Optimal Single-Choice Prophet Inequalities from Samples

TL;DR

This paper improves the understanding of prophet inequalities with samples, showing optimal ratios can be achieved with minimal samples and resolving an open problem for identical distributions.

Contribution

It demonstrates that the optimal single-choice prophet inequality ratio can be achieved with just one sample per distribution and extends results to multiple samples for identical distributions.

Findings

Optimal ratio of 1/2 with one sample per distribution.

Achieves approximately 0.745 ratio with O(n) samples for identical distributions.

Resolves an open problem in prophet inequalities with samples.

Abstract

We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $\varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within $(1+\varepsilon)$, resolving an open problem of Correa, D\"utting, Fischer, and Schewior.