Optimal Stopping with Random Horizon with Application to the Full-Information Best-Choice Problem with Random Freeze
Abstract A general theorem is given showing that an optimal stopping problem with independent random horizon is essentially equivalent to a “discounted” fixed horizon optimal stopping problem derived from it. The result is used to treat the “full information” secretary problem, where a known number of applicants for a job are potentially available. Applicants are interviewed sequentially with no recall, and their X i -values are observed, where the X i are iid random variables from a known continuous distribution. The goal is to pick the applicant with the highest X i value. The “payoff” for this applicant is 1, and for any other applicant is zero. A random “freeze-time” variable M, with known distribution independent of the X i 's makes it impossible to pick an applicant after time M. The optimal rule is described and necessary and sufficient conditions for it to have a “monotone structure” are given. Uniform and geometric freeze variables are discussed, and some asymptotic results are given.