Condorcet loser criterion: Difference between revisions
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In single-winner [[voting system]] theory, the '''Condorcet loser 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'' 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.)
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}}
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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.
== Notes ==
Any voting method that operates by having or being able to be reduced to a final runoff will always pass the Condorcet loser criterion, since either the Condorcet loser is not in the runoff and thus can't win, or is in the runoff and is by definition defeated by their opponent; this includes [[IRV]], [[STAR voting|STAR]], Top-two runoff, etc. 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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