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

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