Condorcet paradox

The Condorcet paradox (also known as voting paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e., not transitive), 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. 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.

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):

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 correlations between the voters' preferences 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.

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

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