The secretary problem on an unknown poset

TL;DR

This paper extends the secretary problem to partially ordered sets, providing algorithms with success probabilities based on the number of maximal elements, and proves bounds for specific cases.

Contribution

It introduces a generalized secretary problem on posets with limited information and establishes success probability bounds, including proofs for certain poset widths.

Findings

Success probability at least 1/e with given info

Conjecture on success probability for k maximal elements

Proved bounds for posets of width k

Abstract

We consider generalizations of the classical secretary problem, also known as the problem of optimal choice, to posets where the only information we have is the size of the poset and the number of maximal elements. We show that, given this information, there is an algorithm that is successful with probability at least $\frac{1}{e}$. We conjecture that if there are $k$ maximal elements and $k \geq 2$ then this can be improved to $\sqrt[k-1]{\frac{1}{k}}$, and prove this conjecture for posets of width $k$. We also show that no better bound is possible.