This paper considers a variation of the classical secretary problem; this variation allows for the interviewer to make an offer to previously interviewed applicants and for the applicant to reject an offer. In this version, the process ends after any offer is made. Two special cases are considered, where the optimal procedure and its probability of success are given for each. Furthermore, the problem is discussed in general and conditions for optimal strategies are given.

The optimal selection of a maximum of a sequence with the possibility of ties is considered. The object is to examine each observation in the sequence of known length n and, based only on the relative rank among predecessors, either to stop and select it as a maximum or to continue without recall. Rules which maximize the probability of correctly selecting a maximum from a sequence with ties are investigated. These include rules which randomly break ties, rules which discard tied observations, and minimax rules based on the atoms of a discrete distribution function. If the sequence is random from F, a random distribution function from a Dirichlet process prior with non-atomic parameter, optimal rules are developed. The limiting behavior of these rules is studied and compared with other rules. The selection of the parameter of the Dirichlet process regulates the ties.

From one point of view this paper adds to a previous formulation of the best-choice problem (Petruccelli (1981)) the possibility that the number of available observations, rather than being known, is a bounded random variable N with known distribution. From another perspective, it expands the formulations of Presman and Sonin (1972) and Rasmussen and Robbins (1975) to include recall and uncertainty of selection of observations. The behaviour of optimal stopping rules is examined under various assumptions on the general model. For optimal stopping rules and their probabilities of best choice relations are obtained between the bounded and unbounded N cases. Two particular classes of stopping rules which generalize the s(r) rules of Rasmussen and Robbins (1975) are considered in detail.

A secretary problem which allows the applicant to refuse an offer of employment with a fixed probability and admits backward solicitations of previous interviewees with known probability of successful solicitation is considered. The optimal strategy that maximizes the probability of employing the best applicant is derived. Two types of probability of successful solicitation, constant and geometric, are discussed in detail.

We consider the problem of maximizing the probability of choosing the largest from a sequence of N observations when N is a bounded random variable. The present paper gives a necessary and sufficient condition, based on the distribution of N, for the optimal stopping rule to have a particularly simple form: what Rasmussen and Robbins (1975) call an s(r) rule. A second result indicates that optimal stopping rules for this problem can, with one restriction, take virtually any form.

A best choice problem is presented which is intermediate between the constraints of the ‘no-information’ problem (observe only the sequence of relative ranks) and the demands of the ‘full-information’ problem (observations from a known continuous distribution). In the intermediate problem prior information is available in the form of a ‘training sample’ of size m and observations are the successive ranks of the n current items relative to their predecessors in both the current and training samples.

Optimal stopping rules for this problem depend on m and n essentially only through m + n; and, as m/(m + n) → t, their success probabilities, P*(m, n), converge rapidly to explicitly derived limits p*(t) which are the optimal success probabilities in an infinite version of the problem. For fixed n, P*(m, n) increases with m from the ‘no-information’ optimal success probability to the ‘full-information’ value for sample size n. And as t increases from 0 to 1, p*(t) increases from the ‘no-information’ limit e–1 ≍ 0·37 to the ‘full-information’ limit ≍0·58. In particular p*(0·5) ≍ 0·50.

This paper explores best choice problems which allow both recall of applicants and uncertainty of a current applicant accepting an offer of employment. Properties of optimal selection procedures are derived for the general case. Optimal procedures and the associated probabilities of obtaining the best applicant are found in two special cases. The results unify and extend those of Yang (1974) and Smith (1975).

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.