Pure strategy solutions of the silent duel on the uniform lattice with identical linear accuracy functions

Анотація

A finite zero-sum game defined on a uniform lattice of the unit square is solved as a game of timing. The game is a 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. 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 the number of possible shooting moments decreased by 2 then the duelist’s optimal behavior is to shoot at the middle of the duel time span. A boundary case of the accuracy factor is determined, where the factor is equal to the reciprocal of the number of possible shooting moments decreased by 2. In this case, the duel has four pure strategy solutions issued by the two last moments of possible shooting. Otherwise, the duel has a single optimal pure strategy situation by when the accuracy factor belongs to the definite nonempty interval. A 4x4 or bigger duel is not solved in pure strategies if this membership is false or the interval in the membership is empty. The trivial duels, with just two or three moments of possible shooting, are always solved in pure strategies.