Previous sequential solutions for selecting the population with the largest center of symmetry amongkpopulations differing only in location have taken the following approach: Let cs denote the event of correct selection and derive a stopping rule which is such that, for α > 0 preassigned, lim infP(cs) ≥ 1 − α as δ → 0 when δ = θ[k]− θ[k−1]and θ[i]is theith smallest center of symmetry. In the present article the fixed δ case is of primary concern, and rules which perform well as α → 0 are sought. A general method of solution is presented and applied to yield a procedure based on the sample means. Comparison with a previously developed rule is made.