Single Sample Prophet Inequality for Uniform Matroids of Rank 2

TL;DR

This paper extends the prophet inequality with a single sample to uniform matroids of rank 2, providing a deterministic mechanism that achieves the optimal 1/2 guarantee, and conjectures further generalization.

Contribution

It introduces a deterministic mechanism for uniform matroids of rank 2 that attains the optimal 1/2 guarantee with a single sample, and establishes its optimality among deterministic methods.

Findings

Achieves 1/2 guarantee for rank 2 uniform matroids with a single sample.

Proves the optimality of the mechanism among deterministic strategies.

Conjectures generalization to all uniform matroids.

Abstract

We study the prophet inequality when the gambler has an access only to a single sample from each distribution. Rubinstein, Wang and Weinberg showed that an optimal guarantee of 1/2 can be achieved when the underlying matroid has rank 1, i.e. in the case of a single choice. We show that this guarantee can be achieved also in the case of a uniform matroid of rank 2 by a deterministic mechanism, and we show that this is best possible among deterministic mechanisms. We also conjecture that a straightforward generalization of our policy achieves the guarantee of 1/2 for all uniform matroids.