A Double Secretary Problem

TL;DR

This paper investigates a double secretary problem with 2n applicants of n qualities, establishing an optimal threshold-based stopping rule and analyzing its limiting behavior compared to the classical secretary problem.

Contribution

It introduces a novel double secretary problem model and proves the optimality of a threshold-based stopping rule, extending classical secretary problem results.

Findings

The optimal stopping rule has a monotone structure.

The threshold-based rule is proven to be optimal.

Limiting behavior of the rule is characterized and compared to CSP.

Abstract

We consider a double secretary problem which contains $2n$ applicants of $n$ different qualities, two of each quality. As in the classical secretary problem (CSP), the applicants are interviewed sequentially in a random order by a manager and the manager wants to find an optimal stopping rule that maximizes the probability of selecting one of the two best applicants. We prove that the problem leads to a monotone structure and so the corresponding one-stage look-ahead rule (which is of threshold type) is optimal. The limiting behavior of the rule and of the maximum probability are studied and compared with that of the CSP.