An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions

TL;DR

This paper introduces an improved prophet inequality of order O(log log m) for subadditive combinatorial auctions, enabling better online decision-making and revenue approximation with simple posted-price mechanisms.

Contribution

It presents the first O(log log m) prophet inequality for subadditive valuations, using a novel primal-dual approach and polynomial-time computable static prices.

Findings

Achieves an O(log log m) approximation for subadditive auctions.

Provides a polynomial-time online policy based on static prices.

Develops a simple incentive compatible mechanism with improved revenue guarantees.

Abstract

Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.