The multi-returning secretary problem

TL;DR

This paper introduces an efficient algorithm for the multi-returning secretary problem, generalizing previous solutions for m=2 to any m, and provides methods to compute optimal thresholds and success probabilities.

Contribution

It offers a novel algorithm to compute optimal thresholds and success probabilities for the multi-returning secretary problem for any number of copies m.

Findings

Provides an efficient algorithm for computing thresholds and success probabilities.

Develops a method to determine asymptotic values using a system of m ODEs.

Extends solutions from m=2 to general m in the multi-returning secretary problem.

Abstract

In this paper we consider the so-called Multi-returning secretary problem, a version of the Secretary problem in which each candidate has $m$ identical copies. The case $m=2$ has already been completely solved by several authors using different methods both the case $m>2$ had not been satisfactorily solved yet. Here, we provide and efficient algorithm to compute the optimal threshold and the probability of success for every $m$. Moreover, we give a method to determine their asymtoptic values based on the solution of a system of $m$ ODEs.