Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Nonstochastic Inputs

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), multidimensional 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.

An O(log log m) prophet inequality for subadditive combinatorial auctions

“…XOS) functions. This leads to "balanced prices" in the sense of [11], where the sum of all prices matches the optimal allocation precisely.…”

“…More generally, the balanced prices framework of [11] entails constructing prices for any fixed valuation profile, such that (a) the prices of any subset of items partially offset the value lost due to allocating these items, and (b) the sum of all prices is upper bounded by the total value of the optimal allocation of all items. With parameters 1/α and β for (a) and (b) this leads to O(1/(αβ)) price-based prophet inequalities.…”

“…Prophet inequalities for XOS and subadditive combinatorial auctions in which agents arrive one by one were previously given in [6], [11] and [21]. For XOS valuations an optimal factor 2 is shown in [6], [11], and this can be improved to 1 − 1/e by additionally assuming agents arrive in random order [21]. For subadditive valuations, Feldman et al [6] give an O(log m) approximation.…”

“…Using Lemma III.3 it is straightforward to prove Theorem III.1 using a hallucination trick similar to that in the price of anarchy literature [20], [31], the literature on algorithmic stability [32], and in the balanced prices framework for prophet inequalities [11].…”

“…Interest in prophet inequalities has surged recently, in part due to applications in pricing and auction design driven by the observation that the fixed threshold can be viewed as a posted price. This has lead to a line of literature studying prophet inequalities for more general instances of the allocation problem, yielding approximatelyoptimal online policies and incentive compatible auctions for increasingly general problem instances with many objects and rich classes of valuation functions (e.g., [4], [5], [6], [7], [8], [9], [10], [11], [12]).…”