Editing Condorcet method (section)

{{Short description|Pairwise-comparison electoral system}}
{{Electoral systems sidebar|expanded=Single-winner}}
[[Image:Preferential ballot.svg|right|thumb|Example Condorcet method voting ballot. Blank votes are equivalent to ranking that candidate last.]]
A '''Condorcet method''' ({{IPAc-en|lang|pron|k|ɒ|n|d|ɔr|ˈ|s|eɪ}}; {{IPA|fr|kɔ̃dɔʁsɛ|lang}}) is an [[election method]] that elects the candidate who wins a [[majority rule|majority of the vote]] in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the ''pairwise champion'' or ''beats-all winner'', is formally called the ''Condorcet winner''<ref>{{cite journal |doi=10.1007/s003550000071 |quote=The Condorcet winner in an election is the candidate who would be able to defeat all other candidates in a series of pairwise elections.|title=Condorcet efficiency: A preference for indifference|year=2001|last1=Gehrlein|first1=William V.|last2=Valognes|first2=Fabrice|journal=Social Choice and Welfare|volume=18|pages=193–205|s2cid=10493112}}</ref> or ''Pairwise Majority Rule Winner'' (PMRW).<ref>{{Cite book |last=Gehrlein |first=William V. |title=Condorcet's paradox |date=2006 |publisher=Springer |isbn=978-3-540-33798-0 |series=Theory and decision library Series C, Game theory, mathematical programming and operations research |location=Berlin Heidelberg |quote=And, this is why the PMRW is commonly referred to as the Condorcet Winner.}}</ref><ref>{{Cite journal |last1=Tideman |first1=T. Nicolaus |last2=Plassmann |first2=Florenz |date=2011 |title=Modeling the Outcomes of Vote-Casting in Actual Elections |url=http://dx.doi.org/10.2139/ssrn.1627787 |journal=SSRN Electronic Journal |doi=10.2139/ssrn.1627787 |issn=1556-5068 |quote=A common definition of a voting cycle is the absence of a strict pairwise majority rule winner (SPMRW) … if no candidate beats all other candidates in pairwise comparisons.|url-access=subscription }}</ref> The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.<ref>{{cite web |last1=Green-Armytage |first1=James |title=Four Condorcet-Hare Hybrid Methods for Single-Winner Elections |date=2011 |s2cid=15220771 |url=http://www.votingmatters.org.uk/ISSUE29/I29P1.pdf |archive-url=https://web.archive.org/web/20130603095453/http://www.votingmatters.org.uk/ISSUE29/I29P1.pdf |archive-date=2013-06-03 |url-status=live }}</ref><ref>{{Cite book |title=The Mathematics of Elections and Voting |last=Wallis |first=W. D. |chapter=Simple Elections II: Condorcet's Method |publisher=[[Springer Nature|Springer]] |year=2014 |isbn=978-3-319-09809-8 |pages=19–32 |doi=10.1007/978-3-319-09810-4_3 |chapter-url=https://link.springer.com/chapter/10.1007/978-3-319-09810-4_3}}</ref>

Some elections may not yield a Condorcet winner because voter preferences may be cyclic—that is, it is possible  that every candidate has an opponent that defeats them in a two-candidate contest.<ref>{{cite journal |jstor=30022874?seq=1 |quote=Condorcet's paradox [6] of simple majority voting occurs in a voting situation [...] if for every alternative there is a second alternative which more voters prefer to the first alternative than conversely.|last1=Gehrlein|first1=William V.|last2=Fishburn|first2=Peter C.|title=Condorcet's Paradox and Anonymous Preference Profiles|journal=Public Choice|year=1976|volume=26|pages=1–18|doi=10.1007/BF01725789|s2cid=153482816}}</ref> The possibility of such cyclic preferences is known as the [[Condorcet paradox]]. However, a smallest group of candidates that beat all candidates not in the group, known as the [[Smith set]], always exists. The Smith set is guaranteed to have the Condorcet winner in it should one exist. Many Condorcet methods elect a candidate who is in the Smith set absent a Condorcet winner, and is thus said to be "Smith-efficient".<ref>{{cite web|url=http://pj.freefaculty.org/Papers/Ukraine/PJ3_VotingSystemsEssay.pdf |title=Voting Systems |first=Paul E. |last=Johnson |date=May 27, 2005|quote=Formally, the '''Smith set''' is defined as the smaller of two sets:<br/>1. The set of all alternatives, X.<br/>2. A subset A ⊂ X such that each member of A can defeat every member of X that is not in A, which we call B=X − A.}}</ref>

Condorcet voting methods are named for the 18th-century French [[mathematician]] and [[philosopher]] Marie Jean Antoine Nicolas Caritat, the [[Marquis de Condorcet]], who championed such systems. However, [[Ramon Llull]] devised the earliest known Condorcet method in 1299.<ref>{{cite journal|author=G. Hägele and F. Pukelsheim|year=2001|title=Llull's writings on electoral systems|url=http://www.math.uni-augsburg.de/stochastik/pukelsheim/2001a.html|journal=Studia Lulliana|volume=41|pages=3–38|archiveurl=https://web.archive.org/web/20060207154726/http://www.math.uni-augsburg.de/stochastik/pukelsheim/2001a.html|archivedate=2006-02-07}}</ref> It was equivalent to [[Copeland's method]] in cases with no pairwise ties.<ref>
{{cite journal|last=Colomer|first=Josep|year=2013|title=Ramon Llull: From Ars Electionis to Social Choice Theory|url=https://www.researchgate.net/publication/220007301|journal=[[Social Choice and Welfare]]|volume=40|issue=2|pages=317–328|doi=10.1007/s00355-011-0598-2|hdl=10261/125715|s2cid=43015882|hdl-access=free}}</ref>

Condorcet methods may use [[Ranked voting|preferential ranked]], [[cardinal voting|rated vote]] ballots, or explicit votes between all pairs of candidates. Most Condorcet methods employ a single round of preferential voting, in which each voter ranks the candidates from most (marked as number 1) to least preferred (marked with a higher number). A voter's ranking is often called their ''order of preference.'' Votes can be tallied in many ways to find a winner. All Condorcet methods will elect the Condorcet winner if there is one. If there is no Condorcet winner different Condorcet-compliant methods may elect different winners in the case of a cycle—Condorcet methods differ on which other criteria they satisfy.

The procedure given in [[Robert's Rules of Order]] for voting on motions and amendments is also a Condorcet method, even though the voters do not vote by expressing their orders of preference.<ref>{{cite journal |doi=10.1007/BF01789561 |quote=Binary procedures of the Jefferson/Robert variety will select the Condorcet winner if one exists|title=Did Jefferson or Madison understand Condorcet's theory of social choice?|year=1992|last1=McLean|first1=Iain|last2=Urken|first2=Arnold B.|journal=Public Choice|volume=73|issue=4|pages=445–457|s2cid=145167169}}</ref> There are multiple rounds of voting, and in each round the vote is between two of the alternatives. The loser (by majority rule) of a pairing is eliminated, and the winner of a pairing survives to be paired in a later round against another alternative. Eventually, only one alternative remains, and it is the winner. This is analogous to a single-winner or round-robin tournament; the total number of pairings is one less than the number of alternatives. Since a Condorcet winner will win by majority rule in each of its pairings, it will never be eliminated by Robert's Rules. But this method cannot reveal a [[voting paradox]] in which there is no Condorcet winner and a majority prefer an early loser over the eventual winner (though it will always elect someone in the [[Smith set]]). A considerable portion of the literature on social choice theory is about the properties of this method since it is widely used and is used by important organizations (legislatures, councils, committees, etc.). It is not practical for use in public elections, however, since its multiple rounds of voting would be very expensive for voters, for candidates, and for governments to administer.