The Secretary Problem with an Unknown Number of Options

A method of selecting the best element from a random sequence of unknown length is investigated. By assuming that the arrival times of the elements are independent identically distributed (i.i.d.) exponential random variables, a procedure is established that maximizes the probability of selecting the best element. Asymptotically for large values of the actual length of the sequence, the optimal probability is 1/e, which is also the corresponding asymptotic optimal value when the length is known. It is shown that the method behaves well even when the actual number of options is comparatively small, and that it is not particularly sensitive to errors in the specification of the arrival rate of the process.