Prophets and Secretaries with Overbooking
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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 ℓ-out-of-k setting, where the decision maker can select up to k elements immediately and irrevocably, but her performance is measured by the top ℓ elements in the selected set. Equivalently, the decision makes can hold up to ℓ elements at any given point in time, but can make up to k-ℓ returns as new elements arrive. We give upper and lower bounds on the competitive ratio of ℓ-out-of-k prophet and secretary scenarios. These include a single-sample prophet algorithm that gives a competitive ratio of 1-ℓ· e^-Θ((k-ℓ)^2/k), which is asymptotically tight for k-ℓ=Θ(ℓ). For secretary settings, we devise an algorithm that obtains a competitive ratio of 1-ℓ e^-k-4ℓ/2+2ℓ - e^-k/6, and show that no secretary algorithm obtains a better ratio than 1-e^-k (up to negligible terms). In passing, our results lead to an improvement of the results of Assaf et al. [2000] for 1-out-of-k prophet scenarios. Beyond the contribution to online algorithms and optimal stopping theory, our results have implications to mechanism design. In particular, we use our prophet algorithms to derive overbooking mechanisms with good welfare and revenue guarantees; these are mechanisms that sell more items than the seller's capacity, then allocate to the agents with the highest values among the selected agents.
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