An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions
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 worstcase competitive analysis, of particular importance in the design and analysis of simple (postedprice) 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 primaldual 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 itemindependence assumption.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.09784
 Bibcode:
 2020arXiv200409784D
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Computer Science  Data Structures and Algorithms