A new method for computing asymptotic results in optimal stopping problems

TL;DR

This paper introduces a new systematic method based on differential equations to compute asymptotic thresholds and success probabilities in various optimal stopping problems, including multiple secretary problem variants.

Contribution

The paper presents a novel differential equation-based approach for asymptotic analysis in optimal stopping problems, covering both classical and new variants.

Findings

Effective computation of asymptotic thresholds and success probabilities

Application to nine classical secretary problem variants

Extension to four new problem variants

Abstract

In this paper, we present a novel method for computing the asymptotic values of both the optimal threshold, and the probability of success in sequences of optimal stopping problems. This method, based on the resolution of a first-order linear differential equation, makes it possible to systematically obtain these values in many situations. As an example, we address nine variants of the well-known secretary problem, including the classical one, that appear in the literature on the subject, as well as four other unpublished ones.