Aggregation of smooth preferences

Abstract

Suppose p is a smooth preference profile (for a society, N) belonging to a domain P N . Let σ be a voting rule, and σ(p)(x) be the set of alternatives in the space, W, which is preferred to x. The equilibrium E(σ(p)) is the set {x∈W:σ(p)(x) is empty}. A sufficient condition for existence of E(σ(p)) when p is convex is that a “dual”, or generalized gradient, dσ(p)(x), is non-empty at all x. Under certain conditions the dual “field”, dσ(p), admits a “social gradient field”Γ(p). Γ is called an “aggregator” on the domain P N if Γ is continuous for all p in P N . It is shown here that the “minmax” voting rule, σ, admits an aggregator when P N is the set of smooth, convex preference profiles (on a compact, convex topological vector space, W) and P N is endowed with a C 1-topology. An aggregator can also be constructed on a domain of smooth, non-convex preferences when W is the compact interval. The construction of an aggregator for a general political economy is also discussed. Some remarks are addressed to the relationship between these results and the Chichilnisky-Heal theorem on the non-existence of a preference aggregator when P N is not contractible.