Condorcet winner criterion: Difference between revisions

The Condorcet criterion can be abstractly modified to be "if the voting method would consider a candidate to be better than all other candidates when compared one-on-one, then it must consider that candidate better than all other candidates." Approval Voting and Score Voting, as well as traditional Condorcet methods pass this abstract version of the criterion, while IRV and STAR Voting don't (since they reduce to Plurality in the 2-candidate case and thus would need to always elect the traditional Condorcet winner in order to pass).<ref>https://rangevoting.org/CondDQ.html</ref>

One logical property (call it the "additive beatpath" property) that all traditional Condorcet methods fail, but which Approval and Score Voting pass is "if a voter with acyclic ranked preferences expresses a preference between two candidates (say A>Z), then the strength of that voter's preference between those two candidates (the amount of support they give to A to help beat Z) must equal the sum of the strengths of preference of all pairwise matchups of candidates that are in a beatpath from A to Z when sequentially going through each pair." In other words, if a voter's cardinally expressed preference is A5 B3 Z2, then under Score Voting the strength of A>Z (5-2=3 points, or 60% of the max score) will always equal the strength of preference of A>B (5-3=2 points/40% support) plus the strength of preference of B>Z (3-2=1 point/20% support), since that is just 3 = 2 + 1. With a traditional Condorcet method, this will fail because A>Z will be evaluated at 100% support, as will A>B and B>Z, and therefore the Condorcet method would give 100% = 100% + 100% which is incorrect. It would appear Borda methods pass this property, as a voter voting A>B>Z would have each candidate receive one point for every rank higher they are than another candidate, and thus a beatpath could be sequentially evaluated and strengths of preference added up to remain consistent. The failure of this property is the cause of Condorcet cycles in traditional Condorcet methods, and Condorcet cycles are the only time where traditional Condorcet methods can fail Favorite Betrayal and Independence of Irrelevant Alternatives, so in some sense, cardinal methods are a special case of Condorcet methods modified to pass the additive beatpath property, and on this basis cardinal methods pass and fail various properties that traditional Condorcet methods don't.

Approval Voting (and thus Score Voting when all voters use only the minimum or maximum score) is equivalent to a traditional Condorcet method where a voter must rank all candidates 1st or 2nd. Score Voting where some voters give some candidates intermediate scores can be treated as Approval Voting using the [[KP transform]], and thus treated as a traditional Condorcet method in the same way as Approval Voting.