Condorcet winner criterion

The Condorcet candidate, Pairwise Champion (PC), or Condorcet winner (CW) of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate.

The Condorcet criterion for a voting system is that it chooses the pairwise champion when one exists. Any method conforming to the Condorcet criterion is known as a Condorcet method.

On a one-dimensional political spectrum, the pairwise champion will be at the position of the median voter.

Mainly because of Condorcet's voting paradox, a pairwise champion will not always exist in a given set of votes.

If the pairwise champion exists, they will be the only candidate in the Smith set; otherwise, the Smith set will have three or more members.

A more general wording of Condorcet criterion definition

Requirements:

1. The voting system must allow the voter to vote as many transitive pairwise preferences as desired. (Typically that's in the form of an unlimited ranking)
2. If there are one or more unbeaten candidates, then the winner should be an unbeaten candidate.

Traditional definition of "beat":

X beats Y iff more voters vote X over Y than vote Y over X.

Alternative definition of "beat" that is claimed to be more consistent with the preferences, intent and wishes of equal-top-ranking voters:

(Argument supporting that claim can be found at the Symmetrical ICT article.)

(X>Y) means the number of ballots voting X over Y.

(Y>X) means the number of ballots voting Y over X.

(X=Y)T means the number of ballots voting X and Y at top

(a ballot votes a candidate at top if it doesn't vote anyone over him/her)

X beats Y iff (X>Y) > (Y>X) + (X=Y)T

With this alternative definition of "beat", FBC and the Condorcet Criterion are compatible.

Majority Condorcet criterion

The majority Condorcet criterion is the same as the above, but with "beat" replaced by "majority-beat", defined to be "X majority-beats Y iff over 50% voters vote X over Y."

Complying methods

Black, Condorcet//Approval, Smith/IRV, Copeland, Llull-Approval Voting, Minmax, Smith/Minmax, ranked pairs and variations (maximize affirmed majorities, maximum majority voting), and Schulze comply with the Condorcet criterion.

It has been recently argued that the definition of the verb "beat" should be regarded as external to the Condorcet Criterion...and that "beat should be defined in a way that interprets equal-top ranking consistent with the actual preferences, intent and wishes of the equal-top-ranking voters. When such a definition of "beat" is used in the Condorcet Criterion definition, then the Condorcet Criterion is compatible with FBC, and there are Condorcet methods that pass FBC. Discussion and arguments on that matter can be found at the Symmetrical ICT article.

Approval voting, Range voting, Borda count, plurality voting, and instant-runoff voting do not comply with the Condorcet Criterion.

Commentary

Non-ranking methods such as plurality and approval cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences. But instant-runoff voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet criterion.

Consider, for example, the following vote count of preferences with three candidates {A,B,C}:

499:	A>B>C
498:	C>B>A
3:	B>C>A

In this case, B is preferred to A by 501 votes to 499, and B is preferred to C by 502 to 498, hence B is preferred to both A and C. So according to the Condorcet criteria, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transferred voted from B; in plurality voting A wins with the most first choices.

Range voting does not comply because it allows for the difference between 'rankings' to matter. E.g. 51 people might rate A at 100, and B at 90, while 49 people rate A at 0, and B at 100. Condorcet would consider this 51 people voting A>B, and 49 voting B>A, and A would win. Range voting would see this as A having support of 5100/100 = 51%, and B support of (51*90+49*100)/100 = 94.9%; range voting advocates would probably say that in this case the Condorcet winner is not the socially ideal winner.

Abstract Condorcet Criterion

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).

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.

References

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