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A version of the secretary problem is considered. The ranks of items, whosevalues are independent, identically distributed random variables$X_1,X_2,...,X_n$ from a uniform distribution on $[0; 1]$, are observedsequentially by the grader. He has to select exactly one item, when it appears,and receives a payoff which is a function of the unobserved realization ofrandom variable assigned to the item diminished by some cost. The methods ofanalysis are based on the existence of an embedded Markov chain and use thetechnique of backward induction. The result is a generalization of theselection model considered by Bearden(2006). The asymptotic behaviour of thesolution is also investigated.

A rank-based selection with cardinal payoffs and a cost of choice