Kelly's conjectures on Condorcet winner probabilities for odd electorates

Abstract

Let $C(m,n)$ denote the probability that, among $n$ independent uniformly random permutations of $m$ labels, there exists a label that precedes every other label in a strict majority of the permutations. We prove that $C(m,n)>C(m,n+2)$ for $m \ge 3$ and odd $n \ge 1$, and that $C(m,n)>C(m+1,n)$ for $m \ge 2$ and odd $n \ge 3$. These results establish Kelly's two monotonicity conjectures for odd electorates. The proofs use integral representations involving probabilities associated with sums of independent Bernoulli random variables.