Matroid Prophet Inequalities

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 of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.