Social homogeneity and Condorcet's paradox

This paper has examined the relationship between social homogeneity measured by σ(p)=p 1 2 + ... + p 6 2 and the likelihood of Condorcet's paradox. Attention was restricted to three-candidate elections. It was shown first that the most general restriction on p vectors that produces a definite inverse relationship between σ(p) and the limit-in-voters probability P ∞ (p) of Condorcet's paradox is the dual culture restriction. We then deleted this restriction to allow any p vector and considered the relationship between σ(p) and the paradox probability when Abrams' positioning effect was removed by averaging the P n (p) values all over rearrangements of the components of p. The resultant averaged probability of Condorcet's paradox with n voters was denoted as Q n (p). Theorem 1 showed that there are p vectors for all odd n ≥ 3 which deny a definite inverse relationship between Q n (p) and σ(p). However, Theorem 2 verified for n=3 that the intervals of possible Q n (p) values for fixed values of σ(p) decrease as σ(p) increases. It was shown also that the latter relationship does not hold for large n although there is a partial tendency for Q n (p) to decrease as σ(p) increases for large n. Copyright Martinus Nijhoff Publishers bv 1980