Condorcet winner criterion

The Condorcet candidate, Pairwise Champion (PC), beats-all winner, or Condorcet winner (CW) of an election is the candidate who is preferred by more voters than any other candidate. This is determined by observing whether more voters rank or score the Condorcet winner higher than each of the other candidates than the other way around.

The Condorcet criterion for a voting system is that it chooses the beats-all winner when one exists. Any method conforming to the Condorcet criterion is known as a Condorcet method.

On a one-dimensional political spectrum, the beats-all winner will be at the position of the median voter.

Mainly because of Condorcet's voting paradox, a beats-all winner will not always exist in a given set of votes. However, there will always be a smallest group of candidates such that more voters prefer anyone in the group over anyone outside of the group. If the beats-all winner exists, they will be the only candidate in this group, which is called the Smith set. Voting methods that always elect from the Smith set are known as "Smith-efficient".

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. However, any voting method that collects enough information to detect pairwise preferences (i.e. scoring or ranking methods) can be "forced" to comply with the Condorcet criterion by automatically electing the Condorcet winner if one exists (or alternatively, eliminating all candidates not in the Smith Set) before doing anything else.

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. In general however, it is expected that the Condorcet winner (and Smith Set candidates in general) will almost always be very high-utility when compared to the utilitarian winner.

Sometimes there is no Condorcet winner, but there may be candidate(s) who are preferred by at least as many voters as all other candidates (i.e. as many voters rank or score them higher or equally as each of the other candidates as the other way around), who are known as weak Condorcet winners. While it may thus seem reasonable that a Condorcet method should pass a condition of always electing solely from the set of weak Condorcet winners when no regular Condorcet winner exists and at least one weak Condorcet winner exists, this guaranteeably leads to failures of reversal symmetry and clone immunity. Example (parentheses are used to indicate implied rankings):

3 A(>B1=B2=B3)

1 B1>B2>B3(>A)

1 B2>B3>B1(>A)

1 B3>B1>B2(>A)

A is the only weak Condorcet winner here, as they tie 3 to 3 when compared to B1, B2, and B3, and each of the latter 3 candidates suffer at least one pairwise defeat, and since there is no regular Condorcet winner (since every candidate has at least one pairwise tie or defeat), by the above-proposed condition A must win. To show a failure of reversal symmetry, suppose the ballots are reversed:

3 B1=B2=B3(>A)

1 A>B3>B2>B1

1 A>B1>B3>B2

1 A>B2>B1>B3

Again A is the only weak CW here, with there being no regular CW, so the same condition holds that A must still win. To show a failure of clone immunity, suppose B2 and B3 drop out of the race:

3 A(>B1) 3 B1(>A)

Now both of A and B1 are weak CWs, because they both pairwise tie each other. In this particular example, since there is nothing that distinguishes either candidate from the other, the neutrality criterion requires that both A and B1 must have an equal probability of winning i.e. both must have a 50% chance. This means that removing clones of B1 increased B1's chances of winning (which were originally at 0%, since A was guaranteed to win earlier i.e. had a 100% chance of winning.)

Multi-winner generalizations

Schulze has proposed a generalization of the Condorcet criterion for multi-winner methods: Suppose all but M+1 candidates are eliminated from the ballots, and the remaining candidates include candidate b. If b is always a winner when electing M winners from the M+1 remaining candidates, no matter who the other M remaining candidates are, then b is an M-seat Condorcet winner.

A method passes the M-seat Condorcet criterion if its M-seat election outcome always contains such a b when he exists, and passes the multi-winner Condorcet criterion if it passes the M-seat Condorcet criterion for all M.

When M=1, the generalization reduces to the ordinary Condorcet criterion as long as the method passes the majority criterion.

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 last. 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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