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 voter preferences are uniformly distributed among the candidates. (This is the "impartial culture" model, which is known to be unrealistic, so, in practice, a Condorcet paradox may be more or less likely than this calculation.) 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. Using the central limit theorem, we show that tends to where is a variable following a Cauchy distribution, which gives (constant quoted in the OEIS). The asymptotic