In Martin Gardner's Mathematical Games column in the February 1960 issue of Scientific American, there appeared a simple problem that has come to be known today as the Secretary Problem, or the Marriage Problem. It has since been taken up and developed by many eminent probabilists and statisticians and has been extended and generalized in many different directions so that now one can say that it constitutes a "field" within mathematics-probability-optimization. The object of this article is partly historical (to give a fresh view of the origins of the problem, touching upon Cayley and Kepler), partly review of the field (listing the subfields of recent interest), partly serious (to answer the question posed in the title), and partly entertainment. The contents of this paper were first given as the Allen T. Craig lecture at the University of Iowa, 1988.

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Who Solved the Secretary Problem

We study the problem of maximizing the probability of stopping at an object which is best in at least one of a given set of criteria, using only stopping rules based on the knowledge of whether the current object is relatively best in each of the criteria. The asymptotic results for the case of independent criteria are shown to hold in certain cases where the componentwise maxima are, pairwise, either asymptotically independent or asymptotically full dependent. An example of the former is a random sample from a bivariate correlated normal distribution; thus our results settle a question posed recently by T. S. Ferguson.

On a best-choice problem with dependent criteria

For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[ q ( R τ )], where R j is the rank of the j th observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q ( r ) = 1 ( r > 1), and Robbins' problem, with q ( r ) = r . As n tends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

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Optimal stopping with rank-dependent loss

A note on monotonicity in optimal multiple stopping problems

Consider a sequence of random vectors Y(1),…, Y(n) in ℛ d , with a known joint distribution, finite expectations, adapted to a filtration ℱ. For a given monotone function h : ℛ d  → ℛ, a randomized stopping rule t ⋆ such that sup t h(E Y(t)) = h(E Y(t ⋆)) is desired. A full solution to this problem is given. In particular we show that there exists a vector of constants a = (a 1,…, a d ) such that the optimizing t ⋆ is optimal also for the one-dimensional optimal stopping problem, where the sequence of rewards is Z(1),…, Z(d) with Z(j) =  . With symmetry assumptions the a vector will be (1,…, 1). The results are extended also to the infinite horizon case and are applied to obtain an explicit solution to the multivariate “house-selling problem.”

Yuliy M Baryshinikov

Papers

On a class of best-choice problems