Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to ma- troids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg [STOC 2012] and Feldman et al. [SODA 2015] show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it’s conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1−1/e)- approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan [SODA 2011] and Esfandiari et al. [ESA 2015] who worked in the special cases where we can fully control the arrival order or when there is only a single item.

Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

Joint work with Soheil Ehsani, MohammadTaghi Hajiaghayi, and Sahil Singla. To appear in SODA 2018.