Before revision:  Second, we consider a case where the  player is limited to playing  completely history-independent strategies and provide a full characterization of the value and optimal strategies in this case. Naturally, this limitation worsens his situation. We deduce a bound on his loss in the general case as well as more precise bounds when more assumptions regarding the game or the switching costs are introduced .
After revision:  Second, we consider a case where the player is limited to playing completely history-independent strategies and 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 .