Published in The Annals of Probability, Volume 19, Issue 1, January 1, 1991, pages 342-353.
Copyright © 1991 Institute of Mathematical Statistics.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
The definitive version is available at https://doi.org/10.1214/aop/1176990548.
For the secretary (or best-choice) problem with an unknown number N of objects, minimax-optimal stop rules and (worst-case) distributions are derived, under the assumption that $N$ is a random variable with unknown distribution, but known upper bound n. Asymptotically, the probability of selecting the best object in this situation is of order of (log n)-1. For example, even if the only information available is that there are somewhere between 1 and 100 objects, there is still a strategy which will select the best item about one time in five.