The paradox of voting: Probability calculations

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.