Preference domains and the monotonicity of condorcet extensions

An alternative is a Condorcet winner if it beats all other alternatives in a pairwise majority vote. A social choice correspondence is a Condorcet extension if it selects the Condorcet winners–and nothing else–whenever a Condorcet winner exists. It is well known that Condorcet extensions are not monotonic (hence, not Nash implementable) when all preferences are admissible, but are implementable when restricted to a domain in which Condorcet winners always exist. We fill the gap by studying the intermediate domains and find that monotonicity is violated on all such domains.