The Secretary Problem with a Stochastic Precursor

2026-05-21

AI Summary

This study investigates a variant of the secretary problem featuring content-free but timing-constrained preamble signals that, while conveying no direct information about item values, are guaranteed to arrive before the optimal item. By integrating optimal stopping theory and probabilistic analysis under both random-order and adversarial-order models, the work demonstrates that such signals can substantially enhance online decision-making performance. Under random arrival order, a single uniformly distributed preamble signal alone boosts the success probability of selecting the optimal item to at least 1/2, with this probability approaching 1 as the signal’s delay increases. In the adversarial setting, signals exhibiting sufficient concentration similarly recover a constant-factor success guarantee. This work extends the notion of “predictions” in learning-augmented algorithms by establishing, for the first time, the efficacy of purely temporal, content-free signals.

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.

Problem

• Secretary Problem
• Stochastic Precursor
• Online Algorithms
• Optimal Stopping
• Temporal Information

Innovation

• stochastic precursor
• online algorithms
• secretary problem
• optimal stopping
• learning-augmented algorithms