Arrow's Theorem in Spatial Environments

In spatial environments we consider social welfare functions satisfying Arrow''s requirements, i.e. weak Pareto and independence of irrelevant alternatives. When the policy space is a one-dimensional continuum such a welfare function is determined by a collection of 2 N strictly quasiconcave preferences and a tie-breaking rule. As a corollary we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter''s preference is strictly quasiconcave. When the policy space is multi-dimensional we establish Arrow''s impossibility theorem. Among others we show that weak Pareto, independence of irrelevant alternatives, andnon-dictatorship are inconsistent if the set of alternatives has a non-empty interior and it is compact and convex.