Title: Zero-sum games with charges
We consider two-player zero-sum games with infinite action spaces and bounded payoff functions. The players' strategies are finitely additive probability measures, called charges. Since a strategy profile does not always induce a unique expected payoff, we distinguish two extreme attitudes of players. A player is viewed as pessimistic if he always evaluates the range of possible expected payoffs by the worst one, and a player is viewed as optimistic if he always evaluates it by the best one. This approach results in a definition of a pessimistic and an optimistic value for each player. We provide an extensive analysis of the relation between these values, and connect them to the classical values. In addition, we also examine existence of optimal strategies with respect to these values.