Condorcet winner criterion: Difference between revisions
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{{Wikipedia}}
[[File:Finding the Condorcet winner.png|thumb|623x623px|Finding the Condorcet winner using [[Pairwise counting|pairwise counting]].]]<!-- {{cleanup|date=February 2020}} -->
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 in [[Pairwise counting|pairwise matchups]]. 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.
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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]].
Mainly because of [[Condorcet paradox|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]]". ▼
On a one-dimensional [[political spectrum]], the beats-all winner will be at the position of the median voter.▼
▲Mainly because of [[Condorcet paradox|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 ==
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# 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)
# If there are one or more unbeaten candidates, then the winner should be an unbeaten candidate. (Though usually this requirement is simply "If there is one candidate who beats all others, then they must win.")
'''Traditional definition of "beat":'''
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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.)▼
{{definition|1=▼
(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 ==
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Black, [[Condorcet//Approval]], Smith/IRV, [[Copeland's method|Copeland]], [[Llull-Approval Voting]], [[Minmax]], Smith/Minmax, [[ranked pairs]] and variations ([[maximize affirmed majorities]], [[maximum majority voting]]), and [[Schulze method|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==
▲On a one-dimensional [[political spectrum]], the beats-all winner will be at the position of the median voter.
Non-ranking methods such as [[plurality voting|plurality]] and [[approval voting|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.
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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. they beat '''or''' tie all other candidates; 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, and so it may be better to say that the set of weak Condorcet winners should have some, but not total priority to win. Example (parentheses are used to indicate implied rankings):<blockquote>3 A(>B1=B2=B3)
1 B1>B2>B3(>A)
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1 A>B2>B1>B3</blockquote>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:<blockquote>3 A(>B1)
3 B1(>A)</blockquote>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.) <ref>https://arxiv.org/abs/1804.02973v6 p. 206-207</ref>
Note that the Condorcet criterion also implies the following criterion which is somewhat related to Independence of Irrelevant Alternatives: removing losing candidates can't change the result of an election if there is a Condorcet winner. <ref>https://arxiv.org/abs/1804.02973 The Schulze Method of Voting p.351 "The Condorcet criterion for single-winner elections (section 4.7) is important because, when there is a Condorcet winner b ∈ A, then it is still a Condorcet winner when alternatives a1,...,an ∈ A \ {b} are removed. So an alternative b ∈ A doesn’t owe his property of being a Condorcet winner to the presence of some other alternatives. Therefore, when we declare a Condorcet winner b ∈ A elected whenever a Condorcet winner exists, we know that no other alternatives a1,...,an ∈ A \ {b} have changed the result of the election without being elected."</ref> In addition, adding candidates who are pairwise beaten by the Condorcet winner (when one exists) can't change the result of the election.
== Multi-winner generalizations ==
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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.
== Notes ==
▲'''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.)
▲{{definition|1=
▲(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.
▲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.
[[Category:Voting system criteria]]
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