The Condorcet paradox (also known as the voting paradox or the paradox of voting) 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.
Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed left-to-right for each voter in decreasing order of preference):
|Voter||First preference||Second preference||Third preference|
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. 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 (namely, –.5), as calculated with Spearman's rank correlation coefficient formula designed by Charles Spearman much later.
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:
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. (In this particular example, though, the incentive is weak, as those who prefer C to A only score C 1 point above A; in a ranked Condorcet method, it's quite possible they would simply equally rank A and C because of how weak their preference is, in which case a Condorcet cycle wouldn't have formed in the first place, and A would've been the Condorcet winner). 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 (x + y – 1). Since the paradox (a majority preferring C over A) requires z < 1/2, a necessary condition for the paradox is that
Likelihood 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. In particular, Andranik Tangian has proved that the probability of Condorcet paradox is negligible in a large society.
Impartial culture model
We can calculate the probability of seeing the paradox for the special case where voters preferences are uniformly distributed among the candidates. (This is the "impartial culture" model, which is known to be unrealistic,:40 so, in practice, a Condorcet paradox may be more or less likely than this calculation.:320)
For voters providing a preference list of three candidates A, B, C, we write (resp. , ) 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 (we double because there is also the symmetric case A> C> B> A). 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:
The sequence seems to be tending towards a finite limit.
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.:78
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%:325 (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).:47 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 (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.
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 (in their biography of Condorcet), 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 lifted", "to be stronger than" in some geometrical constructions.
- Arrow's impossibility theorem
- Kenneth Arrow, Section with an example of a distributional difficulty of intransitivity + majority rule
- Discursive dilemma
- Gibbard–Satterthwaite theorem
- Independence of irrelevant alternatives
- Instant-runoff voting
- Nakamura number
- Quadratic voting
- Rock paper scissors
- Simpson's paradox
- Smith set
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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
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Here, Condorcet notes that we have a 'contradictory system' that represents what has come to be known as Condorcet's Paradox.
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agents' cardinal (utility-based) preferences are embedded into the space of ordinal preferences. This often gives rise to a distortion in the preferences, and hence in the social welfare of the outcome
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This is the fundamental problem with two-way comparisons. There is no accounting for degrees of preference. ... Cycles result from giving equal weight to unequal preferences. ... The paradox obscures the fact that the voters really do prefer one option.
- Kok, Jan; Shentrup, Clay; Smith, Warren. "Condorcet cycles". RangeVoting.org. Retrieved 2017-02-09.
...any method based just on rank-order votes, fails miserably. Range voting, which allows voters to express strength of preferences, would presumably succeed in choosing the best capital A.
- In this example, the available scores are 0–6, and each voter normalizes their max/min scores to this range, while selecting a score for the middle that is proportional to distance.
- Silver, Charles. "The voting paradox", The Mathematical Gazette 76, November 1992, 387–388.
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it is widely acknowledged that the impartial culture is unrealistic ... the impartial culture is the worst case scenario
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Voting theorists generally acknowledge that they consider this model to be unrealisticCite journal requires
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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
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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.
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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.
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