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On optimal allocation of a continuous resource using an iterative approach and total positivity

Part of:Sequential methodsGame theoryStochastic processes

Published online by Cambridge University Press:  01 July 2016

Jay Bartroff,
Larry Goldstein,
Yosef Rinott and
Ester Samuel-Cahn
Jay Bartroff*
Affiliation:
University of Southern California
Larry Goldstein*
Affiliation:
University of Southern California
Yosef Rinott*
Affiliation:
The Hebrew University of Jerusalem
Ester Samuel-Cahn*
Affiliation:
The Hebrew University of Jerusalem
*
∗ Postal address: Department of Mathematics, University of Southern California, 1042 West 36th Place, Los Angeles, CA 90089-1113, USA.
∗ Postal address: Department of Mathematics, University of Southern California, 1042 West 36th Place, Los Angeles, CA 90089-1113, USA.
∗∗ Postal address: Department of Statistics and Center for the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem, 91905, Israel.
∗∗ Postal address: Department of Statistics and Center for the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem, 91905, Israel.
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Abstract

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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 homogeneous Poisson process over a fixed time duration t. Upon encountering an enemy, the aircraft has the choice of spending any amount 0 ≤ y ≤ 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. Conjectures (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 to be continuous, and, for the bomber problem, we prove (A) and (C), while, for the fighter problem, 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 (TP2) in the bomber and fighter problems, respectively. The TP2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves TP2 and other properties.

Keywords

Bomber problemlog-concaveoptimal allocationsequential optimizationtotal positivity

MSC classification

Primary:60G40: Stopping times; optimal stopping problems; gambling theory
Secondary:62L05: Sequential design91A60: Probabilistic games; gambling

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010, pp. 795 - 815
Copyright
Copyright © Applied Probability Trust 2010 

References

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