A new secretary problem with rank-based selection and cardinal payoffs

We present an extension of the secretary problem in which the decision maker (DM) sequentially observes up to n applicants whose values are random variables X₁,X₂,…,X_n drawn i.i.d. from a uniform distribution on 0,1]. The DM must select exactly one applicant, cannot recall released applicants, and receives a payoff of x_t, the realization of X_t, for selecting the tth applicant. For each encountered applicant, the DM only learns whether the applicant is the best so far. We prove that the optimal policy dictates skipping the first sqrt(n)-1 applicants, and then selecting the next encountered applicant whose value is a maximum.