Prophet Inequalities with Limited Information

By Pablo D. Azar, Robert D. Kleinberg, S. Matthew Weinberg. 2013

137 citations recorded by the source corpus.

In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1, ..., Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards V1, ..., Vn are drawn. The assumption that the gambler knows the distribution from which V1, ..., Vn are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related secretary problem to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called "virtual values" even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities.

Subfields: Computer Networks and Communications, Management Science and Operations Research, Marketing, Game & Decision Theory

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Citations in the corpus

Online Adversarial Attacks (2021, background): “Well known single threshold-based algorithms that solve the k-secretary problem include the Virtual, Optimistic Babaioff et al. [2007] and the recent Single-Ref algorithm Albers et al. [2019].”
Secretaries with Advice (2020, unknown): “, [4, 9, 10, 11, 21, 32]).”
Competitive Analysis with a Sample and the Secretary Problem (2019, unknown): “The single sample prophet inequality (see [1] for example) relaxes the assumption that the distributions are known.”
The sample complexity of revenue maximization (2014, result): “est constants like 1 4 . For example, previous work has demonstrated that auctions with reasonably good approximation factors are possible with minimal dependence on the valuation distributions (e.g. [9, 26, 2]) or even, when there is no bidder with a unique valuation distribution, with no dependenc”