The Secretary Problem with a Stochastic Precursor

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.