Editing Condorcet paradox (section)

== 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]].

A second instance of a Condorcet cycle was found in the 2022 District 4 School Director election in Oakland, CA. Manigo was preferred to Hutchinson, Hutchinson to Resnick, and Resnick to Manigo. Like in Minneapolis, the margins were quite narrow: for instance, 11370 voters preferred Manigo to Hutchinson while 11322 preferred Hutchinson to Manigo.<ref name="g816">{{cite arXiv | last=McCune | first=David | title=Ranked Choice Bedlam in a 2022 Oakland School Director Election | date=2023-03-10 | eprint=2303.05985 | class=econ.GN}}</ref>

Another instance of a Condorcet cycle was with the seat of [[Results of the 2014 Victorian state election (Legislative Assembly)#Prahran|Prahran in the 2014 Victorian state election]], with a narrow circular tie between the [[Australian Greens|Greens]], [[Liberal Party of Australia|Liberal]], and [[Australian Labor Party|Labor]] candidates. The Greens candidate, who was initially third on primary votes, defeated the Liberal candidate by less than 300 votes. However, if the contest had been between Labor and Liberal, the Liberal candidate would have won by 25 votes. While a Greens vs Labor count was not conducted, Liberal preferences tend to flow more towards Labor than Greens in other cases ([[Electoral results for the district of Richmond (Victoria)|including in the seat of Richmond in the same election]]), meaning that Labor would have very likely been preferred to the Greens. This means there was a circular pattern, with the Greens preferred over Liberal, who were preferred over Labor, who were preferred over the Greens.