View source for Condorcet paradox

{{About|results that can arise in a collective choice among three or more alternatives|the contention that an individual's vote will probably not affect the outcome|Paradox of voting}}
{{Short description|Self-contradiction of majority rule}}
{{Electoral systems}}

In [[social choice theory]], '''Condorcet's voting paradox''' is a fundamental discovery by the [[Marquis de Condorcet]] that [[majority rule]] is inherently [[contradiction|self-contradictory]]. The result implies that it is logically impossible for any voting system to guarantee that a winner will have support from a majority of voters: for example there can be rock-paper-scissors scenario where a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called '''Condorcet cycles''' or '''cyclic ties'''.

In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, who is preferred by more than half of all voters. Thus, any attempt to ground social decision-making in [[majoritarianism]] must accept such self-contradictions (commonly called [[spoiler effect]]s). Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called [[Condorcet method]]s.

Condorcet's paradox is a special case of [[Arrow's impossibility theorem|Arrow's paradox]], which shows that ''any'' kind of social decision-making process is either self-contradictory, a [[dictatorship mechanism|dictatorship]], or incorporates information about the strength of different voters' preferences (e.g. [[cardinal utility]] or [[rated voting]]).

== History ==
Condorcet's paradox was first discovered by [[Catalonia|Catalan]] [[philosopher]] and [[Theology|theologian]] [[Ramon Llull]] in the 13th century, during his investigations into [[church governance]], but his work was lost until the 21st century. The mathematician and political philosopher [[Marquis de Condorcet]] rediscovered the paradox in the late 18th century.<ref>{{cite book |author=Marquis de Condorcet |author-link=Marquis de Condorcet |url=http://gallica.bnf.fr/ark:/12148/bpt6k417181 |title=Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix |year=1785 |language=fr |format=PNG |access-date=2008-03-10}}</ref><ref>{{Cite book |last1=Condorcet |first1=Jean-Antoine-Nicolas de Caritat |title=The political theory of Condorcet |last2=Sommerlad |first2=Fiona |last3=McLean |first3=Iain |date=1989-01-01 |publisher=University of Oxford, Faculty of Social Studies |location=Oxford |pages=69–80, 152–166 |oclc=20408445 |quote=Clearly, if anyone's vote was self-contradictory (having cyclic preferences), it would have to be discounted, and we should therefore establish a form of voting which makes such absurdities impossible}}</ref><ref>{{Cite journal |last=Gehrlein |first=William V. |year=2002 |title=Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* |journal=Theory and Decision |volume=52 |issue=2 |pages=171–199 |doi=10.1023/A:1015551010381 |issn=0040-5833 |s2cid=118143928 |quote=Here, Condorcet notes that we have a 'contradictory system' that represents what has come to be known as Condorcet's Paradox.}}</ref>

Condorcet's discovery means he arguably identified the key result of [[Arrow's impossibility theorem]], albeit under stronger conditions than required by Arrow: Condorcet cycles create situations where any ranked voting system [[May's theorem|that respects majorities]] must have a [[spoiler effect]].

==Example==
Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows:

{| class="wikitable" style="text-align: center;"
! Voter !! First preference !! Second preference !! Third preference
|- 
! Voter 1 
| A || B || C
|- 
! Voter 2 
| B || C || A
|- 
! Voter 3 
| C || A || B
|}

[[File:Voting Paradox example.png|alt=3 blue dots in a triangle. 3 red dots in a triangle, connected by arrows that point counterclockwise.|thumb|Voters (blue) and candidates (red) plotted in a 2-dimensional preference space. Each voter prefers a closer candidate over a farther. Arrows show the order in which voters prefer the candidates.]]
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B.  However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. Thus the society's preferences show cycling: A is preferred over B which is preferred over C which is preferred over A.

As a result, any attempt to appeal to the principle of [[majority rule]] will lead to logical [[Contradiction|self-contradiction]]. Regardless of which alternative we select, we can find another alternative that would be preferred by most voters.

=== Practical scenario ===
The voters in Cactus County prefer the incumbent [[county executive]] '''Alex''' of the Farmers' Party over rival '''Beatrice''' of the Solar Panel Party by about a 2-to-1 margin. This year a third candidate, '''Charlie''', is running as an independent. Charlie is a wealthy and outspoken businessman, of whom the voters hold polarized views.

The voters divide into three groups:

* Group 1 revere Charlie for saving the high school football team. They rank Charlie first, and then Alex above Beatrice as usual ('''CAB''').
* Group 2 despise Charlie for his sharp business practices. They rank Charlie ''last'', and then Alex above Beatrice as usual ('''ABC''').
* Group 3 are Beatrice's core supporters. They want the Farmers' Party out of office in favor of the Solar Panel Party, and regard Charlie's candidacy as a sideshow. They rank Beatrice first and Alex last as usual, and Charlie second by default ('''BCA''').

Therefore a majority of voters prefer Alex to Beatrice (A > B), as they always have. A majority of voters are either Beatrice-lovers or Charlie-haters, so prefer Beatrice to Charlie (B > C). And a majority of voters are either Charlie-lovers or Alex-haters, so prefer Charlie to Alex (C > A). Combining the three preferences gives us A > B > C > A, a Condorcet cycle.

== Likelihood ==
It is possible to estimate the probability of the paradox by extrapolating from real election data, or using mathematical models of voter behavior, though the results depend strongly on which model is used.

=== Impartial culture model ===
We can calculate the probability of seeing the paradox for the special case where voter preferences are uniformly distributed among the candidates. (This is the "[[impartial culture]]" model, which is known to be a "worst-case scenario"<ref>{{Cite journal|last1=Tsetlin|first1=Ilia|last2=Regenwetter|first2=Michel|last3=Grofman|first3=Bernard|date=2003-12-01|title=The impartial culture maximizes the probability of majority cycles|journal=Social Choice and Welfare|volume=21|issue=3|pages=387–398|doi=10.1007/s00355-003-0269-z|s2cid=15488300|issn=0176-1714|quote=it is widely acknowledged that the impartial culture is unrealistic ... the impartial culture is the worst case scenario}}</ref><ref name=":1">{{Cite book|title=Voting paradoxes and group coherence : the condorcet efficiency of voting rules|last1=Gehrlein|first1=William V.|last2=Lepelley|first2=Dominique|date=2011|publisher=Springer|isbn=9783642031076|location=Berlin|doi=10.1007/978-3-642-03107-6|oclc=695387286|quote=most election results do not correspond to anything like any of DC, IC, IAC or MC ... empirical studies ... indicate that some of the most common paradoxes are relatively unlikely to be observed in actual elections. ... it is easily concluded that Condorcet’s Paradox should very rarely be observed in any real elections on a small number of candidates with large electorates, as long as voters’ preferences reflect any reasonable degree of group mutual coherence}}</ref>{{Rp|40}}<ref name=":0">{{Cite journal|last=Van Deemen|first=Adrian|date=2014|title=On the empirical relevance of Condorcet's paradox|journal=Public Choice|language=en|volume=158|issue=3–4|pages=311–330|doi=10.1007/s11127-013-0133-3|s2cid=154862595|issn=0048-5829|quote=small departures of the impartial culture assumption may lead to large changes in the probability of the paradox. It may lead to huge declines or, just the opposite, to huge increases.}}</ref>{{Rp|320}}<ref>{{Cite journal|last=May|first=Robert M.|date=1971|title=Some mathematical remarks on the paradox of voting|journal=Behavioral Science|volume=16|issue=2|pages=143–151|doi=10.1002/bs.3830160204|issn=0005-7940}}</ref>—most models show substantially lower probabilities of Condorcet cycles.)

For <math> n </math> voters providing a preference list of three candidates A, B, C, we write <math> X_n </math> (resp. <math> Y_n </math>, <math> Z_n </math>) the random variable equal to the number of voters who placed A in front of B (respectively B in front of C, C in front of A). The sought probability is <math> p_n = 2P (X_n> n / 2, Y_n> n / 2, Z_n> n / 2) </math> (we double because there is also the symmetric case A> C> B> A). We show that, for odd <math> n </math>, <math> p_n = 3q_n-1/2 </math> where <math> q_n = P (X_n> n / 2, Y_n> n / 2) </math> which makes one need to know only the joint distribution of <math> X_n </math> and <math> Y_n </math>.

If we put <math> p_{n, i, j} = P (X_n = i, Y_n = j) </math>, we show the relation which makes it possible to compute this distribution by recurrence: <math> p_ { n + 1, i, j} = {1 \over 6} p_ {n, i, j} + {1 \over 3} p_ {n, i-1, j} + {1 \over 3} p_ {n, i, j-1} + {1 \over 6} p_ {n, i-1, j-1} </math>.

The following results are then obtained:
{| class="wikitable"
!<math>n</math>
!3
!101
!201
!301
!401
!501
!601
|-
|<math>p_n</math>
|5.556%
|8.690%
|8.732%
|8.746%
|8.753%
|8.757%
|8.760%
|}
The sequence seems to be tending towards a finite limit.

Using the [[central limit theorem]], we show that <math> q_n </math> tends to <math>q = \frac{1}{4} P\left(|T| > \frac{\sqrt{2}}{4}\right),</math> where <math> T </math> is a variable following a [[Cauchy distribution]], which gives <math>q=\dfrac{1}{2\pi }\int_{\sqrt{2}/4}^{+\infty }\frac{dt}{1+t^{2}}=\dfrac{
\arctan 2\sqrt{2}}{2\pi }=\dfrac{\arccos \frac{1}{3}}{2\pi }</math> (constant [[oeis: A289505|quoted in the OEIS]]).

The asymptotic probability of encountering the Condorcet paradox is therefore <math>{{3\arccos{1\over3}}\over{2\pi}}-{1\over2}={\arcsin{\sqrt 6\over 9}\over \pi}</math> which gives the value 8.77%.<ref>{{Cite journal|last=Guilbaud|first=Georges-Théodule|date=2012|title=Les théories de l'intérêt général et le problème logique de l'agrégation|journal=Revue économique|volume=63|issue=4|pages=659–720|doi=10.3917/reco.634.0659|issn=0035-2764|doi-access=free}}</ref><ref name=":2">{{Cite journal|last=Gehrlein|first=William V.|date=2002-03-01|title=Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*|url=https://doi.org/10.1023/A:1015551010381|journal=Theory and Decision|language=en|volume=52|issue=2|pages=171–199|doi=10.1023/A:1015551010381|s2cid=118143928|issn=1573-7187|quote=to have a PMRW with probability approaching 15/16 = 0.9375 with IAC and UC, and approaching 109/120 = 0.9083 for MC. … these cases represent situations in which the probability that a PMRW exists would tend to be at a minimum … intended to give us some idea of the lower bound on the likelihood that a PMRW exists.}}</ref>

Some results for the case of more than three candidates have been calculated<ref>{{Cite journal|last=Gehrlein|first=William V.|date=1997|title=Condorcet's paradox and the Condorcet efficiency of voting rules|url=https://www.researchgate.net/publication/257651659|journal=Mathematica Japonica|volume=45|pages=173–199}}</ref> and simulated.<ref name=":4">{{Cite journal |last=Merrill |first=Samuel |date=1984 |title=A Comparison of Efficiency of Multicandidate Electoral Systems |url=https://www.jstor.org/stable/2110786 |journal=American Journal of Political Science |volume=28 |issue=1 |pages=23–48 |doi=10.2307/2110786 |jstor=2110786 |issn=0092-5853}}</ref>  The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates:<ref name=":4" />{{Rp|page=28|quote=% Condorcet winners 100.0 91.6 83.4 75.8 64.3 52.5|location=}}
{| class="wikitable"
|+
!3
!4
!5
!7
!10
|-
|8.4%
|16.6%
|24.2%
|35.7%
|47.5%
|}
The likelihood of a Condorcet cycle for related models approach these values for three-candidate elections with large electorates:<ref name=":2" />
* [[Impartial culture#Impartial Anonymous Culture (IAC)|Impartial anonymous culture]] (IAC): 6.25%
* Uniform culture (UC): 6.25%
* Maximal culture condition (MC): 9.17%

All of these models are unrealistic, but can be investigated to establish an upper bound on the likelihood of a cycle.<ref name=":2" />

=== Group coherence models ===
When modeled with more realistic voter preferences, Condorcet paradoxes in elections with a small number of candidates and a large number of voters become very rare.<ref name=":1" />{{Rp|78}}

=== Spatial model ===
A study of three-candidate elections analyzed 12 different models of voter behavior, and found the [[spatial model of voting]] to be the most accurate to real-world [[Ranked voting|ranked-ballot]] election data.  Analyzing this spatial model, they found the likelihood of a cycle to decrease to zero as the number of voters increases, with likelihoods of 5% for 100 voters, 0.5% for 1000 voters, and 0.06% for 10,000 voters.<ref name=":3">{{Citation|last1=Tideman|first1=T. Nicolaus|title=Modeling the Outcomes of Vote-Casting in Actual Elections|date=2012|url=http://link.springer.com/10.1007/978-3-642-20441-8_9|work=Electoral Systems|editor-last=Felsenthal|editor-first=Dan S.|at=Table 9.6 Shares of strict pairwise majority rule winners (SPMRWs) in observed and simulated elections|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|doi=10.1007/978-3-642-20441-8_9|isbn=978-3-642-20440-1|quote=Mean number of voters: 1000 … Spatial model: 99.47% [0.5% cycle likelihood] … 716.4 [ERS data] … Observed elections: 99.32% … 1,566.7 [ANES data] … 99.56%|access-date=2021-11-12|last2=Plassmann|first2=Florenz|editor2-last=Machover|editor2-first=Moshé}}</ref>

Another spatial model found likelihoods of 2% or less in all simulations of 201 voters and 5 candidates, whether two or four-dimensional, with or without correlation between dimensions, and with two different dispersions of candidates.<ref name=":4" />{{Rp|page=31|location=|quote=% Condorcet winners 99+ 99 99+ 99+ 98 98 98 99}}

=== Empirical studies ===
Many attempts have been made at finding empirical examples of the paradox.<ref>{{Cite journal|last=Kurrild-Klitgaard|first=Peter|date=2014|title=Empirical social choice: An introduction|journal=Public Choice|language=en|volume=158|issue=3–4|pages=297–310|doi=10.1007/s11127-014-0164-4|s2cid=148982833|issn=0048-5829}}</ref>  Empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available.

While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.<ref>{{Cite journal |last=Kurrild-Klitgaard |first=Peter |date=2001 |title=An empirical example of the Condorcet paradox of voting in a large electorate |journal=Public Choice |language=en |volume=107 |pages=135–145 |doi=10.1023/A:1010304729545 |issn=0048-5829 |s2cid=152300013}}</ref>

A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4%<ref name=":0" />{{Rp|325}} (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).<ref name=":1" />{{Rp|47}}

An analysis of 883 three-candidate elections extracted from 84 real-world ranked-ballot elections of the [[Electoral Reform Society]] found a Condorcet cycle likelihood of 0.7%.  These derived elections had between 350 and 1,957 voters.<ref name=":3" /> A similar analysis of data from the 1970–2004 [[American National Election Studies]] [[thermometer scale]] surveys found a Condorcet cycle likelihood of 0.4%. These derived elections had between 759 and 2,521 "voters".<ref name=":3" />

Andrew Myers, who operates the [[online poll|Condorcet Internet Voting Service]], analyzed 10,354 nonpolitical CIVS elections and found cycles in 17% of elections with at least 10 votes, with the figure dropping to 2.1% for elections with at least 100 votes, and 1.2% for ≥300 votes.<ref name="CIVS">{{cite conference |last=Myers |first=A. C. |author-link= |date=March 2024 |title=The Frequency of Condorcet Winners in Real Non-Political Elections |url=https://www.cs.cornell.edu/andru/papers/civs24/ |conference=61st Public Choice Society Conference |pages=5 |quote=83.1% … 97.9% … 98.8% … Figure 2: Frequency of CWs and weak CWs with an increasing number of voters}}</ref>

=== Real world instances ===
A database of 189 ranked United States elections from 2004 to 2022 contained only one Condorcet cycle: the [[2021 Minneapolis City Council election#Ward 2|2021 Minneapolis City Council election in Ward 2]], with a narrow circular tie between candidates of the [[Green Party of Minnesota|Green Party]] ([[Cam Gordon]]), the [[Minnesota Democratic–Farmer–Labor Party]], (Yusra Arab) and an independent [[Democratic socialism|democratic socialist]] ([[Robin Wonsley]]).<ref name="GSM2023">{{cite arXiv | last1=Graham-Squire | first1=Adam | last2=McCune | first2=David | title=An Examination of Ranked Choice Voting in the United States, 2004-2022 |eprint=2301.12075v2 | date=2023-01-28 | class=econ.GN}}</ref> Voters' preferences were non-transitive: Arab was preferred over Gordon, Gordon over Wonsley, and Wonsley over Arab, creating a cyclical pattern with no clear winner. Additionally, the election exhibited a [[Negative responsiveness |downward monotonicity]] paradox, as well as a paradox akin to [[Simpson’s paradox]].

==Implications==
[[File:Mexican Standoff.jpg|thumb|Three men portraying a [[Mexican standoff]]. Just as there is no winner in a Mexican standoff with certain combinations of gun-pointings, there is sometimes no [[Condorcet winner|majority-preferred winner]] in a ranked-ballot election.]]When a [[Condorcet method]] is used to determine an election, the voting paradox of cyclical societal preferences implies that the election has no [[Condorcet winner]]: no candidate who can win a one-on-one election against each other candidate. There will still be a smallest group of candidates, known as the [[Smith set]], such that each candidate in the group can win a one-on-one election against each of the candidates outside the group. The several variants of the Condorcet method differ on how they [[Condorcet method#Circular ambiguities|resolve such ambiguities]] when they arise to determine a winner.<ref>{{Cite book|chapter-url=http://www.opentextbookstore.com/mathinsociety/|title=Math in society|last=Lippman|first=David|year=2014|isbn=978-1479276530|chapter=Voting Theory|publisher=CreateSpace Independent Publishing Platform |oclc=913874268|quote=There are many Condorcet methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist.}}</ref> The Condorcet methods which always elect someone from the Smith set when there is no Condorcet winner are known as [[Smith criterion|Smith-efficient]]. Note that using only rankings, there is no fair and deterministic resolution to the trivial example given earlier because each candidate is in an exactly symmetrical situation.

Situations having the voting paradox can cause voting mechanisms to violate the axiom of [[independence of irrelevant alternatives]]—the choice of winner by a voting mechanism could be influenced by whether or not a losing candidate is available to be voted for.

=== Two-stage voting processes ===
One important implication of the possible existence of the voting paradox in a practical situation is that in a paired voting process like those of standard [[parliamentary procedure]], the eventual winner will depend on the way the majority votes are ordered. For example, say a popular bill is set to pass, before some other group offers an amendment; this amendment passes by majority vote. This may result in a majority of a [[legislature]] rejecting the bill as a whole, thus creating a paradox (where a popular amendment to a popular bill has made it unpopular). This logical inconsistency is the origin of the [[poison pill amendment]], which deliberately engineers a false Condorcet cycle to kill a bill. Likewise, the order of votes in a legislature can be manipulated by the person arranging them to ensure their preferred outcome wins.

Despite frequent objections by [[social choice theory|social choice theorists]] about the logically incoherent results of such procedures, and the existence of better alternatives for choosing between multiple versions of a bill, the procedure of pairwise majority-rule is widely-used and is codified into the [[by-law]]s or parliamentary procedures of almost every kind of [[deliberative assembly]].

=== Spoiler effects ===
Condorcet paradoxes imply that [[Majority rule|majoritarian methods]] fail independence of irrelevant alternatives. Label the three candidates in a race [[Rock paper scissors|''Rock'', ''Paper'', and ''Scissors'']]. In one-on-one races, Rock loses to Paper, Paper loses to Scissors, and Scissors loses to Rock. 

[[Without loss of generality]], say that Rock wins the election with a certain method. Then, Scissors is a spoiler candidate for Paper; if Scissors were to drop out, Paper would win the only one-on-one race (Paper defeats Rock). The same reasoning applies regardless of the winner.

This example also shows why Condorcet elections are rarely (if ever) spoiled; spoilers can ''only'' happen when there is no Condorcet winner. Condorcet cycles are rare in large elections,<ref name=":53">{{Cite journal |last=Gehrlein |first=William V. |date=2002-03-01 |title=Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* |url=https://doi.org/10.1023/A:1015551010381 |journal=Theory and Decision |language=en |volume=52 |issue=2 |pages=171–199 |doi=10.1023/A:1015551010381 |issn=1573-7187}}</ref><ref name=":63">{{Cite journal |last=Van Deemen |first=Adrian |date=2014-03-01 |title=On the empirical relevance of Condorcet's paradox |url=https://doi.org/10.1007/s11127-013-0133-3 |journal=Public Choice |language=en |volume=158 |issue=3 |pages=311–330 |doi=10.1007/s11127-013-0133-3 |issn=1573-7101}}</ref> and the [[median voter theorem]] shows cycles are impossible whenever candidates are arrayed on a [[Political spectrum|left-right spectrum]].

==See also==
* [[Arrow's impossibility theorem]]
* [[Discursive dilemma]]
* [[Spoiler effect]]
* [[Independence of irrelevant alternatives]]
* [[Nakamura number]]
* [[Quadratic voting]]
* [[Rock paper scissors]]
* [[Smith set]]

==References==
{{Reflist}}

==Further reading==

* {{cite journal|last1=Garman|first1=M. B.|last2=Kamien|first2=M. I.|year=1968|title=The paradox of voting: Probability calculations|journal=Behavioral Science|volume=13|issue=4|pages=306–316|doi=10.1002/bs.3830130405|pmid=5663897}}
* {{cite journal|last1=Niemi|first1=R. G.|last2=Weisberg|first2=H.|year=1968|title=A mathematical solution for the probability of the paradox of voting|journal=Behavioral Science|volume=13|issue=4|pages=317–323|doi=10.1002/bs.3830130406|pmid=5663898}}
* {{cite journal|last1=Niemi|first1=R. G.|last2=Wright|first2=J. R.|year=1987|title=Voting cycles and the structure of individual preferences|journal=Social Choice and Welfare|volume=4|issue=3|pages=173–183|jstor=41105865|doi=10.1007/BF00433943|s2cid=145654171}}

==External links==
{{subject bar|auto=y|d=y|Politics}}
{{Decision theory paradoxes}}

[[Category:Decision-making paradoxes]]
[[Category:Eponymous paradoxes]]
[[Category:Voting theory]]