On best choice problems having similar solutions
For a best choice problem with a fixed number \(N=n\) of objects with the quality as i.i.d. random variables, having a distribution uniform on \([\Theta-{1\over 2}, \Theta+{1\over 2}]\) with unknown \(\Theta\in{\mathbb R}\), \textit{J. D. Petruccelli} [Ann. Stat. 8, 1171-1174 (1980; Zbl 0459.62071)] gave optimal decision levels and the asymptotically optimal winning probability \((n\rightarrow\infty)\) for getting the best object. At the other side, for random \(N\), uniformly distributed on \(\{1,\ldots,n\}\), and random qualities with known continuous distribution, the author [Stochastic Processes Appl. 24, 293-307 (1987; Zbl 0623.60059)] gave the corresponding solution (see above). Unexpectedly these equal although the problem positions are quite different. The reason, why they equal, remains unknown. The author detected still a similar property for two other problems.
scientific article; zbMATH DE number 1796693