Existence of Equilibrium on a Manifold

Abstract

Existence of equilibrium of a continuous preference relation p or correspondence P on a compact topological space W can be proved either by assuming acyclicity or convexity (no point belongs to the convex hull of its preferred set). Since both properties may well be violated in both political and economic situations, this paper considers instead a “local” convexity property appropriate to a “local” preference relation or preference field. The local convexity property is equivalent to the nonexistence of “local” cycles. When the state space W is a convex set, or is a smooth manifold of a certain topological type, then the “local” convexity property is sufficient to guarantee the existence of a set of critical optima.