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An optimal strategy belongs to the class of strategies defined by a set of threshold numbers <math> (a_1, a_2, ... , a_r)</math>, where <math> a_1<a_2< \cdots <a_r </math>. The first choice is to be used on the first candidates starting with <math>a_1</math>th applicant, and once the first choice is used, second choice is to be used on the first candidate starting with <math>a_2</math>th applicant, and so on.
When <math>r=2 </math>, {{harvnb|Ano|Kakinuma|Miyoshi|2010}} showed that the tight lower bound of win probability is <math> e^{-1}+ e^{-\frac{3}{2}} (n \rightarrow \infty). </math>
For general positive integer <math>r</math>, {{harvnb|Matsui|Ano|2016}} discussed the lower bound of win probability.
When <math> r=3,4,5 </math>, tight lower bounds of win probabilities are equal to <math> e^{-1}+ e^{-\frac{3}{2}}+e^{-\frac{47}{24}} </math>, <math> e^{-1}+e^{-\frac{3}{2}}+e^{-\frac{47}{24}}+e^{-\frac{2761}{1152}} </math> and <math> e^{-1}+e^{-\frac{3}{2}}+e^{-\frac{47}{24}}+e^{-\frac{2761}{1152}}+e^{-\frac{4162637}{1474560}}, </math> respectively.
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