A Single-Shot Noisy Duel with Detection Uncertainty

Duelists, Blue and Red, each with one noisy shot and with accuracies at distance −x of P₁(x) and P₂(x), x ϵ [X, 0], approach each other from an initial distance −X. Red can see Blue and knows the current value of x. Blue detects Red at distance −y, where y has distribution function F(y), F(X) = 0. Subsequent to detection Blue knows the current x. Red is not informed when he has been detected. If Blue fires first at x, he survives with probability P₁(x). If Red fires first, or simultaneously with Blue, at x, Blue survives with probability 1 − P₂(x). The payoff is the probability that Blue survives. This game is solved under the assumptions that the functions P₁, P₂, and F have continuous first derivatives, and that P₁ and P₂ are strictly increasing when not equal to zero or one.