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A secretary problem with uncertain employment and best choice of available candidates. (English) Zbl 0741.90090
The author considers the following secretary problem: \(n\) candidates appear one-by-one in front of an employer in random order with all \(n\)! permutations equally likely. At any time the candidates that have so far appeared can be ranked according to some order of preference. Each candidate may be classified into one of two types: available and unavailable. A candidate is available with a known fixed probability \(p\), an unavailable candidate does not accept an offer of employment, and a candidate to whom an offer is not made cannot be recalled later. When a candidate appears, the employer must decide whether or not to make an offer to the candidate. The author deals with two models according to whether the availability of a candidate can be ascertained before or after giving an offer of employment, and for each model he finds a policy that maximizes the probability of employing the best among the available candidates, based on both the relative ranks and the availabilities observed so far. These models are more realistic than that of M. H. Smith [J. Appl. Probab. 12, 620-624 (1975; Zbl 0313.60033)].
Reviewer: Y.Ohtsubo (Tobata)

90C39 Dynamic programming
60G40 Stopping times; optimal stopping problems; gambling theory
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