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{{Short description|Property of electoral systems}}
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A '''Condorcet winner''' ({{IPA|fr|kɔ̃dɔʁsɛ|lang}}, {{IPAc-en|lang|pron|k|ɒ|n|d|ɔr|ˈ|s|eɪ}}) is a candidate who would receive the support of [[majority|more than half]] of the electorate in a one-on-one race against any one of their opponents. [[Electoral system|Voting systems]] where a majority winner will always win are said to satisfy the '''Condorcet winner criterion'''. The Condorcet winner criterion extends the principle of [[majority rule]] to elections with multiple candidates.<ref>{{Cite journal |last1=Lepelley |first1=Dominique |last2=Merlin |first2=Vincent |date=1998 |title=Choix social positionnel et principe majoritaire |journal=Annales d'Économie et de Statistique |issue=51 |pages=29–48 |doi=10.2307/20076136 |issn=0769-489X |jstor=20076136 }}</ref><ref name="Fishburn 1977">{{Cite journal |last=Fishburn |first=Peter C. |author-link=Peter C. Fishburn |date=1977 |title=Condorcet Social Choice Functions |url=http://epubs.siam.org/doi/10.1137/0133030 |journal=SIAM Journal on Applied Mathematics |language=en |volume=33 |issue=3 |pages=469–489 |doi=10.1137/0133030 |issn=0036-1399 |jstor=2100704 |url-access=subscription}}</ref>

Named after [[Marquis de Condorcet|Nicolas de Condorcet]], it is also called a '''majority winner''', a '''majority-preferred candidate''',<ref>{{Cite journal |last1=Brandl |first1=Florian |last2=Brandt |first2=Felix |last3=Seedig |first3=Hans Georg |date=2016 |title=Consistent Probabilistic Social Choice |url=https://www.econometricsociety.org/doi/10.3982/ECTA13337 |journal=Econometrica |language=en |volume=84 |issue=5 |pages=1839–1880 |arxiv=1503.00694 |doi=10.3982/ECTA13337 |issn=0012-9682}}</ref><ref name=":0">{{Cite journal |last=Sen |first=Amartya |date=2020 |title=Majority decision and Condorcet winners |journal=Social Choice and Welfare |volume=54 |issue=2/3 |pages=211–217 |doi=10.1007/s00355-020-01244-4 |issn=0176-1714 |jstor=45286016}}</ref><ref>{{cite journal |last=Lewyn |first=Michael |title=Two Cheers for Instant Runoff Voting |date=2012 |journal=Phoenix Law Review |volume=6 |pages=117–136 |archive-url=https://digitalcommons.tourolaw.edu/scholarlyworks/776 |archive-date=16 March 2022 |via=''Scholarly Works''. 776 |location=Rochester, NY |language=en |ssrn=2276015 |url=https://tourolaw.elsevierpure.com/en/publications/two-cheers-for-instant-runoff-voting/ |access-date=10 November 2025}}<!-- https://summitlawreview.org/Vol_1_1.pdf --></ref> a '''beats-all''' '''winner''', or '''tournament winner''' (by analogy with [[Round-robin tournament|round-robin tournaments]]). A Condorcet winner may not necessarily always exist in a given electorate: it is possible to have a [[Rock paper scissors|rock, paper, scissors]]-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called [[Voting paradox|Condorcet's voting paradox]],<ref name="Fishburn 1977"/> and is analogous to the counterintuitive [[Intransitive dice|intransitive dice phenomenon]] known in [[probability]]. However, the Smith set, a generalization of the Condorcet criteria that is the smallest set of candidates that are pairwise unbeaten by every candidate outside of it, will always exist.

If voters are arranged on a sole 1-dimensional axis, such as the [[Political spectrum|left-right political spectrum]] for a common example, and always prefer candidates who are more similar to themselves, a majority-rule winner always exists and is the candidate whose ideology is most representative of the electorate, a result known as the [[median voter theorem]].<ref>{{cite journal |last=Black |first=Duncan |author-link=Duncan Black |year=1948 |title=On the Rationale of Group Decision-making |journal=The Journal of Political Economy |volume=56 |pages=23–34 |doi=10.1086/256633 |jstor=1825026 |s2cid=153953456 |number=1}}</ref> However, in real-life political electorates are inherently multidimensional, and the use of a one- or even two-dimensional model of such electorates would be inaccurate.<ref name=":23">{{Cite journal |last1=Alós-Ferrer |first1=Carlos |author-link=Carlos Alós-Ferrer |last2=Granić |first2=Đura-Georg |date=2015-09-01 |title=Political space representations with approval data |url=http://repub.eur.nl/pub/111247 |journal=Electoral Studies |volume=39 |pages=56–71 |doi=10.1016/j.electstud.2015.04.003 |quote=The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space. |hdl-access=free |hdl=1765/111247}}</ref><ref>{{Cite book |last1=Black |first1=Duncan |author-link1=Duncan Black |last2=Newing |first2=R.A. |editor-last1=McLean |editor-first1=Iain S. |editor-link1=:cy:Iain McLean |url=https://books.google.com/books?id=JT-SBgAAQBAJ&q=%22to+be+almost+inevitable%22&pg=PR35 |title=The Theory of Committees and Elections by Duncan Black and Committee Decisions with Complementary Valuation by Duncan Black and R.A. Newing |editor-last2=McMillan |editor-first2=Alistair |editor-last3=Monroe |editor-first3=Burt L. |date=2013-03-09 |publisher=Springer Science & Business Media |isbn=9789401148603 |language=en |quote=For instance, if preferences are distributed spatially, there need only be two or more dimensions to the alternative space for cyclic preferences to be almost inevitable}}</ref> Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1-10% of races.<ref>{{Cite journal |last=Van Deemen |first=Adrian |date=2014-03-01 |title=On the empirical relevance of Condorcet's paradox |journal=Public Choice |language=en |volume=158 |issue=3 |pages=311–330 |doi=10.1007/s11127-013-0133-3 |issn=1573-7101}}</ref> 

Systems that guarantee the election of a Condorcet winners (when one exists) include [[Ranked pairs|Ranked Pairs]], [[Schulze method|Schulze's method]], and the [[Tideman alternative method]]. Methods that do ''not'' guarantee that the Cordorcet winner will be elected, even when one does exist, include [[instant-runoff voting]] (often called [[Ranked-choice voting in the United States|ranked-choice in the United States]]), [[first-past-the-post voting]], and the [[two-round system]]. Most [[Rated voting|rated systems]], like [[score voting]] and [[Highest median voting rules|highest median]], fail the majority winner criterion.

== History ==
Condorcet methods were first studied in detail by [[Ramon Llull]], a Spanish [[philosopher]] and [[Theology|theologian]] in the 13th century, during his investigations into [[church governance]]. Because his manuscript ''Ars Electionis'' was lost soon after his death, his ideas were overlooked for the next 500 years.<ref>{{cite journal |last1=Colomer |first1=Josep M. |title=Ramon Llull: from 'Ars electionis' to social choice theory |journal=Social Choice and Welfare |date=February 2013 |volume=40 |issue=2 |pages=317–328 |doi=10.1007/s00355-011-0598-2 |url=https://link.springer.com/article/10.1007/s00355-011-0598-2|url-access=subscription }}</ref>

The first revolution in [[voting theory]] coincided with the rediscovery of these ideas during the [[Age of Enlightenment]] by [[Marquis de Condorcet|Nicolas de Caritat, Marquis de Condorcet]], a [[mathematician]] and [[Political philosophy|political philosopher]].

== Example ==
Suppose the government comes across a [[Windfall gain|windfall source of funds]] and has three options for what to do with the money: spend it, use it to cut taxes, or use it to pay off the debt. The government holds a vote where it asks citizens which of two options they would prefer and tabulates the results as follows:
{| class="wikitable"
!
!... vs. Spend more
!... vs. Cut taxes
!
|-
! scope="row" |''Pay debt''
|403–305
|496–212
|[[Winning percentage|2–0]] ''{{Tick}}''
|-
! scope="row" |Cut taxes
|522–186
|[[Winning percentage|1–1]]
|
|-
! scope="row" |Spend more
|[[Winning percentage|0–2]]
| colspan="2" |
|}
In this case, the option of paying off the debt is the beats-all winner because repaying debt is more popular than the other two options.

Such a winner will not always exist; in that case, [[tournament solution]]s search for the candidate who is closest to being an undefeated champion. Majority-rule winners can be determined from [[ranking]]s by counting the number of voters who rated each candidate higher than another.

== Desirable properties ==
The Condorcet criterion is related to several other [[voting system criteria]].<!-- TODO: Strategic voting -->
=== Stability (no-weak-spoilers) ===
Condorcet methods are highly resistant to [[spoiler effect]]s. Intuitively, this is because the only way to dislodge a Condorcet winner is by beating them, implying spoilers can exist only if there is no majority-rule winner.

=== Participation ===
One disadvantage of majority-rule methods is they can all theoretically fail the [[participation criterion]] in constructed examples. However, studies suggest this is empirically rare for modern Condorcet methods, like [[ranked pairs]]. One study surveying 306 publicly-available election datasets found no examples of participation failures for methods in the ranked pairs-[[Minimax Condorcet|minimax]] family.<ref>{{cite conference |last1=Mohsin |first1=F. |last2=Han |first2=Q. |last3=Ruan |first3=S. |last4=Chen |first4=P. Y. |last5=Rossi |first5=F. |last6=Xia |first6=L. |date=May 2023 |title=Computational Complexity of Verifying the Group No-show Paradox |conference=Proceedings of the 2023 International Conference on Autonomous Agents and Multiagent Systems |pages=2877–2879}}</ref>

=== Majoritarian criteria ===
The Condorcet criterion implies the [[majority winner criterion|majority criterion]] since a candidate ranked first by a majority is clearly ranked above every other candidate by a majority.

When a Condorcet winner exists, this candidate is also part of the smallest [[mutual majority]] set, so any Condorcet method passes the [[mutual majority criterion]] and [[Condorcet loser]] in elections where a Condorcet winner exist. However, this need not hold in full generality: for instance, the [[minimax Condorcet method]] fails the [[Condorcet loser]] and [[mutual majority]] criteria.

=== Stronger criteria ===
The [[top cycle|Smith criterion]] guarantees an even stronger kind of majority rule. It says that if there is no majority-rule winner, the winner must be in the [[Smith set|top cycle]], which includes all the candidates who can beat every other candidate, either directly ''or'' [[Tournament (graph theory)|indirectly]]. Most, but not all, Condorcet systems satisfy the top-cycle criterion.

== By method ==

=== List ===
{{main|Condorcet method}}

==== Pass ====
Most sensible [[Tournament solution|tournament solutions]] satisfy the Condorcet criterion. Other methods satisfying the criterion include:

* [[Black electoral system|Black]]
* [[Kemeny-Young method|Kemeny-Young]]
* [[Dodgson's method]]
* [[Minimax Condorcet|Minimax]]
* [[Baldwin's method]]
* [[Ranked pairs]]
* [[Schulze method|Schulze]]
* [[Baldwin method|Total Vote Runoff]]
* [[Tideman's alternative method]]

See {{Category link|Condorcet methods}} for more.

==== Fail ====
The following [[Electoral system|voting systems]] do '''not''' satisfy the Condorcet criterion:

* [[Plurality voting]]
* [[Instant-runoff voting]]
* [[Borda count]]
* [[Approval Voting]]
* [[Coombs rule|Coombs' rule]]<ref>{{Cite journal |last1=Felsenthal |first1=Dan |last2=Tideman |first2=Nicolaus |date=2013 |title=Varieties of failure of monotonicity and participation under five voting methods |url=https://ideas.repec.org/a/kap/theord/v75y2013i1p59-77.html |journal=Theory and Decision |language=en |volume=75 |issue=1 |pages=59–77|doi=10.1007/s11238-012-9306-7 }}</ref>
* [[Bucklin voting]] (and the closely related [[Highest median voting rules|median voting]])
* [[Score voting|Score Voting]]

=== Examples of failures ===

==== Plurality voting ====
{{Main|Plurality voting system}}

With plurality voting, the full set of voter preferences is not recorded on the ballot and so cannot be deduced therefrom (e.g. following a real election). Plurality fails the Condorcet criterion because of [[Vote-splitting|vote-splitting effects]]. 

Consider an election in which 30% of the voters prefer candidate A to candidate B to candidate C and vote for A, 30% of the voters prefer C to A to B and vote for C, and 40% of the voters prefer B to A to C and vote for B. Candidate B would win (with 40% of the vote) even though A would be the Condorcet winner, beating B 60% to 40%, and C 70% to 30%.

A real-life example may be the [[2000 United States presidential election in Florida|2000 election in Florida]], where most voters preferred [[Al Gore]] to [[George W. Bush|George Bush]], but Bush won as a result of spoiler candidate [[Ralph Nader]].
==== Instant-runoff voting ====
{{Main|Instant-runoff voting}}

In instant-runoff voting (IRV) voters rank candidates from first to last. The last-place candidate (the one with the fewest first-place votes) is eliminated; the votes are then reassigned to the non-eliminated candidate the voter would have chosen had the candidate not been present.

Instant-runoff does not comply with the Condorcet criterion, i.e. it is possible for it to elect a candidate that could lose in a head to head contest against another candidate in the election. For example, the following vote count of preferences with three candidates {A, B, C}:

* A > B > C: 35
* C > B > A: 34
* B > C > A: 31

In this case, B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34, so B is preferred to both A and C. B must then win according to the Condorcet criterion. Under IRV, B is ranked first by the fewest voters and is eliminated, and then C wins with the transferred votes from B.

Note that 65 voters, a majority, prefer either candidate B or C over A; since IRV passes the [[mutual majority criterion]], it guarantees one of B and C must win. If candidate A, an [[Independence of irrelevant alternatives|irrelevant alternative]] under IRV, was not running, a majority of voters would consider B their 1st choice, and IRV's mutual majority compliance would thus ensure B wins.

One real-life example of instant runoff failing the Condorcet criteria was the [[2009 Burlington mayoral election#Analysis|2009 mayoral election of Burlington, Vermont]].

==== Borda count ====
{{Main|Borda count}}
Borda count is a voting system in which voters rank the candidates in an order of preference. Points are given for the position of a candidate in a voter's rank order. The candidate with the most points wins.

The Borda count does not comply with the Condorcet criterion in the following case. Consider an election consisting of five voters and three alternatives (candidates A, B, and C), with the following votes:

* A > B > C: 3
* B > C > A: 2

In this election, the Borda count awards 2 points for 1st choice, 1 point for second and 0 points for third. Thus, the total points received by each alternative is as follows:

* A: (3 * 2) + (0 * 1) + (2 * 0) = 6 + 0 + 0 = 6
* B: (2 * 2) + (3 * 1) + (0 * 0) = 4 + 3 + 0 = 7
* C: (0 * 3) + (2 * 1) + (3 * 0) = 0 + 2 + 0 = 2

With 7 points, B is the Borda count winner; however, the fact that A is preferred by three of the five voters to all other alternatives makes it a beats-all champion, and the required winner to satisfy the Condorcet criterion.

==== Bucklin/Median ====
{{Main|Highest median voting rules|Bucklin voting}}
Highest medians is a system in which the voter gives all candidates a rating out of a predetermined set (e.g. {"excellent", "good", "fair", "poor"}). The winner of the election would be the candidate with the best median rating. Consider an election with three candidates A, B, C.

* 35 voters rate candidate A "excellent", B "fair", and C "poor",
* 34 voters rate candidate C "excellent", B "fair", and A "poor", and
* 31 voters rate candidate B "excellent", C "good", and A "poor".

B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34. Hence, B is the beats-all champion. But B only gets the median rating "fair", while C has the median rating "good"; as a result, C is chosen as the winner by highest medians.

==== Approval voting ====
Main article: [[Approval voting]]

Approval voting is a system in which the voter can approve of (or vote for) any number of candidates on a ballot. Approval voting fails the Condorcet criterion

Consider an election in which 70% of the voters prefer candidate A to candidate B to candidate C, while 30% of the voters prefer C to B to A. If every voter votes for their top two favorites, Candidate B would win (with 100% approval) even though A would be the Condorcet winner.

==== Score voting ====
{{Main|Score voting}}
Score voting is a system in which the voter gives all candidates a score on a predetermined scale (e.g. from 0 to 5). The winner of the election is the candidate with the highest total score. Score voting fails the Condorcet criterion. For example:
{| class="wikitable sortable"
!{{Diagonal split header|Votes|Candidates}}
!A
!B
!C
|-
|45
|5/5
|1/5
|0/5
|-
|40
|0/5
|1/5
|5/5
|-
|15
|2/5
|5/5
|4/5
|-
!Average
!2.55
!1.6
!2.6
|}
Here, C is declared winner, even though a majority of voters would prefer B; this is because the supporters of C are much more enthusiastic about their favorite candidate than the supporters of B. The same example also shows that [[STAR voting |adding a runoff]] does not always cause score to comply with the criterion (as the Condorcet winner B is not in the top-two according to score).

==Further reading==

* {{cite book |last=Black |first=Duncan |author-link=Duncan Black |url=https://archive.org/details/theoryofcommitte0000blac |title=The Theory of Committees and Elections |publisher=[[Cambridge University Press]] |year=1958 |url-access=registration}}
* {{cite book |last=Farquharson |first=Robin |author-link=Robin Farquharson |title=Theory of Voting |publisher=Blackwell |year=1969 |isbn=0-631-12460-8 |location=Oxford}}
* {{cite book |last=Sen |first=Amartya Kumar |author-link=Amartya Sen |title=Collective Choice and Social Welfare |publisher=Holden-Day |year=1970 |isbn=978-0-8162-7765-0}}

== See also ==
{{Portal|Politics|Mathematics
}}

* [[Condorcet loser criterion]]
* [[Condorcet method]]
* [[Multiwinner voting]] - contains information on some multiwinner variants of the Condorcet criterion.

==References==
<references responsive="1"></references>
{{voting systems}}
<references group="note" />
[[Category:Electoral system criteria]]