Roberts’ weak welfarism theorem: a minor correction

Abstract Roberts’ “weak neutrality” or welfarism” theorem concerns Sen social welfare functionals which are defined on an unrestricted domain of utility function profiles and satisfy independence irrelevant alternatives, the Pareto condition, a form weak continuity. Roberts (Rev Econ Stud 47(2):421–439, 1980) claimed that induced ordering states has one-way representation by continuous, monotonic real-valued Euclidean space interpersonal vectors—that is, increase in this is sufficient, but may not be necessary, for strict preference. A counter-example shows continuity insufficient; minor strengthening to pairwise proposed instead its sufficiency demonstrated.