The Spend-It-All Region and Small Time Results for the Continuous Bomber Problem

Abstract

A problem of optimally allocating partially effective ammunition \(x\) to be used on randomly arriving enemies in order to maximize an aircraft's probability of surviving for time~\(t\), known as the Bomber Problem, was first posed by \citet{Klinger68}. They conjectured a set of apparently obvious monotonicity properties of the optimal allocation function \(K(x,t)\). Although some of these conjectures, and versions thereof, have been proved or disproved by other authors since then, the remaining central question, that \(K(x,t)\) is nondecreasing in~\(x\), remains unsettled. After reviewing the problem and summarizing the state of these conjectures, in the setting where \(x\) is continuous we prove the existence of a ``spend-it-all'' region in which \(K(x,t)=x\) and find its boundary, inside of which the long-standing, unproven conjecture of monotonicity of~\(K(\cdot,t)\) holds. A new approach is then taken of directly estimating~\(K(x,t)\) for small~\(t\), providing a complete small-\(t\) asymptotic description of~\(K(x,t)\) and the optimal probability of survival.