Prophet Inequality on I.I.D. Distributions: Beating 1-1/e with a Single Query

TL;DR

This paper introduces a new model for the prophet inequality problem where an oracle provides quantile queries, and demonstrates that with minimal queries, the algorithm can surpass the classic 1-1/e competitive ratio, achieving up to 0.6718.

Contribution

The paper proposes a novel quantile-query model for prophet inequalities and develops algorithms that beat the 1-1/e benchmark with only one or two queries.

Findings

A two-threshold algorithm achieves a 0.6786 competitive ratio.

A single-query observe-and-accept algorithm achieves a 0.6718 ratio.

The algorithms outperform the classic 1-1/e benchmark.

Abstract

In this work, we study the single-choice prophet inequality problem, where a gambler faces a sequence of~$n$ online i.i.d. random variables drawn from an unknown distribution. When a variable reveals its value, the gambler needs to decide irrevocably whether or not to accept the value. The goal is to maximize the competitive ratio between the expected gain of the gambler and that of the maximum variable. It is shown by Correa et al. that when the distribution is unknown or only $o(n)$ uniform samples from the distribution are given, the best an algorithm can do is $1/e$-competitive. In contrast, when the distribution is known or $\Omega(n)$ uniform samples are given, the optimal competitive ratio of 0.7451 can be achieved. In this paper, we study a new model in which the algorithm has access to an oracle that answers quantile queries about the distribution and investigate to what extent we can use a small number of queries to achieve good competitive ratios. We first use the answers from the queries to implement the threshold-based algorithms and show that with two thresholds our algorithm achieves a competitive ratio of $0.6786$. Motivated by the two-threshold algorithm, we design the observe-and-accept algorithm that requires only a single query. This algorithm sets a threshold in the first phase by making a query and uses the maximum realization from the first phase as the threshold for the second phase. It can be viewed as a natural combination of the single-threshold algorithm and the algorithm for the secretary problem. By properly choosing the quantile to query and the break-point between the two phases, we achieve a competitive ratio of $0.6718$, beating the benchmark of $1-1/e$.