Prophets and Secretaries with Overbooking

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 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. For l-out-of- k prophet scenarios we provide a single-sample algorithm with competitive ratio 1-l· e-Θ((k-l)2/k) . The algorithm is a single-threshold algorithm, which sets a threshold that equals the (l+k/2)th highest sample, and accepts all values exceeding this threshold, up to reaching capacity k . On the other hand, we show that this result is tight if the number of possible returns is linear in l (i.e., k-l =Θ(l)). In particular, we show that no single-sample algorithm obtains a competitive ratio better than 1 - 2-(2k+1)/k+1 . We also present a deterministic single-threshold algorithm for the 1-out-of- k prophet setting which obtains a competitive ratio of 1-3/2 · e-s/k 6, knowing only the distribution of the maximum value. This result improves the result of [Assaf & Samuel-Cahn, J. of App. Prob., 2000].