How to Hire Secretaries with Stochastic Departures

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 nonincreasing in the time. Our second main result is proving that when the number of candidates is large, a single threshold policy is almost optimal.

Windowed Prophet Inequalities

“…In this setting [HK15] examine a sliding window input model similar to ours, and find and analyze the optimal stopping rule as a function of window size. Additionally, [KPV19] study an even more general model where samples disappear after a random amount of time, and characterize optimal stopping rules in this setting. However, the differences between the prophet inequalities problem and the secretary problem make it difficult to apply techniques from these papers to our setting.…”