Uniformly bounded sufficient sets and quasi‐extreme social welfare functions

The set of alternatives is infinite. If the social welfare function is transitive‐valued and minimal sufficient sets are uniformly bounded, then there are arbitrarily large finite subsets of the feasible set, and a rich sub‐domain of profiles, within which a reduction in the scope of someone's dictatorial power must be accompanied by an equal increase in the fraction of the pairs that are socially ordered without consulting anyone's preferences.