In social choice theory, a Condorcet paradox (or voting paradox) is a situation where majority rule behaves in a way that is self-contradictory. In such a situation, every possible choice is rejected by the electorate in favor of another, because there is always some other outcome that a majority of voters consider to be better.

## History

Condorcet's paradox was first discovered by Spanish philosopher and 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.[1][2][3]

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 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:

Voter First preference Second preference Third preference
Voter 1 A B C
Voter 2 B C A
Voter 3 C A B

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 majority rule will cause self-contradictory behavior. Regardless of which alternative we select, we can find another alternative that would be preferred by most voters.

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"[4][5]: 40 [6]: 320 [7]—most models show substantially lower probabilities of Condorcet cycles.)

For ${\displaystyle n}$ voters providing a preference list of three candidates A, B, C, we write ${\displaystyle X_{n}}$ (resp. ${\displaystyle Y_{n}}$, ${\displaystyle Z_{n}}$) 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 ${\displaystyle p_{n}=2P(X_{n}>n/2,Y_{n}>n/2,Z_{n}>n/2)}$ (we double because there is also the symmetric case A> C> B> A). We show that, for odd ${\displaystyle n}$, ${\displaystyle p_{n}=3q_{n}-1/2}$ where ${\displaystyle q_{n}=P(X_{n}>n/2,Y_{n}>n/2)}$ which makes one need to know only the joint distribution of ${\displaystyle X_{n}}$ and ${\displaystyle Y_{n}}$.

If we put ${\displaystyle p_{n,i,j}=P(X_{n}=i,Y_{n}=j)}$, we show the relation which makes it possible to compute this distribution by recurrence: ${\displaystyle p_{n+1,i,j}={1 \over 6}p_{n,i,j}+{1 \over 3}p_{n,i-1,j}+{1 \over 3}p_{n,i,j-1}+{1 \over 6}p_{n,i-1,j-1}}$.

The following results are then obtained:

${\displaystyle n}$ 3 101 201 301 401 501 601
${\displaystyle p_{n}}$ 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 ${\displaystyle q_{n}}$ tends to ${\displaystyle q={\frac {1}{4}}P\left(|T|>{\frac {\sqrt {2}}{4}}\right),}$ where ${\displaystyle T}$ is a variable following a Cauchy distribution, which gives ${\displaystyle q={\dfrac {1}{2\pi }}\int _{{\sqrt {2}}/4}^{+\infty }{\frac {dt}{1+t^{2}}}={\dfrac {\arctan 2{\sqrt {2}}}{2\pi }}={\dfrac {\arccos {\frac {1}{3}}}{2\pi }}}$ (constant quoted in the OEIS).

The asymptotic probability of encountering the Condorcet paradox is therefore ${\displaystyle {{3\arccos {1 \over 3}} \over {2\pi }}-{1 \over 2}={\arcsin {{\sqrt {6}} \over 9} \over \pi }}$ which gives the value 8.77%.[8][9]

Some results for the case of more than three candidates have been calculated[10] and simulated.[11] The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates:[11]:

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:[9]

All of these models are unrealistic, and are investigated to establish an upper bound on the likelihood of a cycle.[9]

### 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.[5]: 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-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.[12]

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.[11]:

### Empirical studies

Many attempts have been made at finding empirical examples of the paradox.[13] 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.[14]

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%[6]: 325  (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).[5]: 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. 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".[12]

A database of 189 ranked United States election from 2004 to 2022 contained only one Condorcet cycle: the 2021 Minneapolis Ward 2 city council election.[15] While this indicates a very low rate of Condorcet cycles (0.5%), it's possible that some of the effect is due to general two-party domination.

Andrew Myers, who operates the 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.[16]

## Implications

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 resolve such ambiguities when they arise to determine a winner.[17] The Condorcet methods which always elect someone from the Smith set when there is no Condorcet winner are known as 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 two-stage voting process, the eventual winner may depend on the way the two stages are structured. For example, suppose the winner of A versus B in the open primary contest for one party's leadership will then face the second party's leader, C, in the general election. In the earlier example, A would defeat B for the first party's nomination, and then would lose to C in the general election. But if B were in the second party instead of the first, B would defeat C for that party's nomination, and then would lose to A in the general election. Thus the structure of the two stages makes a difference for whether A or C is the ultimate winner.

Likewise, the structure of a sequence of votes in a legislature can be manipulated by the person arranging the votes, to ensure a preferred outcome.

### Spoiler effects

Condorcet paradoxes imply majoritarian methods fail independence of irrelevant alternatives. Label the three candidates in a race Rock, Paper, and Scissors. In a one-on-one race, Rock loses to Paper, Paper to Scissors, etc.

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,[18][19] and the median voter theorem shows cycles are impossible whenever candidates are arrayed on a left-right spectrum.

## References

1. ^ Marquis de Condorcet (1785). Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (PNG) (in French). Retrieved 2008-03-10.
2. ^ Condorcet, Jean-Antoine-Nicolas de Caritat; Sommerlad, Fiona; McLean, Iain (1989-01-01). The political theory of Condorcet. Oxford: University of Oxford, Faculty of Social Studies. pp. 69–80, 152–166. OCLC 20408445. 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
3. ^ Gehrlein, William V. (2002). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 0040-5833. S2CID 118143928. Here, Condorcet notes that we have a 'contradictory system' that represents what has come to be known as Condorcet's Paradox.
4. ^ Tsetlin, Ilia; Regenwetter, Michel; Grofman, Bernard (2003-12-01). "The impartial culture maximizes the probability of majority cycles". Social Choice and Welfare. 21 (3): 387–398. doi:10.1007/s00355-003-0269-z. ISSN 0176-1714. S2CID 15488300. it is widely acknowledged that the impartial culture is unrealistic ... the impartial culture is the worst case scenario
5. ^ a b c Gehrlein, William V.; Lepelley, Dominique (2011). Voting paradoxes and group coherence : the condorcet efficiency of voting rules. Berlin: Springer. doi:10.1007/978-3-642-03107-6. ISBN 9783642031076. OCLC 695387286. 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
6. ^ a b Van Deemen, Adrian (2014). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3–4): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 0048-5829. S2CID 154862595. 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.
7. ^ May, Robert M. (1971). "Some mathematical remarks on the paradox of voting". Behavioral Science. 16 (2): 143–151. doi:10.1002/bs.3830160204. ISSN 0005-7940.
8. ^ Guilbaud, Georges-Théodule (2012). "Les théories de l'intérêt général et le problème logique de l'agrégation". Revue économique. 63 (4): 659. doi:10.3917/reco.634.0659. ISSN 0035-2764.
9. ^ a b c Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187. S2CID 118143928. 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.
10. ^ Gehrlein, William V. (1997). "Condorcet's paradox and the Condorcet efficiency of voting rules". Mathematica Japonica. 45: 173–199.
11. ^ a b c Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786.
12. ^ a b Tideman, T. Nicolaus; Plassmann, Florenz (2012), Felsenthal, Dan S.; Machover, Moshé (eds.), "Modeling the Outcomes of Vote-Casting in Actual Elections", Electoral Systems, Berlin, Heidelberg: Springer Berlin Heidelberg, Table 9.6 Shares of strict pairwise majority rule winners (SPMRWs) in observed and simulated elections, doi:10.1007/978-3-642-20441-8_9, ISBN 978-3-642-20440-1, retrieved 2021-11-12, 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%
13. ^ Kurrild-Klitgaard, Peter (2014). "Empirical social choice: An introduction". Public Choice. 158 (3–4): 297–310. doi:10.1007/s11127-014-0164-4. ISSN 0048-5829. S2CID 148982833.
14. ^ Kurrild-Klitgaard, Peter (2001). "An empirical example of the Condorcet paradox of voting in a large electorate". Public Choice. 107: 135–145. doi:10.1023/A:1010304729545. ISSN 0048-5829. S2CID 152300013.
15. ^ Graham-Squire, Adam; McCune, David (2023-01-28). "An Examination of Ranked Choice Voting in the United States, 2004-2022". arXiv:2301.12075v2 [econ.GN].
16. ^ Myers, A. C. (March 2024). The Frequency of Condorcet Winners in Real Non-Political Elections. 61st Public Choice Society Conference. p. 5. 83.1% … 97.9% … 98.8% … Figure 2: Frequency of CWs and weak CWs with an increasing number of voters
17. ^ Lippman, David (2014). "Voting Theory". Math in society. ISBN 978-1479276530. OCLC 913874268. 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.
18. ^ Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
19. ^ Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.