Preferences, Summation, and Social Welfare Functions

This paper gives a necessary and sufficient condition for the following proposition, in which \preceq and \preceq _i for i - 1,..., n are weak orders on a finite set X. There are real-valued functions f₁, f₂,..., f_n on X such that, for all x and y in X and i in {1,...,n}x\preceq _i y if and only if f_i(x) < f_i(y), and x \preceq y if and only if f_l(x) + \cdots + f_n(x) < f_l(y) + \cdots + f_n(y). The condition can be viewed as an extension of a simple unanimity or dominance condition.