Middle is the Invariant Unique Optimal Time Moment in Exponentially-Concave-Reward 1-Bullet Progressive Silent Duel

The finite 1-bullet progressive silent duel is considered, in which each of the two duellists shoots with exponentially-concave reward. The duel models a system of one-decision-making competition between two identical intelligent competitors (duellists) through a quantized time span, in which the duellist benefits from shooting as late as possible but only by acting first. The duel is silenced because the duellist does not learn about the action of the other duellist until the very end moment of the duel. Thus, the duel is a symmetric matrix game whose optimal value is 0, and the set of optimal strategies is the same for both duellists, regardless of the duel size and how time is quantized. The duel time quantization is such that time progresses by the geometrical progression pattern, according to which every following time moment is the partial sum of the respective geometric series. In this duel, the duellist has the invariant unique optimal time moment, which is the duel middle, regardless of the number of time moments. Usually, the system manager benefits namely from such a solution, by which the manager latently forces both competitors to act at the invariant optimal moment, maintaining the system more stable and controllable.