The Existence of Value and Saddle Point in Games of Fixed Duration

Differential games of fixed duration are defined. The definition of strategy follows that of Friedman, while the definition of payoff follows that of Krasovskii and Subbotin. It is shown by relatively elementary methods that games of fixed duration which satisfy the Isaacs condition have values and saddle points. It is also shown under appropriate hypotheses on the data of the problem that if the Isaacs condition holds, then the value is uniformly Lipschitz continuous on bounded sets and satisfies the Isaacs equation at all points of differentiability. The relationship of the value as defined here to other values is studied.