Condorcet paradox
The Condorcet paradox in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic, even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority wishes can be in conflict with each other: Majorities prefer, for example, candidate A over B, B over C, and yet C over A. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
Thus an expectation that transitivity on the part of all individuals' preferences should result in transitivity of societal preferences is an example of a fallacy of composition.
The paradox was independently discovered by Lewis Carroll and Edward J. Nanson, but its significance was not recognized until popularized by Duncan Black in the 1940s.
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. A paradoxical feature of relations between the voters' preferences described above is that although the majority of voters agree that A is preferable to B, B to C, and C to A, all three coefficients of rank correlation between the preferences of any two voters are negative, as calculated with Spearman's rank correlation coefficient formula designed by Charles Spearman much later.
Cardinal ratings
Note that in the graphical example, the voters and candidates are not symmetrical, but the ranked voting system "flattens" their preferences into a symmetrical cycle. Cardinal voting systems provide more information than rankings, allowing a winner to be found. For instance, under score voting, the ballots might be:| A | B | C | |
| 1 | 6 | 3 | 0 |
| 2 | 0 | 6 | 1 |
| 3 | 5 | 0 | 6 |
| Total: | 11 | 9 | 7 |
Candidate A gets the largest score, and is the winner, as A is the nearest to all voters. However, a majority of voters have an incentive to give A a 0 and C a 10, allowing C to beat A, which they prefer, at which point, a majority will then have an incentive to give C a 0 and B a 10, to make B win, etc.. So though the cycle doesn't occur in any given set of votes, it can appear through iterated elections with strategic voters with cardinal ratings.
Necessary condition for the paradox
Suppose that x is the fraction of voters who prefer A over B and that y is the fraction of voters who prefer B over C. It has been shown that the fraction z of voters who prefer A over C is always at least. Since the paradox requires z < 1/2, a necessary condition for the paradox is thatLikelihood of the paradox
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 voters preferences are uniformly distributed between 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 objects have been calculated.
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.Empirical studies
Many attempts have been made at finding empirical examples of the paradox.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%.. On the other hand, the 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.
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 such that each candidate in the group can win a one-on-one election against each other candidate however, which is known as the Smith set. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. 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.
Contrary to a widespread opinion promoted among others by Élisabeth Badinter and Robert Badinter, this paradox calls into question only the coherence of certain voting systems and not that of democracy itself.
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.
The structure of the Condorcet paradox can be reproduced in mechanical devices demonstrating intransitivity of relations such as "to rotate faster than", "to lift and be not be lifted", "to be stronger than" in some geometrical constructions.