Social equilibrium and cycles on compact sets

One proof of existence of general equilibrium assumes convexity and continuity of a preference correspondence on a compact convex feasible set W. Here the existence of a local equilibrium for a preference field which satisfies, not convexity, but the weaker local acyclicity is shown. The theorem is then applied to a voting game, σ, without veto players. It is shown that if the dimension of the policy space is no greater than ν(σ) ? 2, where ν(σ) is the Nakamura number of the game, then no local cycles may occur and a local equilibrium must exist. With convex preferences, then, there will exist a choice of the game from W.