Prophet Inequalities via Linear Programming

TL;DR

This paper introduces a linear programming-based method to derive prophet inequalities for stopping problems involving polyhedral constraints, providing both known and new results in the field.

Contribution

It presents a systematic LP-based technique for deriving prophet inequalities in polyhedral stopping problems, including new bounds for on-line polymatroids.

Findings

Proved a 1/2-prophet inequality for on-line polymatroids.

Established a composition property for Minkowski sums of polyhedra.

Derived several known and new prophet inequalities using the proposed method.

Abstract

Prophet inequalities bound the expected reward that can be obtained in a stopping problem by the optimal reward of its corresponding off-line version. We propose a systematic technique for deriving prophet inequalities for stopping problems associated with selecting a point in a polyhedron. It utilizes a reduced-form linear programming representation of the stopping problem. We illustrate the technique to derive a number of known results as well as some new ones. For instance, we prove a $\frac{1}{2}$-prophet inequality when the underlying polyhedron is an on-line polymatroid; one whose underlying submodular function depends upon the realized rewards. We also demonstrate a composition by the Minkowski sum property. If an $r-$ prophet inequality holds for polyhedra $P^1$ and $P^2$, it also holds for their Minkowski sum.