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A Secretary Problem with Uncertain Employment and Best Choice of Available Candidates

A finite number of candidates appear one-by-one in random order with all permutations equally likely. We are able, at any time, to rank the candidates that have so far appeared according to some order of preference. Each candidate may be classified into one of two types independent of the other candidates: available or unavailable. An unavailable candidate does not accept an offer of employment. The goal is to find a strategy that maximizes the probability of employing the best among the available candidates based on both the relative ranks and the availabilities observed so far. According to when the availability of a candidate can be ascertained, two models are considered. The availability is ascertained only by giving an offer of employment MODEL 1, while the availability is ascertained just after the arrival of the candidate MODEL 2.