Approximate results for a generalized secretary problem

TL;DR

This paper investigates approximate strategies for a generalized secretary problem, demonstrating that simple double-level policies closely match optimal solutions through exact and asymptotic analysis.

Contribution

It introduces and analyzes approximate policies using one or two position thresholds combined with level ranks, simplifying the optimal policy computation.

Findings

Double-level policy is highly accurate compared to optimal policies.

Asymptotic results show effectiveness of approximate policies as n grows large.

Exact analysis confirms the approximation quality for finite n.

Abstract

A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b is a preassigned (natural) number. It is known, already for a long time, that for the optimal policy one needs to compute b position thresholds, for instance via backwards induction. In this paper we study approximate policies, that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n tends to infinity) results, which show that the double-level policy is an extremely accurate approximation.