Some mathematical remarks on the paradox of voting

When a group of m individuals endeavours to choose a winner from a set of n alternatives by making all possible pairwise comparisons among the alternatives (using simple majority rule), there exists the possibility that no outright winner will emerge, e.g., a beats b, b beats c, c beats a. This phenomenon is called the paradox of voting, and it has been shown to have relevance in many contexts in the behavioural sciences. In the first part of this paper, we prove the conjecture that the paradox probability tends to unity as n ∞ for all m(m ≧ 3): however, this limit is seen to be attained very slowly when m is small. We also note that the paradox probability is exactly doubled when the number of alternatives increases from 3 to 4, irrespective of the value of m. This work is restricted to the conventional case of an impartial culture, wherein all alternatives are intrinsically equally favoured. In the second part, we give a simple yet accurate approximate formula for the paradox probability with n = 3 in a quite arbitrary culture, when the individual alternatives are not intrinsically equally favoured. The character of these results is discussed, and some specific examples are considered in detail.