Politics via math?

When I learn a new thing, I like to see what sorts of unexpected things I can model with it. I’m dabbling in some less intuitive properties of high dimensional spaces right now and the concentration of measure is a pretty core part of that. This post is about a particularly amusing model with some interesting results I came up with earlier today.

Concentration of Measure

If you take a unit ball (the surface of a sphere) in N dimensions, nearly all the points lie on the equator if N is sufficiently large. Note that the equator is relative because I haven’t told you which side is north! That’s the concentration of measure. Now let’s go to our model.

Model

Let’s represent people’s political beliefs as vectors in some high dimensional space. It’ll look like [pro-life/pro-choice, pro-2A/anti-2A, ...] and so on where each coordinate is $\pm 1$. We’d have one coordinate for every version of every position (so pro-life is different from pro-life-with-rape-clause, and so on), so this would actually be a very high dimensional space. Since everything is a one-hot vector, we can consider just the directions of the vectors (e.g. the surface of the sphere).

A society is represented by a collection of vectors on the surface of that sphere (one for each member). A more homogeneous society is more concentrated and more diverse society is more uniformly distributed.

The consequence is: as a society becomes more diverse, the elected officials become more orthogonal to their constituents.

Conclusion

As a government grows in scope, it loses the ability to represent its citizens. This is pretty obvious, but it’s nice that we can show this mathematically (via an admittedly limited model).