Condorcet Winner Probabilities - A Statistical Perspective

Abstract

A Condorcet voting scheme chooses a winning candidate as one who defeats all others in pairwise majority rule. We provide a review which includes the rigorous mathematical treatment for calculating the limiting probability of a Condorcet winner for any number of candidates and value of $n$ odd or even and with arbitrary ran k order probabilities, when the voters are independent. We provide a compact and complete Table for the limiting probability of a Condorcet winner with three candidates and arbitrary rank order probabilities. We present a simple proof of a result of May to show the limiting probability of a Condorcet winner tends to zero as the number of candidates tends to infinity. We show for the first time that the limiting probability of a Condorcet winner for any given number of candidates $m$ is monotone decreasing in $m$ for the equally likely case. This, in turn, settles the conjectures of Kelly and Buckley and Westen for the case $n \to \infty$. We prove the validity of Gillett's conjecture on the minimum value of the probability of a Condorcet winner for $m=3$ and any $n$. We generalize this result for any $m$ and $n$ and obtain the minimum solution and the minimum probability of a Condorcet winner.