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

TL;DR

This paper proves the 'spend-it-all' conjecture for the Bomber Problem by solving the integral equation for optimal survival probability in certain regions, establishing where full expenditure of ammunition is optimal.

Contribution

It completes the proof of the conjecture, precisely characterizing the boundary where the optimal strategy involves spending all available ammunition.

Findings

Exact solutions for the integral equation in specific regions

Complete proof of the 'spend-it-all' conjecture

Characterization of the boundary for optimal ammunition expenditure

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 $K(x,t)=x$.