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A Double Secretary Problem

Shoou-Ren Hsiau, Yi-Shen Lin

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.

A Double Secretary Problem

Abstract

We consider a double secretary problem which contains applicants of 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.

Paper Structure

This paper contains 4 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The Monotone Case Problem
  3. Comparing with the CSP and Concluding Remarks
  4. Proof of Lemma \ref{['lem 2.1']}

Key Result

Theorem 2.1

The stopping rule is optimal for the problem 2.2, i.e. the optimal rule stops on the first $T_j$ with $D_{T_j}\ge r_n$ and $S_{T_j}=2$.

Theorems & Definitions (5)

  • Theorem 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1