In spatial environments, social welfare functions must meet certain criteria like weak Pareto and independence of irrelevant alternatives. When the policy space is one-dimensional, a welfare function is determined by a collection of preferences and a tie-breaking rule. For odd numbers of voters, simple majority voting is transitive only if each voter's preference is strictly quasi-concave. In multi-dimensional policy spaces, Arrow's impossibility theorem holds, showing that certain criteria like weak Pareto, independence of irrelevant alternatives, and non-dictatorship cannot all be satisfied if the set of alternatives is compact and convex.