Social welfare functions for economic environments with and without the pareto principle

It is shown for the case of private goods economies that every social welfare function satisfying a weak nonimposition condition and the independence of irrelevant alternatives axiom is of one of the following forms. It is either null, or the class of decisive coalitions is an ultrafilter, or the class of anti-decisive coalitions is an ultrafilter. In the case of a private goods economy with finitely many traders, the latter conditions imply the existence of either a dictator or anti-dictator. By requiring the Pareto principle as well, it is easily seen that the social welfare function must be dictatorial.