The best-choice secretary problem with random freeze on jobs

We consider the best-choice secretary problem, with a known number, n, of applicants, and a random, independent “freeze” variable M, with known distribution. No hiring is possible after time M. The goal is to choose the best among the n applicants, where the decisions must be made depending only on the relative ranks of the applicants observed so far. A necessary and sufficient condition is given for the optimal rule to have the “simple” structure: let k∗ — 1 applicants pass, and stop with the first applicant (if any) from the k∗th onward, who is better than all previous observed candidates. For uniform, geometric and Poisson freeze variables the optimal rules are simple. Some asymptotic results (as n → ∞), and minimax results are also discussed.