Partially ordered secretaries

TL;DR

This paper extends the classical secretary problem to partially ordered sets, providing a strategy that guarantees a success probability of 1/e for selecting a maximal element in an online setting.

Contribution

It introduces a generalized secretary problem for partial orders and presents a strategy that achieves the optimal success probability of 1/e.

Findings

Achieves success probability 1/e for arbitrary partial orders.

Generalizes the classical secretary problem to more complex structures.

Provides a strategy applicable to any finite partial order.

Abstract

The elements of a finite nonempty partially ordered set are exposed at independent uniform times in $[0,1]$ to a selector who, at any given time, can see the structure of the induced partial order on the exposed elements. The selector's task is to choose online a maximal element. This generalizes the classical linear order secretary problem, for which it is known that the selector can succeed with probability $1/e$ and that this is best possible. We describe a strategy for the general problem that achieves success probability $1/e$ for an arbitrary partial order.