Prophet Secretary

Optimal stopping theory is a powerful tool for analyzing scenarios such as online auctions in which we generally require optimizing an objective function over the space of stopping rules for an allocation process under uncertainty. Perhaps the most classic problems of stopping theory are the prophet inequality problem and the secretary problem. The classical prophet inequality states that by choosing the same threshold

OPT

/2 for every step, one can achieve the tight competitive ratio of 0.5. On the other hand, for the basic secretary problem, the optimal strategy achieves the tight competitive ratio of 1/

e

 ≈ 0.36

In this paper, we introduce

prophet secretary

, a natural combination of the prophet inequality and the secretary problems. We show that by using a single uniform threshold one cannot break the 0.5 barrier of the prophet inequality for the prophet secretary problem. However, we show that

using

n

distinct non-adaptive thresholds one can obtain a competitive ratio that goes to (1 − 1/

e

 ≈ 0.63) as

n

grows; and

no online algorithm can achieve a competitive ratio better than 0.73.

Our results improve the (asymptotic) approximation guarantee of single-item sequential posted pricing mechanisms from 0.5 to (1 − 1/

e

) when the order of agents (customers) is chosen randomly.

We also consider the minimization variants of stopping theory problems and in particular the prophet secretary problem. Interestingly, we show that, even for the simple case in which the input elements are drawn from identical and independent distributions (i.i.d.), there is no constant competitive online algorithm for the minimization variant of the prophet secretary problems. We extend this hardness result to the minimization variants of both the prophet inequality and the secretary problem as well.