The Secretary Problem with a Stochastic Precursor

• URL: http://arxiv.org/abs/2605.22653v1
• Date: Thu, 21 May 2026 15:56:09 GMT
• Title: The Secretary Problem with a Stochastic Precursor
• Abstract summary: This paper shows that predictions can also be valuable solely due to their arrival time.<n>We study the fundamental secretary problem augmented with a precursor: a content-free signal that is guaranteed to arrive no later than the best item.<n>We characterize optimal policies in the random-order and adversarial-order models.
• Score: 49.51793477594992
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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