A variant of the Secretary Problem: the Best or the Worst

TL;DR

This paper explores a modified secretary problem where the employer aims to hire either the best or the worst candidate, analyzing complete and probabilistic applicant scenarios with additional interview-dependent payoffs.

Contribution

It introduces a novel variant of the secretary problem focusing on selecting the extreme candidates and analyzes its solutions under various information and distribution assumptions.

Findings

Optimal strategies for complete information case

Analysis under uniform and Poisson applicant distributions

Impact of interview-dependent payoffs on decision rules

Abstract

We consider a variant of the secretary problem in which the candidates state their expected salary at the interview, which we assume is in accordance with their qualifications. The goal is for the employer to hire the best or the worst (cheapest), indifferent between the two cases. We focus on the complete information variant as well as on the cases when the number of applicants is a random variable with a uniform distribution $U[1,n]$ or with a Poisson distribution of parameter $\lambda$. Moreover, we also study two variants of the original problem in which we consider payoffs depending on the number of conducted interviews.