Bayesian Optimal Stopping with Maximum Value Knowledge

TL;DR

This paper studies an optimal stopping problem with correlated offers, assuming only the distribution of the maximum value, and shows asymptotic optimality of a threshold strategy based on the maximum's distribution.

Contribution

It introduces a novel approach to optimal stopping with limited correlation information, establishing asymptotic optimality and convergence guarantees for threshold strategies.

Findings

Deterministic threshold strategy is asymptotically optimal for sublinear maximum value growth.

Provides a tight quadratic convergence bound for smooth maximum value distributions.

Enhances understanding of prophet inequalities with correlated values.

Abstract

We consider an optimal stopping problem with n correlated offers where the goal is to design a (randomized) stopping strategy that maximizes the expected value of the offer in the sequence at which we stop. Instead of assuming to know the complete correlation structure, which is unrealistic in practice, we only assume to have knowledge of the distribution of the maximum value of the sequence, and want to analyze the worst-case correlation structure whose maximum follows this distribution. This can be seen as a trade-off between the setting in which no distributional information is known, and the Bayesian setting in which the (possibly correlated) distributions of all the individual offers are known. As our first main result we show that a deterministic threshold strategy using the monopoly price of the distribution of the maximum value is asymptotically optimal assuming that the expectation of the maximum value grows sublinearly in n. In our second main result, we further tighten this bound by deriving a tight quadratic convergence guarantee for sufficiently smooth distributions of the maximum value. Our results also give rise to a more fine-grained picture regarding prophet inequalities with correlated values, for which distribution-free bounds often only yield a performance guarantee that is of the order 1/n.