Condorcet loser criterion: Difference between revisions
Added information about runoff methods not electing honest Condorcet losers.
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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 ==
Note that it is very unlikely for a candidate with any significant amount of support to be a CL, even if they have significantly less support than other candidates, because they will still beat any candidates with negligible support i.e. [[Write-in candidate|Write-in candidat]]<nowiki/>es marked on only one ballot. Because of this, it is common to discuss the CL 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 on Warren Smith's and Anthony Gierzynski's flawed analysis.|last=FairVote.org|first=|date=|website=FairVote|url-status=live|archive-url=|archive-date=|access-date=2020-05-10|quote=In fact, Bucklin, Approval and Range voting quite possibly would have elected Kurt Wright (the Condorcet-loser among the top three)}}</ref>[[File:Alternative way to find Smith set ranking.png|thumb|633x633px|Note that no candidate is a CL here, because the candidate(s) who lose the most head-to-head matchups (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.
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