Prophet Inequality from Samples: Is the More the Merrier?

TL;DR

This paper investigates how additional samples affect the prophet inequality problem, showing that more samples do not always improve the competitive ratio and analyzing the performance of ordinal static threshold algorithms.

Contribution

It proves that having more samples than one per distribution can reduce the optimal competitive ratio and analyzes the best static threshold algorithms within this setting.

Findings

Optimal ratio of 1/2 is unattainable with more samples.

Best static threshold algorithm achieves a 0.433 ratio.

Provides an alternative proof of Rubinstein et al.'s main result.

Abstract

We study a variant of the single-choice prophet inequality problem where the decision-maker does not know the underlying distribution and has only access to a set of samples from the distributions. Rubinstein et al. [2020] showed that the optimal competitive-ratio of $\frac{1}{2}$ can surprisingly be obtained by observing a set of $n$ samples, one from each of the distributions. In this paper, we prove that this competitive-ratio of $\frac{1}{2}$ becomes unattainable when the decision-maker is provided with a set of more samples. We then examine the natural class of ordinal static threshold algorithms, where the algorithm selects the $i$-th highest ranked sample, sets this sample as a static threshold, and then chooses the first value that exceeds this threshold. We show that the best possible algorithm within this class achieves a competitive-ratio of $0.433$. Along the way, we utilize the tools developed in the paper and provide an alternative proof of the main result of Rubinstein et al. [2020].