In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of n candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the kth best candidate for k ≥ 2. While the optimal stopping rule for k=1,2 (and all n ≥ 2) is known to be of threshold type (involving one threshold), we solve the case k=3 (and all n ≥ 3) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for p(k, n), the maximum probability of selecting the k-th best of n candidates. It is shown that (i) p(1, n) = p(n, n) > p(k, n) for 1<k<n, (ii) p(k, n) ≥ p(k, n + 1), (iii) p(k, n) ≥ p(k + 1, n + 1) and (iv) p(k, ∞): = lim n→∞p(k, n) is decreasing in k.

We consider a nonnegative discrete time and bounded horizon process X for which 0 is an absorbing state and extend it by a random variable that is independent of X. We find a sufficient condition for the resulting process to satisfy, after a canonical time rescaling, the hypothesis of the monotone case theorem. If X describes a secretary type search on a poset with one maximal element or if we consider X with no extension then this condition assumes an especially simple log-concavity type form.

We consider the problem of maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon of length N, where m is a predetermined integer. A prior is given for N. It is known that, when N is degenerate, i.e. P{N = n} = 1 for a given n > m, the sum-the-multiplicative-odds theorem gives the solution and shows that the optimal rule is a threshold rule, i.e. it stops on the first success appearing after a given stage. However, when N is nondegenerate, the optimal rule is not necessarily a threshold rule. So our main concern in Section 2 is to give a sufficient condition for the optimal rule to be a threshold rule when N is a bounded random variable such that P{N ≤ n} = 1. Application will be made to the usual (discrete arrival time) secretary problem with a random number N of applicants in Section 3. When N is uniform or curtailed geometric, the optimal rules are shown to be threshold rules and their asymptotic results are obtained. We also examine, as a nonhomogeneous Poisson process model, an intermediate prior that allows N to be uniform or degenerate. In Section 4 we consider a continuous arrival time version of the secretary problem with a random number M of applicants. It is shown that, whatever the distribution of M, we can win with probability greater than or equal to um*, where um* is, as given in (1.4), the asymptotic win probability of the usual secretary problem when N degenerates to n and n → ∞.

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.