Abstract
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.
Citation
Theodore P. Hill. Ulrich Krengel. "Minimax-Optimal Stop Rules and Distributions in Secretary Problems." Ann. Probab. 19 (1) 342 - 353, January, 1991. https://doi.org/10.1214/aop/1176990548
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