The best choice problem with an unknown number of objects |
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| Authors: | Aarni Lehtinen |
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| Institution: | (1) Department of Mathematics, University of Jyväskylä, Seminaarinkatu 15, 40100 Jyväskylä, Finland |
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| Abstract: | The secretary problem with a known prior distribution of the number of candidates is considered. Ifp(i)=p(N=i),i   ,  ]   , where=inf{i :p(i) > 0} and=sup{i :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=min satisfies asymptotically (as   ) the equationj=exp
![$${{\left {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i) \log (i)/i} } \right)} \right]} \mathord{\left/ {\vphantom {{\left {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i) \log (i)/i} } \right)} \right]} {\left. {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i)/i} } \right) - 1} \right]}}} \right. \kern-\nulldelimiterspace} {\left. {\left( {\sum\limits_{i = max(\alpha ,j)}^\beta {p(i)/i} } \right) - 1} \right]}}$$](/content/t4981ql417kk6k25/186_2005_Article_BF01415532_TeX2GIFIE1.gif)
.The probability of selecting the best object by the corresponding policy will be (j-1)
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. |
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| Keywords: | Secretary problem optimal stopping rule |
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