Preferences, Summation, and Social Welfare Functions

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 f₁, f₂,…, f_n on X such that, for all x and y in X and i in {1,…,n}x⪯_iy 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.