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{{Wikipedia}}<!-- {{cleanup|date=February 2020}} -->
An [[electoral system]] satisfies the "'''Condorcet winner criterion"''', also known as the '''Condorcet criterion''', if it always chooses the Condorcet winner when one exists. The "Condorcet winner" is sometimes referred to as the "'''Condorcet candidate"''', "'''
The '''Condorcet winner 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]]. Though Condorcet winner criterion is sometimes referred to as simply the "Condorcet criterion", it's important not to confuse the Condorcet winner criterion with the "[[Condorcet loser criterion]]" .
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]]".
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As an example, if there are 3 candidates, with head-to-head matchups indicating a 51% majority prefers the second candidate over the first, and a 43% plurality prefer the second over 37% preferring the third (with 20% of voters having no preference), then the second candidate gets more votes than their competitors in all matchups and so they are the Condorcet winner.
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Suppose the voters had been polled on their preferences among the candidates, and the following preferences in head-to-head matchups are produced between French Fries (FF), Hamburger (H), and Cookies (C) (FF>C shows the number of voters who prefer FF over C, for example):
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FF (French Fries) is the CW here.
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'''Requirements:'''
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'''Traditional definition of "beat":'''
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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." Thus, a '''majority Condorcet winner''' is a Condorcet winner who majority-beats all other candidates.
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[[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.
== Occurrences in real elections ==
Most real elections have a Condorcet winner. Andrew Myers, who operates the [[online poll|Condorcet Internet Voting Service]], found that 83% of the nonpolitical CIVS elections with at least 10 votes had a Condorcet winner, with the figure rising to 98.8% for elections with at least 300 votes.<ref name="CIVS">{{cite conference |url=https://www.cs.cornell.edu/andru/papers/civs24/ |title=The Frequency of Condorcet Winners in Real Non-Political Elections |last=Myers |first=A. C. |author-link=https://www.cs.cornell.edu/andru/ |date=March 2024 |conference=61st Public Choice Society Conference}}</ref>
A database of 189 ranked United States election from 2004 to 2022 contained only one Condorcet cycle: the [[2021 Minneapolis Ward 2 city council election]].<ref name="GSM2023">{{cite arXiv | last=Graham-Squire | first=Adam | last2=McCune | first2=David | title=An Examination of Ranked Choice Voting in the United States, 2004-2022 |eprint=2301.12075v2 | date=2023-01-28 | class=econ.GN}}</ref> While this indicates a very high rate of Condorcet winners, it's possible that some of the effect is due to general [[two-party domination]].
==Commentary==
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On a one-dimensional [[political spectrum]], the beats-all winner will be at the position of the median voter. See the following example:
[[File:Left,_Center,_Right.png]]
Supposing this is the voter distribution, with three candidates '''L'''eft, '''C'''enter, and '''R'''ight, with voters concentrated in each area of the distribution as given, with the yellow voters preferring Left, the green preferring Center, and the blue preferring Right. Then the Condorcet winner is Center, because a majority - the Center voters plus the Left voters - prefers Center to Right, and another majority prefers Center to Left.
[[Instant-runoff voting]], which does not pass the Condorcet criterion, fails to detect Center's support and thus Center is eliminated early. However, if the majority preferring Center to the IRV winner knew this in advance, they could have used [[compromising]] strategy to force Center to be elected anyway. See below.
Note that the "Center" candidate is only the center of the voters' distribution. If this were a conservative party primary, the center of the distribution would likely be a conservative candidate, not a centrist.
===Equilibrium point for various voting methods===
The [[Bipartisan set]] (a subset of the [[Smith set]]) is the common [[equilibrium]] point of most voting methods. This is because a majority/plurality of voters have no incentive to deviate towards another candidate. The Condorcet criterion can thus be considered a type of [[Declared strategy voting|automatic strategy]], which reduces the need for [[compromising]] strategy by electing candidates who could have won with majority-strength compromising.
35: A>B|>C
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31: C>B|>A
B is the CW. If voters approve everyone they ranked before the "|", then B is approved by all voters, and wins. If any of the three groups of voters here raises their approval threshold (only approves their 1st choice), then another group has an incentive to maintain their approval threshold where it is i.e. if C-top voters stop approving B, then the 69 voters who prefer B>C have an incentive to move their approval thresholds between B and C to ensure B is approved by a majority and C is not. Note that this requires both accurate polling and coordinated [[Strategic voting|strategic voting]].
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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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<blockquote><table>
<tr><td align="right">499:</td><td align="left">A>B>C</td></tr>
<tr><td align="right">498:</td><td align="left">C>B>A</td></tr>
<tr><td align="right">3:</td><td align="left">B>C>A</td></tr>
</table></blockquote>
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. Note that B and C are a [[Mutual majority criterion|mutual majority]], so most majority rule-based methods would rule A out of winning. If A drops out, then B becomes the majority's 1st choice; so this is an example of IRV failing [[Independence of irrelevant alternatives|independence of irrelevant alternatives]].
See [[Score voting#Majority-related criteria]] to see how Score can fail the Condorcet criterion. In general however, it is expected that the Condorcet winner (and Smith Set candidates in general) will
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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.
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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)
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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 name="Schulze 2018 p206">
Weak CWs have also been called Condorcet non-losers, with the requirement that they always win when they exist being called Exclusive-Condorcet. <ref>{{Cite web|url=http://www.votingmatters.org.uk/ISSUE3/P5.HTM#r2|title=Voting matters, Issue 3: pp 8-15|last=|first=|date=|website=www.votingmatters.org.uk|url-status=live|archive-url=|archive-date=|access-date=2020-05-09|quote=Exclusive-Condorcet (see Fishburn[2]). If there is a Condorcet non-loser, then at least one Condorcet non-loser should be elected.}}</ref>
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Schulze has proposed a generalization of the Condorcet criterion for multi-winner methods:<ref name="Schulze 2018">{{cite
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.
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Note that Bloc Ranked Pairs and Bloc Score voting (if scored methods are considered) would pass this criterion, though they are not proportional, and the latter is not a Condorcet method in the single-winner case. So it may make more sense to consider Schulze's criterion as one of several that a multi-winner method ought to pass to be considered a Condorcet multi-winner or Condorcet PR method, rather than the definitive one.
In addition to Schulze's generalization, Gehrein, and Aziz ''et al.'' have proposed different multi-winner generalizations, based on the concept of stability.<ref name="Aziz">{{cite arXiv | last1=Aziz | first1=Haris | last2=Elkind | first2=Edith | last3=Faliszewski | first3=Piotr | last4=Lackner | first4=Martin | last5=Skowron | first5=Piotr | title=The Condorcet Principle for Multiwinner Elections: From Shortlisting to Proportionality | date=2017-01-27 | eprint=1701.08023 | class=cs.GT}}</ref>
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See [[Self-referential Smith-efficient Condorcet method]].
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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.
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Every Condorcet method is susceptible to burial in at least some elections, and Condorcet is also incompatible with [[later-no-harm]] and [[later-no-help]]. However, if a method
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The Condorcet criterion and methods that pass it have been criticized for certain reasons. Some common arguments are:
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This criticism can easily be averted if the voters on either side of the Democrats simply refuse to rank the Democrats above the other side..
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It is rather common for pollsters to do head-to-head matchup polling to see who is likely to win in an [[FPTP]] general election. Condorcet polling can be done in a similar way, except more efficiently, by allowing polled voters to rank or rate the candidates.<ref>{{Cite journal|last=Potthoff|first=Richard F.|date=2011-07-01|title=Condorcet Polling|url=https://doi.org/10.1007/s11127-010-9646-1|journal=Public Choice|language=en|volume=148|issue=1|pages=67–86|doi=10.1007/s11127-010-9646-1|issn=1573-7101}}</ref>
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One strategy common to most Condorcet methods is to prevent a candidate from being a Condorcet winner by [[burying]] them (giving them a pairwise defeat against another candidate).
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The [[tied at the top]] rule redefines the Condorcet beat relation so that methods using it can pass Condorcet whenever there are no equal-rank, and in addition passes the [[favorite betrayal criterion]]. Doing so in effect trades some Condorcet winner compliance for FBC compliance.
[[Category:Voting system criteria]]
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Analogues to the Condorcet criterion have been proposed in non-voting contexts; it appears in many places when discussing how to aggregate ranked information. It has been used to discuss search engine rankings <ref name="Dwork Kumar Naor Sivakumar 2001 p. ">{{
==References==
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