Conditions for Quasi-Stationarity of the Bayes Rule in Selection Problems with an Unknown Number of Rankable Options

Abstract

In the so called secretary problem, if an unknown number $N$ of options arrive at i.i.d. times with a known continuous distribution, then only the geometric, among proper distributions on $N$, has the property that the stopping risk depends just on the elapsed time and not on the number of arrivals so far. But even with such a prior, the optimal rule may, in general, depend on the number of arrivals so far. The optimal rule is closely related to the optimal policy in the Gianini and Samuels infinite secretary problem, except for a linear change in the time scale which depends only on the parameter of the prior, and not on the loss function.