  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.
  In this paper, we introduce Prophet Secretary, a natural combination of the
prophet inequality and the secretary problems. An example motivation for our
problem is as follows. Consider a seller that has an item to sell on the market
to a set of arriving customers. The seller knows the types of customers that
may be interested in the item and he has a price distribution for each type:
the price offered by a customer of a type is anticipated to be drawn from the
corresponding distribution. However, the customers arrive in a random order.
Upon the arrival of a customer, the seller makes an irrevocable decision
whether to sell the item at the offered price. We address the question of
finding a strategy for selling the item at a high price.
  We show that by using a uniform threshold one cannot break the 0.5 barrier.
However, we show that i) using n distinct non-adaptive thresholds one can
obtain a competitive ratio that goes to (1-1/e) as n grows; and ii) no online
algorithm can achieve a competitive ratio better than 0.75. 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.
