The Existence and Uniqueness of Nash Equilibrium Point in an m-player Game “Shoot Later, Shoot First!”

We consider the following “silent duel” of m players with a possible economic interpretation. Each player has one “bullet”, which she can shoot at any time during the time interval 0,1]. The probability that the i-th player hits the “target” at moment t is given by an increasing accuracy function f _i (t). The winner is the player who hits the target first. Under natural assumptions on the functions f _i (t) we prove the existence and uniqueness of a Nash equilibrium point in this game, and we provide an explicit construction of this equilibrium. This construction allows us to obtain exact solutions for many specific examples. Some of them are presented.This work was partly supported by RBRF grants 03-01-00479.