Prophets and Secretaries with Overbooking
Tomer Ezra, Michal Feldman, Ilan Nehama

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Abstract
The prophet and secretary problems demonstrate online scenarios involving the optimal stopping theory. In a typical prophet or secretary problem, selection decisions are assumed to be immediate and irrevocable. However, many online settings accommodate some degree of revocability. To study such scenarios, we introduce the setting, where the decision maker can select up to elements immediately and irrevocably, but her performance is measured by the top elements in the selected set. Equivalently, the decision makes can hold up to elements at any given point in time, but can make up to returns as new elements arrive. We give upper and lower bounds on the competitive ratio of -out-of- prophet and secretary scenarios. These include a single-sample prophet algorithm that gives a competitive ratio of $1-\ell\cdot…
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