Ensuring every candidate wins under positional voting

Given a fixed set of voter preferences, different candidates may win outright given different scoring rules. We investigate how many voters are able to allow all n candidates to win for some scoring rule. We will say that these voters impose a disordering on these candidates. The minimum number of voters it takes to impose a disordering on 3 candidates is 9. For 4 candidates, 6 voters are necessary, for 5 candidates, 4 voters are necessary, and it takes only 3 voters to disorder 9 candidates. In general, we prove that m voters can disorder n candidates when m and n are both greater than or equal to 3, except when m = 3 and n ≤ 8, when n = 3 and m ≤ 8, and when n = 4 and m equals 4 or 5. 1. Background Information Saari, in his paper “Millions of election outcomes from a single profile” [3], proved for n candidates that it is possible to create a profile leading to (n − 1)(n − 1)! different strict positional election rankings. (Recall that tallying votes using a positional rule assigns specific weights to candidates based on how they are positioned on the ballot. For example, the plurality vote is when the top-ranked candidate is given a weight of one and every other candidate is given a weight of zero. The Borda count is when the top-ranked candidate is given a weight of n, the second-ranked candidate is given a weight of n−1, through the lastranked candidate, who receives a weight of 1.) In other words, even if the voter’s preferences remain the same, it is possible that different scoring rules can produce a large number of different election outcomes. Saari’s result shows that even with as few as 10 candidates, the number of possible election outcomes for one profile can be over one million! This highlights that the choice of the positional rule used in the election is very important. Yet how do we know that this result is relevant to everyday elections? If the number of voters necessary to produce a profile with different outcomes is large, then we can dismiss Saari’s result as irrelevant to most organizations. While in this article I do not explain how many voters are necessary to create a profile with (n − 1)(n − 1)! election outcomes, I characterize how many voters are required to create a profile under which every candidate will win outright under some positional rule. I show that the required number of voters is smaller than one might expect. For example, if there are nine or more voters, then they can vote in such a way that any of three candidates might win under a different positional rule. A particularly surprising conclusion is that as the number of candidates increases, the number of voters required to construct such a profile decreases! A complete explanation for this counterintuitive result is given in the paper, but the basic intuition behind this assertion is that as the number of candidates increases, the number of degrees of freedom in the normalized positional rule increases. With three candidates, there is but one degree of freedom; whereas 2000 Mathematics Subject Classification. Primary 91B12; Secondary 52C35, 91B14.