Optimality of penultimate and final time moments in 1-bullet silent duels with generalized exponentially-convex rewards

Анотація

Silent duels are a class of timing games that model competitive interaction among a group of participants through a given time span under informational uncertainty and reward limitation. The circumstances of the interaction are such that the participant does not learn about actions of the other participants until the duel end, and the participant benefits from acting as late as possible but only by acting first. The finite 1-bullet silent duel is considered, in which the duel time span is equidistantly quantized, and each of the two duelists has a generalized exponentially-convex reward function. In this paper, optimality of penultimate and final time moments is investigated, which are of particular interest in modeling systems, where actions (decisions) are forced to be made as late as possible. The duel is a symmetric matrix game whose optimal value is 0, and each of the duelists has the same optimal behavior, whether it is in pure or mixed strategies. It is proved that the final time moment is single optimal in $3 \times 3$ duels, whichever the factor of reward steepness is. It is also ascertained that in bigger duels the final time moment is single optimal if the factor of reward steepness is higher than the unique root of an exponential equation. If the factor is the root, both penultimate and final time moments become optimal.