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Prophet Inequalities via Linear Programming

Halil I. Bayrak, Mustafa Ç. Pınar, Rakesh Vohra

TL;DR

This work develops a systematic linear-programming framework to derive prophet inequalities for online stopping problems that amount to selecting a point in a polyhedron. By using a reduced-form LP that links online interim allocations with the offline prophet solution, the authors derive several general results, including a $\frac{1}{K+1}$-prophet inequality for problems with $K$ constraints, a $\frac{1}{2}$-prophet bound for polymatroid constraints, a $\frac{1}{2}$ bound for on-line polymatroids where the constraint changes with realized rewards, and a $\frac{1}{n}$ bound for correlated rewards. They also establish a Minkowski-sum composition property: if each component polyhedron supports a prophet inequality with factor $p_m$, their nonnegative Minkowski sum supports a prophet inequality with factor $\min_m p_m$. The framework offers a unified toolkit for deriving prophet inequalities across diverse stopping problems and suggests extensions to nonuniform scalings and random horizons. Overall, the paper advances understanding of online versus offline performance gaps using a clean LP-based methodology with broad applicability.

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.

Prophet Inequalities via Linear Programming

TL;DR

This work develops a systematic linear-programming framework to derive prophet inequalities for online stopping problems that amount to selecting a point in a polyhedron. By using a reduced-form LP that links online interim allocations with the offline prophet solution, the authors derive several general results, including a -prophet inequality for problems with constraints, a -prophet bound for polymatroid constraints, a bound for on-line polymatroids where the constraint changes with realized rewards, and a bound for correlated rewards. They also establish a Minkowski-sum composition property: if each component polyhedron supports a prophet inequality with factor , their nonnegative Minkowski sum supports a prophet inequality with factor . The framework offers a unified toolkit for deriving prophet inequalities across diverse stopping problems and suggests extensions to nonuniform scalings and random horizons. Overall, the paper advances understanding of online versus offline performance gaps using a clean LP-based methodology with broad applicability.

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 -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 prophet inequality holds for polyhedra and , it also holds for their Minkowski sum.
Paper Structure (9 sections, 10 theorems, 53 equations)

This paper contains 9 sections, 10 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Minkowski Sums
  4. Applications
  5. $K$ Many Constraints
  6. Polymatroids
  7. On-line Polymatroids
  8. Correlated Rewards
  9. Concluding Remarks

Key Result

Lemma 1

$\mathbf Q^n$ is implementable if and only if $Q_{i+1}(r_{i+1}) \leq h_{i+1}(\mathbf Q^i)$ for all rewards $r_{i+1} {\in R_{i+1}}$ and all $i \in [n-1]$.

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Example 1
  • ...and 10 more