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

罗伯茨的“弱中立性”或“弱福利主义”定理涉及森社会福利泛函，其定义在效用函数轮廓的不受限制域上，并满足不相关备选方案的独立性、帕累托条件和弱连续性形式。罗伯茨 (Rev Econ Stud 47(2):421–439, 1980) 声称社会状态的诱导福利排序有一个单向表示，它是由在人际效用向量的欧几里德空间上定义的连续单调实值福利函数表示的——也就是说，对于社会严格偏好来说，这个福利函数的增加是足够的，但可能不是必要的。反例表明弱连续性是不够的；相反，建议对成对连续性进行小幅加强，并证明其充分性。