Repeated Games with Switching Costs: Stationary vs History Independent Strategies

TL;DR

This paper analyzes zero-sum repeated games with switching costs, showing the existence of stationary strategies based on previous actions and characterizing their values, including cases with history-independent strategies and bounds on performance loss.

Contribution

It provides a full characterization of game values and strategies for both stationary and history-independent cases, highlighting robustness and limitations.

Findings

Stationary strategies depend only on the previous action.

Full characterization of the game value and optimal strategies.

Bounds on the loss when restricting to history-independent strategies.

Abstract

We study zero-sum repeated games where the minimizing player has to pay a certain cost each time he changes his action. Our contribution is twofold. First, we show that the value of the game exists in stationary strategies, depending solely on the previous action of the minimizing player, not the entire history. We provide a full characterization of the value and the optimal strategies. The strategies exhibit a robustness property and typically do not change with a small perturbation of the switching costs. Second, we consider a case where the minimizing player is limited to playing simpler strategies that are completely history-independent. Here too, we provide a full characterization of the (minimax) value and the strategies for obtaining it. Moreover, we present several bounds on the loss due to this limitation.