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In this paper an expression for determining the frequency of the “voting paradox” as a function of the number of voters, the number of alternatives to be voted upon, and cultural characteristics of the voters, is derived. In the analysis a voter's ranking of the given alternatives is represented by a skewsymmetric matrix whose elements also satisfy a certain transitivity condition. The collective ranking of the alternatives is obtained under simple majority rule by summation of the actual individual rankings and consequently can likewise be represented by a skewsymmetric matrix. It is then shown that the existence of the paradox corresponds to the nonexistence of a row in the matrix denoting the collective ranking whose entries are all nonnegative. Cultural effects are incorporated by assigning to each possible individual ranking of alternatives a probability. It is then shown that when all possible rankings of three alternatives are equally likely and the number of voters becomes very large, the probability of the paradox approaches a definite limit, vig .087. Exact values for the probability of the paradox for several cases in which the number of alternatives exceeds three are also presented.
Garman et al. (Mon,) studied this question.