On Optimal Allocation of a Continuous Resource Using an Iterative Approach and Total Positivity

We study a class of optimal allocation problems, including the well-known Bomber Problem, with the following common probabilistic structure. An aircraft equipped with an amount~$x$ of ammunition is intercepted by enemy airplanes arriving according to a homogenous Poisson process over a fixed time duration~$t$. Upon encountering an enemy, the aircraft has the choice of spending any amount~$0\le y\le x$ of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of $y$. Two different goals have been considered in the literature concerning the optimal amount~$K(x,t)$ of ammunition spent: (i)~Maximizing the probability of surviving for time~$t$, which is the so-called Bomber Problem, and (ii) maximizing the number of enemy airplanes shot down during time~$t$, which we call the Fighter Problem. Several authors have attempted to settle the following conjectures about the monotonicity of $K(x,t)$: [A] $K(x,t)$ is decreasing in $t$, [B] $K(x,t)$ is increasing in $x$, and [C] the amount~$x-K(x,t)$ held back is increasing in $x$. [A] and [C] have been shown for the Bomber Problem with discrete ammunition, while [B] is still an open question. In this paper we consider both time and ammunition continuous, and for the Bomber Problem prove [A] and [C], while for the Fighter we prove [A] and [C] for one special case and [B] and [C] for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 ($\mbox{TP}_2$) in the Bomber and Fighter Problems, respectively. The $\mbox{TP}_2$ property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves $\mbox{TP}_2$ and other properties.