Optimal selection of the $k$th best candidate
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 $k$.
 Publication:

arXiv eprints
 Pub Date:
 December 2016
 DOI:
 10.48550/arXiv.1701.00052
 arXiv:
 arXiv:1701.00052
 Bibcode:
 2017arXiv170100052L
 Keywords:

 Mathematics  Probability;
 60G40;
 62L15