Prophet Secretary for Combinatorial Auctions and Matroids

TL;DR

This paper introduces improved prophet inequalities for matroids and combinatorial auctions under random arrival order, achieving a -approximation, advancing the understanding of stochastic selection in complex combinatorial settings.

Contribution

It presents the first -approximation prophet inequalities for matroids and combinatorial auctions with random arrival order, extending prior work from single items and controlled arrival sequences.

Findings

Achieved -approximation for matroids and combinatorial auctions.

Developed a threshold-based technique converting discrete problems into continuous models.

Improved upon previous bounds for random arrival models in complex combinatorial settings.

Abstract

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 matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} 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 \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} 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.