Condorcet winner criterion: Difference between revisions

The Condorcet[[Bipartisan set]] (a subset of the winner/[[Smith set]]) is athe common [[equilibrium]] point inof manymost 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.

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]].

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 almost alwaysusually be very high-utility, wheneven comparedif they are not the highest-utility tocandidate (the utilitarian winner).

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 name="Schulze 2018 with footnote">https://arxiv.org/abs/1804.02973{{cite arXiv | last=Schulze | first=Markus | title=The Schulze Method of Voting p| date=2018-03-15 | eprint=1804.02973 |page=351|class=cs.GT}} "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.

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">https://arxiv.org/abs/{{cite arXiv | last=Schulze | first=Markus | title=The Schulze Method of Voting | date=2018-03-15 | eprint=1804.02973v602973 p|page=206–207|class=cs. 206-207GT}}</ref>

Schulze has proposed a generalization of the Condorcet criterion for multi-winner methods:<ref name="Schulze 2018">{{cite webarXiv | last=Schulze | first=Markus | title=The Schulze Method of Voting | website=arXiv.org | date=2018-03-15 | urleprint=https://arxiv.org/abs/1804.02973v602973 | access-datepage=2020-02-11351|pageclass=351cs.GT}}</ref> Suppose all but M+1 candidates are eliminated from the ballots, and the remaining candidates include candidate ''b''. If ''b'' is always a winner when electing M winners from the M+1 remaining candidates, no matter who the other M remaining candidates are, then ''b'' is an M-seat Condorcet winner.

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. ">{{Citecite conference web|url last=http://www10.org/cdrom/papers/577/Dwork |title first=RankCynthia Aggregation| Methodslast2=Kumar for| thefirst2=Ravi Web|last last3=Naor |first first3=Moni |date last4=Sivakumar |website first4=D. |url-status title=liveRank aggregation methods for the Web |archive-url publisher=ACM |archive publication-dateplace=New York, NY, USA |access-date year=}}</ref>2001 and| metasearchdoi=10.1145/371920.372165 <ref>{{Cite| web|url=http://www.ccseecs.neuharvard.edu/home~michaelm/jaaCS222/IS4200.10X1/resources/condorcetrank.pdf|title}}</ref> and metasearch<ref name=Condorcet"Montague FusionAslam for2002 Improvedp. ">{{cite conference Retrieval| last=Montague | first=Mark |date last2=Aslam |website first2=Javed A. |url-status title=liveCondorcet fusion for improved retrieval |archive-url publisher=ACM |archive publication-dateplace=New York, NY, USA |access- date=2002-11-04 | doi=10.1145/584792.584881 | url=http://www.ccs.neu.edu/home/jaa/IS4200.10X1/resources/condorcet.pdf}}</ref>. The concept of a [[Smith set ranking]] (which is sometimes referred to as the "extended/generalised Condorcet criterion") helps structure how the ranking of all options should be, rather than only the winner.