Condorcet paradox
In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently 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 scenarios 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. Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called Condorcet methods.
Condorcet's paradox is a special case of Arrow's paradox, which shows that any kind of social decision-making process is either self-contradictory, a dictatorship, or incorporates information about the strength of different voters' preferences.
History
Condorcet's paradox was first discovered by Catalan 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.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 prefer B to C and only one voter 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 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.
- Group 2 despise Charlie for his sharp business practices. They rank Charlie last, and then Alex above Beatrice as usual.
- 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.
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.For voters providing a preference list of three candidates A, B, C, we write the random variable equal to the number of voters who placed A in front of B. The sought probability is . We show that, for odd, where which makes one need to know only the joint distribution of and.
If we put, we show the relation which makes it possible to compute this distribution by recurrence:.
The following results are then obtained:
| 3 | 101 | 201 | 301 | 401 | 501 | 601 | |
| 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 tends to where is a variable following a Cauchy distribution, which gives .
The asymptotic probability of encountering the Condorcet paradox is therefore which gives the value 8.77%.
Some results for the case of more than three candidates have been calculated and simulated. The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates:
The likelihood of a Condorcet cycle for related models approach these values for three-candidate elections with large electorates:
- Impartial anonymous culture : 6.25%
- Uniform culture : 6.25%
- Maximal culture condition : 9.17%
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.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.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.
Empirical studies
Many attempts have been made at finding empirical examples of the paradox. 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 very few examples have been found in larger groups, although some have been identified.
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%.
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".
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.
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 in Ward 2, with a narrow circular tie between candidates of the Green Party, the Minnesota Democratic–Farmer–Labor Party, and an independent democratic socialist. 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 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.
Another instance of a Condorcet cycle was with the seat of Prahran in the 2014 Victorian state election, with a narrow circular tie between the Greens, Liberal, and 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, 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.