Aggregation of smooth preferences

Suppose p is a smooth preference profile (for a society, N) belonging to a domain PN. Let be a voting rule, and (p)(x) be the set of alternatives in the space, W, which is preferred to x. The equilibrium E((p)) is the set {x∈W:(p)(x) is empty}. A sufficient condition for existence of E((p)) when p is convex is that a "dual", or generalized gradient, d(p)(x), is non-empty at all x. Under certain conditions the dual "field", d(p), admits a "social gradient field" (p). is called an "aggregator" on the domain PN if is continuous for all p in PN. It is shown here that the "minmax" voting rule, , admits an aggregator when PN is the set of smooth, convex preference profiles (on a compact, convex topological vector space, W) and PN is endowed with a C1-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 PN is not contractible.