1. We describe four collective choice rules. Each of the rules satisfies three of the axioms while violating the fourth. This is sufficient to prove the independence of the axioms. Rule 1. For all x, y ? X, let xR 0 y if and only if |{i ? N : xR i y}| ? |{i ? N : yR i x}|. This rule clearly satisfies weak Pareto, independence of irrelevant alternatives, and non dictatorship, but violates monotonicity. Rule 2. Let d ? N . For all x, y ? X, let xR 0 y if and only if xR d y. This rule clearly satisfies monotonicity, weak Pareto, and independence of irrelevant alternatives, but violates non-dictatorship;Because R 0 is transitive, it follows that xP 0 z. But this means that xD S 1 z, which implies, by Lemma 1, that S 1 is a dictator. This violates the non-dictatorship axiom, and concludes the impossibility proof. Independence of the Axioms
2. Arrow's theorem: Unusual domains and extended codomains;J.-P Barthelemy;Mathematical Social Sciences,1982
3. Transitive multi-stage majority decisions with quasi-transitive individual preferences;R Batra;Econometrica,1972