Optimal selection of the $k$-th best candidate

TL;DR

This paper extends the classical secretary problem to the case of selecting the k-th best candidate, deriving explicit optimal stopping rules for k=3 and analyzing probabilities of success for various k and n.

Contribution

It provides the first explicit optimal stopping rule involving two thresholds for selecting the 3rd best candidate, expanding understanding of the k-th best secretary problem.

Findings

Explicit optimal stopping rule for k=3 involving two thresholds.

Proved inequalities for the probability of successfully selecting the k-th best.

Demonstrated that success probability decreases with increasing k and n.

Abstract

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of $n$ candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the $k$-th best candidate for $k\ge 2$. While the optimal stopping rule for $k=1,2$ (and all $n\ge 2$) is known to be of threshold type (involving one threshold), we solve the case $k=3$ (and all $n\ge 3$) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for $p(k,n)$, the maximum probability of selecting the $k$-th best of $n$ candidates. It is shown that (i) $p(1,n)=p(n,n)>p(k,n)$ for $1<k<n$, (ii) $p(k,n)\ge p(k,n+1)$, (iii) $p(k,n)\ge p(k+1,n+1)$, and (iv) $p(k,\infty):=\lim_{n\to \infty} p(k,n)$ is decreasing in $k$.