Optimality of internal time moments in 1-bullet silent duels with generalized exponentially-convex rewards

Abstract

The finite 1-bullet silent duel is a 0-value timing game, in which the duel time span is equidistantly quantized, and each of the two duelists has a generalized exponentially-convex reward function. The duel is a symmetric matrix game, and each of the duelists has the same set of optimal strategies. Such a set can consist of either optimal time moments or of probabilistic mixtures over the finite set of successive time moments of possible acting. A particular interest is in optimality of internal time moments, wherein no duel starting and duel final moments are considered. However, the starting moment is never optimal in duels with generalized exponentially-convex rewards, whichever the factor of reward steepness is. Next, there is no optimal internal moment in $3 \times 3$ duels. In $4 \times 4$ duels, the second time moment is never optimal, and the third time moment is optimal only if the reward steepness factor does not exceed a unique positive root of an algebraic equation with a sum of two exponential functions. The conditions of when an internal time moment is optimal in bigger duels are specified as well, where another algebraic equation with a sum of four exponential functions is included. In general, if the number of all time moments is even, then no optimal time moments exist in the first half of the duel span. This is also true if the number of all time moments is odd, where the time moment being right in the duel middle is non-optimal as well.