Condorcet method: Difference between revisions
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All Condorcet methods pass the [[mutual majority criterion]] when there is a Condorcet winner. This is because the CW is guaranteed to be a member of any set of candidates that can pairwise beat all candidates not in the set, and the mutual majority set is such a set, because all candidates in it are ranked by a majority over all candidates not in the set. [[Smith-efficient]] Condorcet methods always pass the [[mutual majority criterion]].
Most Condorcet methods allow for equal-ranking. Because of this, it is possible to vote [[Approval voting]]-style. In fact, if all voters vote Approval-style, the Smith set will only have candidates who pairwise tie, rather than who have [[Condorcet cycle|Condorcet cycles]]. And in fact, if every voter ranks a candidate either 1st or last with a probability proportional to their cardinal [[utility]] for that candidate, then you get a [[Smith set ranking]] mirroring the [[Score voting]] ranking with probability approaching 1 when there are many voters. This is because if 100 voters consider a candidate a 3 out of 10, then if they use a 30% probability of ranking that candidate 1st, otherwise ranking them last, then it is very likely the candidate will end up ranked 1st on 30 of their ballots and last on 70, similar to being approved by only 30 of them. This mitigates one common utilitarian concern with Condorcet, that it might let a majority force its weak preference onto the minority; this is because voters with weak preferences may be willing to equally rank candidates in order to allow voters with stronger preferences to have the deciding vote. See also the [[KP transform]], which can be used to model transforming cardinal utilities into ranked ballots by making Score ballots into Approval ballots, and then Approval ballots into ranked ballots.
One concern with Condorcet methods is that it is very difficult to do [[pairwise counting]] for elections with 10 of more candidates, since that is at least (0.5*10*((10-1)=9))=45 pairwise matchups to record the details of. Allowing write-in candidates makes things even more complex. One possible solution would be to have a primary beforehand using a voting method better than [[FPTP]] to pick 5 top candidates, and then only allow voters to rank those top 5. For all other candidates, they'd be able to approve or score each of them. The rated information could then be used to elect someone other than one of the top 5 when the non-top 5 candidates have significantly higher ratings, but otherwise only elect one of the top 5. The primary itself could be made slightly semi-proportional as well.
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