Editing Condorcet paradox (section)

=== 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" />