Let Xi ≥ 0 be independent, i = 1,…, n, with known distributions and let Xn*= max(X1,…,Xn). The classical ‘ratio prophet inequality’ compares the return to a prophet, which is EXn*, to that of a mortal, who observes the Xis sequentially, and must resort to a stopping rule t. The mortal's return is V(X1,…,Xn) = max EXt, where the maximum is over all stopping rules. The classical inequality states that EXn* < 2V(X1,…,Xn). In the present paper the mortal is given k ≥ 1 chances to choose. If he uses stopping rules t1,…,tk his return is E(max(Xt1,…,Xtk)). Let t(b) be the ‘simple threshold stopping rule’ defined to be the smallest i for which Xi ≥ b, or n if there is no such i. We show that there always exists a proper choice of k thresholds, such that EXn* ≤ ((k+1)/k)Emax(Xt1,…,Xtk)), where ti is of the form t(bi) with some added randomization. Actually the thresholds can be taken to be thej/(k+1) percentile points of the distribution of Xn*, j = 1,…,k, and hence only knowledge of the distribution of Xn* is needed.

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.

A ‘Secretary Problem’ with no recall but which allows the applicant to refuse an offer of employment with a fixed probability 1 – p, (0 < p < 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.