A Proof of the Bomber Problem's Spend-It-All Conjecture

Abstract

The Bomber Problem concerns optimal sequential allocation of partially effective ammunition \(x\) while under attack from enemies arriving according to a Poisson process over a time interval of length \(t\). In the doubly-continuous setting, in certain regions of \((x,t)\)-space we are able to solve the integral equation defining the optimal survival probability and find the optimal allocation function \(K(x,t)\) exactly in these regions. As a consequence, we complete the proof of the "spend-it-all" conjecture of Bartroff et al. (2010b) which gives the boundary of the region where \(K(x,t)=x\).