Prophet Inequalities with Delayed and Uncertain Acceptance
Emile Martinez, Felipe Garrido-Lucero, Umberto Grandi, Sebastian Pérez-Salazar
We introduce the *prophet inequality with delayed and uncertain acceptance* model, a variant of the classical prophet inequality in which a decision-maker sequentially evaluates options whose acceptance is uncertain and revealed only after a random delay. At each time step, the decision-maker observes the realized value of the arriving option and must irrevocably decide whether to select it or continue searching. Once an option is selected, the process is suspended for a delay drawn from a known distribution with expectation $d$, during which no other option can be considered. When the delay expires, the selection succeeds with a known probability. If it succeeds, the decision-maker receives the realized value and the process terminates; otherwise, the search resumes. In addition to the online decision-maker, we consider two stronger benchmarks: the value-aware decision-maker, who knows all value realizations in advance but not the acceptance outcomes, and the prophet, who knows both the values and the acceptance outcomes. We show that, in the worst case, both decision-makers achieve a $1/(d + 2)$ fraction of the prophet's value, while the decision-maker achieves half of the value-aware decision-maker's value. We then turn to the no-delay case, where we establish an equivalence between our model and the classical prophet inequality, and show that the value-aware decision-maker beats the $1/2$ barrier against the prophet whenever all acceptance probabilities are bounded away from $0$.
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