How to Hire Secretaries with Stochastic Departures

TL;DR

This paper extends the secretary problem to a stochastic setting where candidates stay for random durations, characterizing optimal policies and showing that simple threshold strategies are nearly optimal with many candidates.

Contribution

It provides a characterization of optimal policies in a stochastic secretary problem and demonstrates the near-optimality of threshold policies for large candidate pools.

Findings

Optimal policy depends only on current time and number of candidates.

Policy is monotone non-decreasing in the number of candidates.

Single threshold policy is nearly optimal when the candidate number is large.

Abstract

We study a generalization of the secretary problem, where decisions do not have to be made immediately upon candidates' arrivals. After arriving, each candidate stays in the system for some (random) amount of time and then leaves, whereupon the algorithm has to decide irrevocably whether to select this candidate or not. The goal is to maximize the probability of selecting the best candidate overall. We assume that the arrival and waiting times are drawn from known distributions. Our first main result is a characterization of the optimal policy for this setting. We show that when deciding whether to select a candidate it suffices to know only the time and the number of candidates that have arrived so far. Furthermore, the policy is monotone non-decreasing in the number of candidates seen so far, and, under certain natural conditions, monotone non-increasing in the time. Our second main result is proving that when the number of candidates is large, a single threshold policy is almost optimal.