Choosing the best of the current crop

Abstract

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.