The Best-or-Worst and the Postdoc problems with random number of candidates

TL;DR

This paper analyzes two variants of the Secretary problem with unknown number of candidates, providing optimal strategies and success probabilities for cases where candidate counts follow uniform or Poisson distributions.

Contribution

It extends previous models by considering random candidate counts and derives optimal threshold strategies and success probabilities.

Findings

Optimal threshold strategies are identified for both distributions.

Asymptotic success probabilities are derived.

Results relate to and extend existing secretary problem research.

Abstract

In this paper we consider two variants of the Secretary problem: The Best-or-Worst and the Postdoc problems. We extend previous work by considering that the number of objects is not known and follows either a discrete Uniform distribution $\mathcal{U}[1,n]$ or a Poisson distribution $\mathcal{P}(\lambda)$. We show that in any case the optimal strategy is a threshold strategy, we provide the optimal cutoff values and the asymptotic probabilities of success. We also put our results in relation with closely related work.