Games of Timing

5.1 Non-zero Sum Silent DuelConsider the following non-zero sum duel. Two players (player 1 and 2), starting at time t = 0 at a unit distance before each one’s target, walk toward each one’s target at a constant unit speed with no opportunity to retreat. They will reach their targets at time t = 1. Each player has a gun with one bullet, which may be shot at any time in [0, 1]. Each player selects a time to shoot. The accuracies of shooting are described by the accuracy functions A₁(x) and A₂(x), where A_i(x) is the probability of hitting by player i his target if he shoots at time x. These functions are differentiable and strictly increasing in [0, 1] such that A_i(0) = 0, A_i(1) = 1 and A_i′(x) > 0 for x ∊ (0, 1), where i = 1, 2. The accuracy functions are fixed and known beforehand to both players. As soon as one of the players hits his target, the contest is over and the first player hitting his target gets payoff 1, and his opponent gets payoff zero. If none of the players hit their targets their payoffs are zero. If both players hit their target at the same time they share their payoffs. Suppose that both players have silent guns, so if player 1 and player 2 shoot at time x and y, respectively, their payoffs are given as follows $$\begin{array}{*{20}{c}} {{{M}_{1}}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{{A}_{1}}(x)} \hfill & {for x < y,} \hfill \\ {{{P}_{1}}(x)} \hfill & {for x = y,} \hfill \\ {(1 - {{A}_{2}}(y)){{A}_{1}}(x)} \hfill & {for x > y,} \hfill \\ \end{array} } \right.} \\ {{{M}_{2}}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{{A}_{2}}(y)} \hfill & {for y < x,} \hfill \\ {{{P}_{2}}(y)} \hfill & {for y = x,} \hfill \\ {(1 - {{A}_{1}}(x)){{A}_{2}}(y)} \hfill & {for y > x,} \hfill \\ \end{array} } \right.} \\ \end{array}$$ where P_i (x) ∊ [0, A_i(x)) for x ∊ (0, 1], P_i(0) = 0, for example, if the players share the payoff equally when they both hit the targets, then