Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

TL;DR

This paper introduces a unified framework for stochastic online maximization problems using price-based algorithms, extending prophet inequalities to complex combinatorial settings and connecting them with mechanism design.

Contribution

It develops a general extension theorem for prophet inequalities, simplifying and unifying existing results while enabling new applications in combinatorial markets and mechanism design.

Findings

Unified framework for prophet inequalities and posted price mechanisms

Improved approximation guarantees for combinatorial markets and matroids

Connection between smooth mechanisms and posted price approaches

Abstract

We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.