A Single-Shot Noisy Duel with Detection Uncertainty

Duelists, Blue and Red, each with one noisy shot and with accuracies at distance − x of P 1 ( x ) and P 2 ( 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 P 1 ( x ). If Red fires first, or simultaneously with Blue, at x , Blue survives with probability 1 − P 2 ( x ). The payoff is the probability that Blue survives. This game is solved under the assumptions that the functions P 1 , P 2 , and F have continuous first derivatives, and that P 1 and P 2 are strictly increasing when not equal to zero or one.