The Topological Equivalence of the Pareto Condition and the Existence of a Dictator

Abstract

The paper studies two standard properties of rules for aggregating individual into social preferences: non-dictatorship and the Pareto condition. Together with the condition of independence of irrelevant alternatives, these are the three basic axioms of Arrow's social choice paradox.

We prove the topological equivalence between the Pareto condition and the existence of a dictator for continuous rules. The axiom of independence of irrelevant alternatives is not required.

The results use a topological framework for aggregation introduced in Chichilnisky (1980), but under different conditions. ln Chichilnisky (1980) rules are anonymous and respect of unanimity. Since anonymity is strictly stronger than the condition of non-dictatorship, while respeet of unanimity is strictly weaker than the Pareto conditions the two sets of conditions are nocomparable.