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Prophet inequalities compare the expected performance of an online algorithmfor a stochastic optimization problem to the expected optimal solution inhindsight. They are a major alternative to classic worst-case competitiveanalysis, of particular importance in the design and analysis of simple(posted-price) incentive compatible mechanisms with provable approximationguarantees.  A central open problem in this area concerns subadditive combinatorialauctions. Here $n$ agents with subadditive valuation functions compete for theassignment of $m$ items. The goal is to find an allocation of the items thatmaximizes the total value of the assignment. The question is whether thereexists a prophet inequality for this problem that significantly beats the bestknown 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 isalso constructive, resulting in an online policy that takes the form of staticand anonymous item prices that can be computed in polynomial time givenappropriate query access to the valuations. As an application of our approach,we construct a simple and incentive compatible mechanism based on posted pricesthat achieves an $O(\log \log m)$ approximation to the optimal revenue forsubadditive valuations under an item-independence assumption.

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