Pure Strategy Saddle Points in the Generalized Progressive Discrete Silent Duel with Identical Linear Accuracy Functions

Abstract

A finite zero-sum game defined on a subset of the unit square is considered. The game is a generalized progressive discrete silent duel, in which the kernel is skew-symmetric, and the players, referred to as duelists, have identical linear accuracy functions featured with an accuracy proportionality factor. As the duel starts, time moments of possible shooting become denser by a geometric progression. Apart from the duel beginning and end time moments, every following time 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. If the accuracy factor is not less than 1, the duelist’s optimal strategy is the middle of the duel time span. If the factor is less than 1, the duel solution is not always a pure strategy saddle point. In a boundary case, when the accuracy factor is equal to the inverse numerator of the ratio that is the time moment preceding the duel end moment, the duel has four pure strategy saddle points which are of the mentioned time moments. For a trivial game, where the duelist possesses just one moment of possible shooting between the duel beginning and end moments, and the accuracy factor is 1, any pure strategy situation, not containing the duel beginning moment, is optimal.