Sequential Search With Ordinal Ranks and Cardinal Values: An Infinite Discounted Secretary Problem

We consider an extension of the classical secretary problem where a decision maker observes only the relative ranks of a sequence of up to N applicants, whose true values are i.i.d. U[0,1] random variables. Applicants arrive according to a homogeneous Poisson Process, and the decision maker seeks to maximize the expected time-discounted value of the applicant who she ultimately selects. This provides a straightforward and natural objective while retaining the structure of limited information based on relative ranks. We derive the optimal policy in the sequential search, and show that the solution converges as N goes to infinity. We compare these results with a closely related full information problem in order to quantify these informational limitations.