In this paper we study the fighter problem with discrete ammunition. An
aircraft (fighter) equipped with n anti-aircraft missiles is intercepted
by enemy airplanes, the appearance of which follows a homogeneous Poisson
process with known intensity. If j of the n missiles are spent at
an encounter, they destroy an enemy plane with probability a(j),
where a(0) = 0 and {a(j)} is a known, strictly increasing
concave sequence, e.g.
a(j) = 1 - qj, 0 < q < 1.
If the enemy is not destroyed, the enemy shoots the fighter down with known
probability 1 - u, where 0 ≤ u ≤ 1. The goal of the
fighter is to shoot down as many enemy airplanes as possible during a given
time period [0, T]. Let K(n, t) be the smallest
optimal number of missiles to be used at a present encounter, when the fighter
has flying time t remaining and n missiles remaining. Three
seemingly obvious properties of K(n, t) have been
conjectured: (A) the closer to the destination, the more of the n
missiles one should use; (B) the more missiles one has; the more one should
use; and (C) the more missiles one has, the more one should save for possible
future encounters. We show that (C) holds for all
0 ≤ u ≤ 1, that (A) and (B) hold for the `invincible
fighter' (u = 1), and that (A) holds but (B) fails for the `frail
fighter' (u = 0); the latter is shown through a surprising
counterexample, which is also valid for small u > 0 values.