The Secretary Problem with a Stochastic Precursor

TL;DR

This paper investigates how the timing of a stochastic precursor signal influences optimal stopping strategies in the secretary problem, revealing that even content-free timing information can significantly improve success probabilities.

Contribution

It introduces the concept of stochastic precursors in the secretary problem, demonstrating their impact on success probabilities in both random and adversarial order models.

Findings

A single uniformly timed precursor raises success probability to at least 1/2.

Success probability approaches 1 with increasingly late precursors.

Sufficiently concentrated precursors provide constant success guarantees in adversarial order.

Abstract

In learning-augmented online algorithms, predictions are usually valued for what they say: a value estimate, a solution, or an algorithmic recommendation. This paper shows that predictions can also be valuable solely due to their arrival time. We study the fundamental secretary problem augmented with a stochastic precursor: a content-free signal that is guaranteed to arrive no later than the best item, but is otherwise stochastically timed. The signal does not carry any additional information; nevertheless, its timing alone changes the structure of optimal stopping. We characterize optimal policies in the random-order and adversarial-order models. In random order, a single uniformly timed precursor already gives success probability at least $\frac12$, improving on the classic $\frac1e$ benchmark. With increasingly late precursors, the success probability approaches $1$. In adversarial order, for which traditional models do not admit strong guarantees, sufficiently concentrated precursors recover constant success guarantees. Our results show that such novel forms of asynchronous temporal information are a distinct and powerful form of advice in online decision making and may also be effective for other problems.