There can be equitable social welfare orders but no fair voting system

Abstract

In the context of intergenerational equity it is natural to ask that social welfare orders have th properties of finite anonymity and strong Pareto efficiency. Such social welfare orders can be constructed from nonprincipal ultrafilters over the set of natural numbers, and the existence of ultrafilters of this kind is known to be equivalent to the existence of a fair voting system for a countably infinite population in the sense of Arrow's theorem. Unfortunately, the existence of this variety of social welfare order, which we call anonymous Pareto, implies the existence of nonmeasurable sets of real numbers. For some, such as Litak, this entails that such social welfare orders are inadmissible for use in theoretical economics because they are non-constructive in nature. The authors seek to give nuance to this view, proving that anonymous Pareto and other social welfare orders can exist in the absence of nonprincipal ultrafilters over the set of natural numbers.