An electron among rn ~> 3 candidates has a Condorcet winner if some candidate can defeat (or tie) every other candidate under pairwise majority voting. The election of a Condorcet winner is appealing for several reasons, including its adherence to the principle of selection according to the will of the majority and its promotion of social stability in that no other alternative can displace a Condorcet winner by a direct vote between the two. However, it is easy to specify voters' preferences for which every candidate can be defeated by some other candidate, and in this case Condorcet's paradox is said to occur. Because commonly-used procedures for selecting one candidate from a field of rn candidates are more susceptible to instability and strategic manipulation when Condorcet's paradox occurs, it is important to have some idea of the likelihood of the paradox in various types of situations. In this paper we shall examine this likelihood as a function of a simple measure of social homogeneity for three-candidate elections.
In general, social homogeneity assesses the extent to which voters have similar preferences and is maximized when all voters have the same preference order on the candidates. Hence it seems reasonable to expect a relationship between social homogeneity and the likelihood of Condorcet's paradox. To describe previous work on such a relationship for various measures of social homogeneity, we shall assume an odd number n of voters each of whom has one of the rn ! linear orders on the rn ~> 3 candidates as his or her preference order. The linear orders or strict rankings will be indexed by k from 1 to rn ! with n k the number of individuals whose preference order is ranking number k. Clearly nl + β’..