Matroid Prophet Inequalities

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.

💡 Research Summary

The paper extends the classic prophet inequality—originally stating that a gambler who knows the distributions of a sequence of independent non‑negative random variables can achieve at least half the expected reward of a prophet who knows the realized values—to settings where both the gambler and the prophet are subject to combinatorial constraints expressed by matroids. In the standard prophet inequality the gambler makes a single selection (rank‑one matroid). Here the authors consider a general matroid M = (E, I) and allow the gambler to select a set S ⊆ E that must be independent in M, while the prophet may select the optimal independent set O maximizing the sum of the realized values.

The main technical contribution is twofold. First, they prove that for any matroid the gambler can guarantee an expected reward of at least ½ · E