Arrovian Aggregation of Convex Preferences and Pairwise Utilitarianism

We consider social welfare functions that satisfy Arrow’s classic axioms of independence of irrelevant alternatives and Pareto optimality when the outcome space is the convex hull over some finite set of alternatives. Individual and collective preferences are assumed to be continuous and convex, which guarantees the existence of maximal elements and the consistency of choice functions that return these elements, even without insisting on transitivity. We provide characterizations of both the domains of preferences and the social welfare functions that allow for Arrovian aggregation. The domains allow for arbitrary preferences over alternatives, which in turn completely determine an agent’s preferences over all outcomes. The only Arrovian social welfare functions on these domains constitute an intriguing combination of utilitarianism and pairwiseness. When also assuming anonymity, Arrow’s impossibility turns into a complete characterization of a unique social welfare function, which can be readily applied in settings that allow for lotteries or divisible goods such as time or money.