The Limits of Acyclic Social Choice and Nash Implementability PRELIMINARY AND INCOMPLETE

I prove that there is no systematic rule for aggregating individual preferences that satisfies the standard axioms of Independence, Pareto, and Acyclicity, that avoids making any individual a weak dictator, and that is sensitive to substantial changes in individual preferences. When there are three or more alternatives, the latter axiom requires that a preference reversal in the same direction by one fourth of all individuals is sufficient to break social indifference; when there are four or more alternatives, a higher threshold of one third of all individuals can be used. These results substantially strengthen the acyclicity theorem of Mas-Colell and Sonnenschein (1972). Corresponding to any social choice rule is a ‘‘revealed social preference’’ rule that satisfies Independence and Acyclicity; when we impose monotonicity as well, further axioms on the social choice rule translate to properties of revealed social preferences, and we can apply the acyclicity theorems. We conclude, broadly speaking, that equilibrium outcome correspondences must either concentrate power in small groups or be insensitive to substantial changes in individual preferences.