Pure strategy solutions in the progressive discrete silent duel with quadratic accuracy symmetry and shooting uniform jitter

Abstract

A generalized class of the discrete game of timing is solved, where possible shooting moments are uniformly jittered. This is a finite zero-sum game defined on a symmetric lattice of the unit square. The game is a progressive discrete silent duel whose kernel is skew-symmetric, and the duelist having a single bullet shoots with quadratic accuracy. As the duel starts, possible shooting moments become denser by a geometric progression, where every following moment, apart from the duel beginning and end moments, 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. The 3x3 duel always has a pure strategy solution, whichever the jitter is. As the duel becomes bigger, an open interval of pure strategy solution non-existence appears. The endpoints of the open interval are irrational. The 4x4 duel has three jitter intervals, within which it has a pure strategy solution, whose optimal strategies can be only either a jittered middle or three-quarters of the duel time span, and the duel end moment. Bigger duels have two jitter intervals, within which a single pure strategy solution exists, but a jittered middle of the duel time span is never optimal. The 4x4 duel has two open intervals of the jitter, within which it does not have a pure strategy solution. Bigger duels have just a single open interval of the jitter, where no pure strategy solution exists. The left endpoint of this interval depends on the number of possible shooting moments.