The best choice problem with an unknown number of objects

Abstract

The secretary problem with a known prior distribution of the number of candidates is considered. Ifp(i)=p(N=i),i ∈ [α, β] ∩ ℕ, whereα=inf{i ∈ℕ:p(i) > 0} andβ=sup{i ∈ℕ:p(i)≳0}, is the prior distribution of the numberN of candidates it will be shown that, if the optimal stopping rule is of the simple form, then the optimal stopping indexj=minΓ satisfies asymptotically (asβ → ∞) the equationj=exp $${{\left[ {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i) \log (i)/i} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i) \log (i)/i} } \right)} \right]} {\left. {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i)/i} } \right) - 1} \right]}}} \right. \kern-\nulldelimiterspace} {\left. {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i)/i} } \right) - 1} \right]}}$$ .The probability of selecting the best object by the corresponding policy will be (j-1) $$\sum\limits_{i = \max (\alpha ,j)}^\beta {p(i)/i} $$ p(i)/i. We also give an example of the distributionp for which the optimal stopping rule consists of a stopping set with two islands. We present an asymptotical solution for this example.