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.

Funding: The project leading to this publication has received funding from the French government under the “France 2030” investment plan managed by the French National Research Agency [Grant ANR-17-EURE-0020] and from the Excellence Initiative of Aix-Marseille University–A*MIDEX. Y. Tsodikovich was supported in part by the Israel Science Foundation [Grants 2566/20, 1626/18, and 448/22]. X. Venel acknowledges the financial support of the French National Research Agency through Project CIGNE [ANR-15-CE38-0007-01].