Limits of acyclic voting.

Assuming three or more alternatives, there is no systematic rule for aggregating individual preferences that satisfies acyclicity and the standard independence and Pareto axioms, that avoids making some voter a weak dictator, and that is minimally responsive to changes in voter preferences. The latter axiom requires that a preference reversal in the same direction by roughly one third of all voters is sufficient to break social indifference. This result substantially strengthens classical acyclicity theorems of Mas-Colell and Sonnenschein (1972) and Schwartz (1986). When the set of alternatives is large, cycles become intuitively easier to construct, the acyclicity axiom has greater bite, and the responsiveness threshold can be increased to two less than the number of individuals, which yields the weakest logically possible responsiveness axiom.