Posted Prices, Smoothness, and Combinatorial Prophet Inequalities

We present a general framework for proving combinatorial prophet inequalities and con-structing posted-price mechanisms. Our framework applies to stochastic welfare optimizationproblems, in which buyers arrive sequentially and make utility-maximizing purchases. Our anal-ysis takes the form of an extension theorem: we derive sufficient conditions for achieving welfarebounds in the special case of deterministic valuations, then prove that these bounds extenddirectly to stochastic settings. Furthermore, our welfare bounds compose in the sense that thewelfare guarantees are preserved when buyers participate in many optimization problems simul-taneously. Our sufficient conditions have a natural economic interpretation, and our approachis closely connected to the smoothness framework for bounding the price of anarchy of mecha-nisms. We show that many smooth mechanisms can be recast as posted price mechanisms withcomparable performance guarantees. We illustrate the power of our framework in a range ofapplications, including combinatorial auctions, matroids, and sparse packing programs, wherewe unify and improve many of the previously known results. Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK, Email:[email protected] School of Computer Science, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel. Email:[email protected] Planck Institute for Informatics and Saarland University, Saarland Informatics Campus, Campus E1 4,66123 Saarbrücken, Germany. Email: [email protected]. Supported in part by the DFG throughCluster of Excellence MMCI. Part of this work was done while the author was visiting Simons Institute for the Theoryof Computing.Microsoft Research, 1 Memorial Drive #1, Cambridge, MA 02142, USA. Email: [email protected]