Pure strategy solutions in progressive discrete silent duel with linear accuracy and compactified shooting moments

Анотація

A zero-sum gamedefined 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. The twoduelistshaveidentical linear accuracyfunctionsvaried by the positiveaccuracyproportionalityfactor. As the duelstarts, time moments of possibleshootingbecomedenser in accordancewith a pattern, whereevery next moment is a fractionwhose numerator and denominator, beinggreater by 1, areincreased by 1 compared to the preceding moment. Due to the skew-symmetry, both the duelistshave the same optimalstrategies and the gameoptimal value is 0. For nontrivialgames, where the duelistpossessesmorethanjust one moment of possibleshootingbetween the duelbeginning and end moments and the accuracyproportionalityfactor is not less than 1, the single optimal pure strategy is to shoot at the middle of the duel time span. As the factorbecomes less than 1, only the twothirds of the duel time span and the duelvery end can be optimal pure strategies, for which the factorshould be equal to $\dfrac{1}{2}$ or not exceed the reciprocal of the number of shootingmomentsdecreased by 2. Progressive discrete silent duelswithfourshootingmoments and greaterare not solved in pure strategies as the respective accuracyproportionalityfactor, being less than 1, occupies at least 50\,\% of interval$\left( {0;\;1} \right)$. As the duel size increases, this pure strategyinsolvabilitypercentagegrows by the same pattern that time moments of possibleshootingbecomedenser.