SAVE Simulated Spatial Models

Spatial models represent voter ideals and alternatives as positions in an issue space, which in this simulation is a 2-dimensional Euclidean field. This representation of the electorate has a lot going on. I'll start by pointing out a few basics.

1. Each electorate is randomly generated. The two main electorate distributions are uniform and normal. In a uniform distribution the voters are equally likely to appear anywhere in the field. In a normal distribution, the voters are more likely to be near the center and less likely to be near the edges. The cycle elections are severely constrained in order to generate the extreme preference profile cycle you see with the Original cycle button in Preference Profiles: Figure 1. What I find interesting about the Cycle elections is how many of them look almost reasonable, as if they could show up naturally.
2. While the voters are generated randomly, the initial alternative are not. In any real voting situation, which is what I'm trying to model, every single choice we have is proposed by someone. If I want to propose an alternative and I don't know what anyone else wants, I'd probably just propose my ideal solution. I would expect anyone else would do the same. Accordingly, each of the three alternatives are always the ideals of the first three randomly generated voters. (Which is why each of the alternative circles is centered on a voter ideal circle.
3. Each of the lines is generated by a pair of alternatives: A and B, A and C, and B and C. If a voter is on the A side of the AB line, that voter considers A to be closer to their own ideal than B is, and will thus voter for A over B when given that choice. The fact that it is very unlikely for a voter to be exactly centered on a line is one justification for people working with preference profiles to ignore ties.
4. Speaking of preference profiles, each region bounded by the lines corresponds to a single preference order. If a voter is on the A side of the AB line, the C side of the AC line, and the C side of the BC line, the voter has the preference order C≻A≻B. These lines also show that some preference profiles are simply not possible in certain electorates, particularly when the three lines do not intersect within the frame.
5. In addition to the majority decisions, tournament digraph, and Copeland scores, the spatial model lets us introduce a new metric: average distance. The field is a 400x400 grid, with voters confined to the inner 350x350 section (to allow room for alternative labels). The average distance score is simply the average distance between an alternative and the voters in the electorate. The order of the Average Distance scores tends to agree with the order of the Copeland scores. But this metric introduces a new concept that is simply not modeled in tournaments or preference profiles. The average distance is a measure of how good an alternative is for the electorate. The lower the average distance the better the alternative is.

So far, we've just looked at static metrics. But spatial models allow for exploring ideas in many ways that just aren't possible with tournaments or preference profiles. For example, let's do a quick experiment on what our metrics tell us.

Pick an electorate, any electorate. Now move alternative C out of the way in the emptiest bottom corner, and move "A" as far as it will go to the top left corner and B as far as it will go to the top right corner. Focus on just A and B. Your model should be in one of three states: A beats B, A loses to B, or they tie. If it's a tie, just pick one, otherwise move the originally loosing alternative about a quarter of the way across the top, directly toward the other alternative. The moving alternative just picked up more votes. Note the vertical line midway between A and B, As that line crosses voters, they change their preference between A and B and the moving alternative picks up another vote.

If you push it close enough, the moving alternative might be able to pick up all the votes and get a 100:0, totally unanimous victory. (Although if you push beyond that, you might see a 0:0 tie when the two alternatives coincide and no one bothers to vote.

Now, return your moving alternative to its original corner and look at the average distance scores for A and B. It's very likely that the winning alternative has a better average distance score. Do the same slow move toward the other alternative, but this time focus on the average distance score of the moving alternative. What you'll see is the moving alternative's average distance score starts decreasing (getting better) until somewhere in the middle where it flattens out, and then as you continue moving to the other alternative the average score increases (gets worse) until the two alternatives coincide and get the same average distance score.

A second exercise is to start with any electorate and its three alternatives. The idea is to pick a losing alternative and move it until it is a Condorcet winner (beating both of the other alternatives). Then repeat the process with one of the new losers. Continue this and see where you end up. It might help to look at the distance scores.

Other things to note are that when you move voters, their votes do not change until they cross a line. When you move any alternative, however, two of the lines move and can easily change many votes. If you move one alternative directly toward another alternative, the line between them stays in the same orientation,