Weber's optimal stopping problem and generalizations

Abstract

One way to interpret the classical secretary problem (CSP) is to consider it as a special case of the following problem. We observe $n$ independent indicator variables $I_1,I_2,\dotsc,I_n$ sequentially and we try to stop on the last variable being equal to 1. If $I_k=1$ it means that the $k$-th observed secretary has smaller rank than all previous ones (and therefore is a better secretary). In the CSP $p_k=E(I_k)=1/k$ and the last $k$ with $I_k=1$ stands for the best candidate. The more general problem of stopping on a last "1" was studied by Bruss(2000). In what we will call Weber's problem the variables $I_k$ can take more than two values and we try to stop on the last occurence of \textit{one} of these values. Notice that we do not know in advance the value taken by the variable on which we stop. We can solve this problem in some cases and provide algorithms to compute the optimal stopping rule. These cases carry enough generality to be applicable in concrete situations.