Preferences, Summation, and Social Welfare Functions

Abstract

This paper gives a necessary and sufficient condition for the following proposition, in which ⪯ and ⪯ i for i = 1,…, n are weak orders on a finite set X . There are real-valued functions f 1 , f 2 ,…, f n on X such that, for all x and y in X and i in {1,…, n } x ⪯ i y if and only if f i ( x ) ≤ f i ( y ), and x ⪯ y if and only if f l ( x ) + ⋯ + f n ( x ) ≤ f l ( y ) + ⋯ + f n ( y ). The condition can be viewed as an extension of a simple unanimity or dominance condition.