Pure strategy saddle point in progressive discrete silent duel with quadratic accuracy functions of the players

Анотація

A zero-sum game defined on a finite subset of the unit square is considered. The game is a progressive discrete silent duel, in which the kernel is skew-symmetric. As the duel starts, time moments of possible shooting become denser by a geometric progression. Apart from the duel beginning and end moments, every following moment is the partial sum of the respective geometric series. Due to the skew-symmetry, both the duelists have the same optimal strategies and the game optimal value is 0. It is proved that the solution of a progressive discrete silent duel, with identical accuracy functions of the duelists, is a pure strategy saddle point. For nontrivial games, where the duelist possesses more than just one moment of possible shooting between the duel beginning and end moments, the saddle point is single. Moreover, the solution renders the game into an invariant decision. For the linear accuracy, whose value is not less than its time moment, the optimal strategy is the middle of the duel time span. For the quadratic accuracy the optimal strategy is the middle of the second half of the duel time span. If the linear accuracy value is less than its time moment, the middle of the duel time span is never optimal.