Matroid Prophet Inequalities
Robert Kleinberg, S. Matthew Weinberg

TL;DR
This paper extends the prophet inequality to multiple selections under matroid constraints, showing the gambler can achieve at least half the prophet's reward, with applications to mechanism design and revenue approximation.
Contribution
It generalizes the prophet inequality to multiple selections with matroid constraints, establishing tight bounds and applications to mechanism design.
Findings
Gambler can achieve at least half of the prophet's reward under matroid constraints.
The ratio between prophet and gambler's reward is tight at O(p) for p intersecting matroids.
Results lead to constant-factor approximations for Bayesian optimal revenue in multi-parameter settings.
Abstract
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAuction Theory and Applications · Advanced Bandit Algorithms Research · Consumer Market Behavior and Pricing
