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A variant of the Secretary Problem: the Best or the Worst

We consider a variant of the secretary problem in which the candidates state their expected salary at the interview, which we assume is in accordance with their qualifications. The goal is for the employer to hire the best or the worst (cheapest), indifferent between the two cases. We focus on the complete information variant as well as on the cases when the number of applicants is a random variable with a uniform distribution $U[1,n]$ or with a Poisson distribution of parameter $λ$. Moreover, we also study two variants of the original problem in which we consider payoffs depending on the number of conducted interviews.