The solution of a generalized secretary problem via analytic expressions

TL;DR

This paper derives explicit analytic formulas for the success probability of the optimal policy in a generalized secretary problem, enabling sequential calculation of the optimal stopping thresholds for any number of top candidates.

Contribution

It provides the first finite analytic expressions for success probability and optimal thresholds in the generalized secretary problem for arbitrary k, extending previous results limited to small k.

Findings

Derived explicit success probability formulas for the optimal policy.

Developed a method to compute optimal thresholds sequentially from k to 1.

Extended analytic solutions to any number of top candidates k.

Abstract

Given integers $1\leq k<n$, the Gusein-Zade version of a generalized secretary problem is to choose one of the $k$ best of $n$ candidates for a secretary, which are interviewing in random order. The stopping rule in the selection is based only on the relative ranks of the successive arrivals. It is known that the best policy can be described by a non--decreasing sequence $(s_1, \ldots, s_k)$ of integers with $l\leq s_l<n$ for every $1\leq l\leq k$, and conversely, any such a sequence determines the general structure of the best policy. We found a finite analytic expression for the probability of success when using the optimal policy with a sequence $(s_1, \ldots, s_k)$. We also study the problem of the construction of the optimal sequence, i.e. a sequence which maximizes the corresponding probability of success. We discovered finite analytic expressions which enable to calculate the elements $s_l$ of an optimal sequence one by one, from $l=k$ to $l=1$. Until now, such expressions were derived separately, and only for the values $k\leq 3$.