The Secretary Problem with a Hazard Rate Condition

Abstract

In the classical secretary problem, the objective is to select the candidate of maximum value among a set of n candidates arriving one by one. The value of the candidates come from an unknown distribution and is revealed at the time the candidate arrives, at which point an irrevocable decision on whether to select the candidate must be made. The well-known solution to this problem, due to Dynkin, waits for n/e steps to set an “aspiration level” equal to the maximum value of the candidates seen, and then accepts the first candidate whose value exceeds this level. This guarantees a probability of at least 1/e of selecting the maximum value candidate, and there are distributions for which this is essentially the best possible. One feature of this algorithm that seems at odds with reality is that it prescribes a long waiting period before selecting a candidate. In this paper, we show that if a standard hazard rate condition is imposed on the distribution of values, the waiting period falls from n/e to n/log(n), meaning that it is enough to observe a diminishingly small sample to set the optimal aspiration level. This result is tight, as both the hazard condition and the optimal sampling period bind exactly for the exponential distribution.