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Added information about runoff methods not electing honest Condorcet losers.
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{{Wikipedia}}{{Image frame|width=500|content=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):
{{Wikipedia}}
 
FF>H:51, FF>C:60
In single-winner [[voting system]] theory, the '''Condorcet loser criterion''' is a measure for differentiating voting systems. It implies the [[majority loser criterion]].▼
 
H>FF:49, H>C:70
A [[voting system]] complying with the Condorcet loser criterion will never allow a ''Condorcet loser'' to win. A Condorcet loser is a candidate who can be defeated in a [[Condorcet method|head-to-head competition]] against each other candidate. (Not all elections will have a Condorcet loser since it is possible for three or more candidates to be mutually defeatable in different head-to-head competitions. However, there is always a Smith loser set, which is the smallest group of candidates such that any of them can be defeated by any candidate not in the group.)▼
 
C>FF:40, C>H:20
A slightly weaker (easier to pass) version is the majority Condorcet loser criterion, which requires that a candidate who can be defeated ''by a majority'' in a head-to-head competition against each other candidate, lose. It is possible for a system, such as Majority Judgment, which allows voters not to state a preference between two candidates, to pass the MCLC but not the CLC.{{citation needed|date=January 2012}}
 
If for each pair of candidates, we subtract the number of votes preferring the second candidate over the first from the number of votes preferring the first to the second, then we'll know which one won the head-to-head matchup.
 
(Margins)
 
FF>H:2, FF>C:20
 
H>FF:-2 (Loss), H>C:20
 
<p style="border: 5px dotted red;">C>FF:-20 (Loss), C>H:-50 (Loss) </p>
The Condorcet loser (if one exists) will be the candidate who got a minority of votes (as indicated by the negative margin) in all of their head-to-head matchups.
 
C (Cookies) is the CL here.
 
▲|caption=Example of finding the Condorcet loser|border=|max-width=}}In single-winner [[voting system]] theory, the '''Condorcet loser criterion''' (or anti-Condorcet criterion) is a measure for differentiating voting systems. It implies the [[majority loser criterion]].
 
▲A [[voting system]] complying with the Condorcet loser criterion will never allow a ''Condorcet loser'' (anti-[[Condorcet candidate]]) to win. A Condorcet loser is a candidate who can be defeated in a [[Condorcet method|head-to-head competition]] against each other candidate. (Not all elections will have a Condorcet loser since it is possible for three or more candidates to be mutually defeatable in different head-to-head competitions. However, there is always a [[Smith loser set]], which is the smallest group of candidates such that any of them can be defeated by any candidate not in the group.)
 
Compliant methods include: [[two-round system]], [[instant-runoff voting]] (AV), [[contingent vote]], [[borda count]], [[Schulze method]], [[ranked pairs]], and [[Kemeny-Young method]].
 
Noncompliant methods include: [[plurality voting]], [[Contingent vote#Supplementary Vote|supplementary voting]], [[Contingent vote#Sri Lankan contingent vote|Sri Lankan contingent voting]], [[approval voting]], [[range voting]], [[Bucklin voting]] and [[minimax Condorcet]].
 
== Implications ==
 
In an election with only two candidates, the Condorcet loser criterion implies the [[majority criterion]]. Given a three-candidate Condorcet cycle, it's always possible to eliminate a candidate who didn't win so that the winner changes. Thus the Condorcet loser criterion is incompatible with [[independence of irrelevant alternatives]].
 
== Examples ==
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Ranked pairs work by "locking in" strong victories, starting with the strongest, unless that would contradict an earlier lock.
Assume that the Condorcet loser is X. For X to win, ranked pairs must lock a preference of X over some other candidate Y (for at least one Y) before it locks Y over X. But since X is the Condorcet loser, the victory of Y over X will be greater than that of X over Y, and therefore Y over X will be locked first, no matter what other candidate Y is. Therefore, X cannot win.
 
== Runoffs and honest Condorcet loser compliance ==
 
Any voting method or algorithm that operates by eventually reducing the number of candidates to two, where the candidate who beats the other one-on-one becomes the winner of the election, will always pass the Condorcet loser criterion. By definition, the Condorcet loser can't win the finalist match-up even if chosen to be one of the two finalists. Such methods include [[IRV]], [[STAR voting|STAR]], the [[contingent vote]], etc.
 
Voting methods with a manual two-candidate runoff round pass a strategy-immune Condorcet loser criterion: if the voters are honest, they pass Condorcet loser as usual. But if the voters are strategic, and everybody participates in both rounds, then the method as a whole will never elect the honest Condorcet loser. This happens because two-candidate majority elections are strategy-proof: there's never any incentive to vote for X when you prefer Y to X, if the only two candidates are Y and X. The honest Condorcet loser thus can't win even if he gets to the second round. One example of such a method is [[top-two runoff]].
 
If the voters change their minds between the rounds, the "honest Condorcet loser" is the candidate who, according to the voters' opinion at the time they vote in the second round, is the Condorcet loser.
 
== Notes ==
 
AnyNote votingthat methodit thatis operatesvery byunlikely havingfor ora beingcandidate ablewith toany besignificant amount of reducedsupport to be a finalCL, runoffeven willif alwaysthey passhave thesignificantly Condorcetless losersupport criterionthan other candidates, sincebecause eitherthey thewill Condorcetstill loserbeat isany notcandidates with negligible support i.e. [[Write-in thecandidate|Write-in runoffcandidat]]<nowiki/>es andmarked thuson can'tonly winone ballot. Because of this, orit is incommon to discuss the runoffCL criterion when only looking at matchups between the major candidates. <ref>{{Cite web|url=https://www.fairvote.org/more-on-warren-smiths-and-anthony-gierzynskis-flawed-analysis|title=More ison byWarren definitionSmith's defeatedand byAnthony theirGierzynski's opponent;flawed thisanalysis.|last=FairVote.org|first=|date=|website=FairVote|url-status=live|archive-url=|archive-date=|access-date=2020-05-10|quote=In includesfact, [[IRV]]Bucklin, Approval and Range voting quite possibly would have elected Kurt Wright (the Condorcet-loser among the top three)}}</ref>[[STARFile:Alternative votingway to find Smith set ranking.png|STAR]]thumb|633x633px|Note that no candidate is a CL here, Topbecause the candidate(s) who lose the most head-to-twohead runoff,matchups etc(G and F) pairwise tie each other. See [[Smith set ranking]]. ]]A generalization of the Condorcet loser criterion is the Smith loser criterion: a candidate in the [[Smith loser set]] (the smallest group of candidates such that more voters prefer anyone not in the group over anyone in the group) should never win unless all candidates are in the Smith loser set. The [[Smith criterion]] implies the Smith loser criterion, since the Smith set only overlaps with the Smith loser set when both sets include all candidates.<ref>[[Talk:Condorcet ranking]] Look for "I think I can prove" on the message with a timestamp date of 21 February 2020.</ref> The Smith loser criterion implies the Condorcet loser criterion, since a Condorcet loser, when they exist, will always be the only candidate in the Smith loser set. Many non-Smith efficient methods that pass the Condorcet loser criterion fail the Smith loser criterion.
 
== See also ==
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{{voting systems}}
[[Category:Voting system criteria]]
[[Category:Condorcet-related concepts]]
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