Cycle-adjusted quota-Condorcet
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q-Condorcet methods use quotas other than 50% to declare a Condorcet winner; for example, a 2/3-Condorcet method declares a candidate to be the winner if they defeat every other candidate by a margin of 2/3. By Nakamura's theorem, the q-Condorcet winner is guaranteed to be acyclic for all voter profiles if and only if
q = (n_candidates - 1) / n_candidates. The same quota also guarantees that a q-Condorcet method is participation-consistent.Working on this more. Right now I have some interesting questions, like: What if q depends on the number of ballots involved in cycles? Could some method satisfy Condorcet-like properties for a "mostly acyclic" electorate, but otherwise fall back on some other method? And do so in a way that still satisfies participation?
This seems like a nice way to smoothly interpolate between Condorcet and non-Condorcet methods (like score), depending on whether the optimality criteria for Condorcet are satisfied.
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In related news, it looks like ranked pairs satisfies participation when there are at most 3 candidates in the Smith set (where it reduces to Minimax). See here for why 3-candidate Minimax is participation-consistent. This can fail for Minimax, though, because it fails ISDA.