Shelf Life of Candidates in the Generalized Secretary Problem

TL;DR

This paper analyzes a variation of the secretary problem called the duration problem, determining optimal thresholds for selecting candidates to maximize their time of possession, with asymptotic results for large candidate pools.

Contribution

It introduces threshold-based strategies for the duration problem and derives asymptotic thresholds and expected shelf life for large numbers of objects.

Findings

Optimal thresholds are approximately 0.417188N and 0.120381N for large N.

Asymptotic mean shelf life of selected objects is about 0.403827N.

Threshold strategies maximize the duration of possession in the secretary problem variant.

Abstract

A version of the secretary problem called the duration problem, in which the objective is to maximize the time of possession of relatively best objects or the second best, is treated. It is shown that in this duration problem there are threshold numbers $(k_1^\star,k_2^\star)$ such that the optimal strategy immediately selects a relatively best object if it appears after time $k_1^\star$ and a relatively second best object if it appears after moment $k_2^\star$. When number of objects tends to infinity the thresholds values are $\lfloor 0.417188N\rfloor$ and $\rfloor 0.120381N\rfloor$, respectively. The asymptotic mean time of shelf life of the object is $0.403827N$.