The Principles Behind Voting Systems

There are other principles of voting systems that are not mentioned here. I like to divide voting systems into these categories: linear/nonlinear and preferential/nonpreferential.

By "linear" I mean that the votes are, or are converted to, numeric scores that are simply added, with the highest score selecting the winner. Plurality, approval, range, and the Borda count are all linear; STV is nonlinear.

A preferential system allows voters to rank all the candidates. Borda and STV are preferential; the others are not.

It can be shown, for example, that if we restrict ourselves to linear preferential systems, the Borda count is optimal among these, given another assumption which I'll come to in a moment. If we model the preference of each voter for each candidate as a real number in [0, 1], then the voter's collective preferences are given by a point in the unit n-cube. A preferential system divides the unit n-cube into n! regions and asks each voter to select one of them as best representing their collective preferences. A linear preferential system then converts these to numerical scores and adds them up. The Borda count, for example, assigns a number to each candidate on each ballot which is linearly related to that candidate's position in the voter's ordering; a ballot ranking the candidates in the order A, B, C might be scored as A = 2, B = 1, C = 0. The scores for each candidate are totalled across all ballots, and the candidate with the highest score wins.

It turns out that the coordinates of the barycenters of the regions of the n-cube (where the center of mass would be under the assumption of uniform density) follow the same linear pattern; for example, the preference A > B > C picks out a region of a 3-cube whose barycenter is at (A = 3/4, B = 1/2, C = 1/4). So, if we start with the assumption that a voter's preference for each candidate is uniformly distributed on [0, 1], and we use preferential ballots, the Borda count provides the best estimate of the mean barycenter of the preferences of all voters. (We can apply any linear function to the barycenter coordinates without changing the results of the election; in this case I have used f(x) = 4x - 1.)

Approval voting, instead of dividing up the unit n-cube into the diagonal-faced regions of a preferential system, simply divides it into 2 ^ n equally sized, smaller n-cubes. Each coordinate of the barycenter of one of the smaller n-cubes is either 1/4 or 3/4; again, we can linearly map these into 0 and 1. Range voting is a further refinement of the same idea that uses k ^ n smaller n-cubes; usually k is a relatively small number like 5 or 7.

... Whoops, I need to go and can't write as much here as I would like. But maybe this will be interesting enough.