A Duel Involving False Targets

A duel is initated by an attacker at some time t in [−T, 0]. The defender possesses weapons at −T and encounters “false” targets that occur at time t with probability density λ(t) and are classified as “real” with probability c_fa. At the time of attack, the probabilities that the attacker is detected and classified as real are D and c_aa (D, c_aa, c_fa are constant). If the defender responds with one of his k weapons at the time of attack, he survives with probability p_k, and, if he does not respond, he survives with probability q_k; q_k < p_k, p_k ≦ p_k+1, q_k ≦ q_k+1 for k = 0, 1, 2, …. The payoff is the defender's survival probability. Both players are informed of the current time, the defender's weapon level, λ, D, c_aa, C_fa, p_k and q_k (k = 0, 1, 2, …). The attacker selects a time of attack to minimize the payoff. He may change the attack time as the defender expends weapons against false targets. The defender responds to classifications so as to maximize the payoff. This paper derives an iterative system of first-order differential equations whose unique solution V₁(t), V₂(t), …, V_k(t), … at time t is the value of the game when the defender has 1, 2, …, k, … weapons, respectively, at time t. It expresses the optimal strategies in terms of the values, and determines the limit of V_k(t) as k → ∞ with k / ∫_t⁰λ(s) ds held constant.