An Adaptive Algorithm for `the Secretary Problem': Alternate Proof of the Divergence of a Maximizer Sequence

Authors

Andrew Benfante, Old Dominion UniversityFollow
Xiang Xu, Old Dominion UniversityFollow

Disciplines

Analysis | Other Applied Mathematics | Probability

Publication Date

2023

Document Type

Article

DOI

10.25778/w3tr-5449

Abstract

This paper presents an alternate proof of the divergence of the unique maximizer sequence {𝑥∗ 𝑛} of a function sequence {𝐹𝑛(𝑥)} that is derived from an adaptive algorithm based on the now classic optimal stopping problem, known by many names but here ‘the secretary problem’. The alternate proof uses a result established by Nguyen, Xu, and Zhao (n.d.) regarding the uniqueness of maximizer points of a generalized function sequence {𝑆𝜇,𝜎 𝑛 } and relies on the strict monotonicity of 𝐹𝑛(𝑥) as 𝑛 increases in order to show divergence of {𝑥∗ 𝑛}. Towards this, limits of the exponentiated Gaussian CDF are established as well as a closed form of 𝐹′ 𝑛 (𝑥), the derivative of the sequence’s function. The proof is elementary but nontrivial. The result in Nguyen et al. (n.d.) relies heavily on a technical lemma but, here, the proof is more transparent and relies solely on fundamentals.

Recommended Citation

Benfante, Andrew and Xu, Xiang (2023) "An Adaptive Algorithm for `the Secretary Problem': Alternate Proof of the Divergence of a Maximizer Sequence," OUR Journal: ODU Undergraduate Research Journal: Vol. 10, Article 4.
DOI: 10.25778/w3tr-5449
Available at: https://digitalcommons.odu.edu/ourj/vol10/iss1/4