Condorcet Winner Probabilities - A Statistical Perspective

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 rank 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 → ∞. 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.