one of the players pays a certain cost each time he changes his action. We derive the properties of the value and optimal strategies as a function of the ratio between the switching costs and the stage payoffs. In particular, the strategies exhibit a robustness property and typically do not change with a small perturbation of this ratio. Our analysis extends partially to the case where the players are limited to simpler strategies that are history independent-namely, static strategies. In this case, we also characterize the (minimax) value and the strategies for obtaining it.