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Kenneth Arrow, Geometry, and Political Parties

My college friends and I often engage in email conversations spanning 50+ messages about some sophisticated topic. Recently, we discussed politics and political parties: whether or not the participants identify with a party, why or why not, the two party system, etc.

I’d been testing out a theory in my head for why political parties exist, and this conversation made me recall it. The reasoning is logical and geometric, and the result is essentially a defense of the two party system.

We start with the observation that there are a lot of things people care about: health care, national security, taxes, guns, abortion — to name just a few high profile issues. The challenge, then, is to make the decisions of the state that align in some “best” way with the preferences of the citizenry.

Well, sadly, we know that there is no “best” way to do this: every voting method is unfair in some way. This is the result that won Kenneth Arrow his Nobel Prize. But this isn’t the only interesting way this theory applies: so far we’ve only applied it directly to people — it works the same way when applied to parties.

So if there’s no way to directly vote in such a way that expresses society’s wishes, let’s examine how we might get close. As noted above, we have some high-dimensional “issue space” and we might parameterize this space with some distance function that expresses how “close” two positions are, which could be something like “how willing someone with position A would be to tolerate voting for a candidate of position B”. “Wants to require stricter background checks” is probably quite close to “wants to ban assault rifles”, and so on. Every voter is somewhere on this position space — or more accurately, every voter occupies a subspace of this issue space.

The issue space is HUGE; for example, ten issues each with three possible positions (a vast oversimplification of course) is 3^10 possible combinations, or about 60,000. Twenty issues with ten possible positions is over 10 trillion combinations. Far too vast for any person’s ability to comprehend.

What political parties do is engage in natural competition to find the best real estate to build their platform. So the function that political parties perform, seen in this light, is quite natural and obviously extremely valuable. They solve an enormous complexity problem by collapsing the issue/decision space from an incomprehensibly large space to something tractable.

So we’re now in a situation where we’re voting for large swaths of the issue space instead of specific positions in it. That’s what political parties did for us. But Arrow’s theorem tells us that any time there are three or more distinct choices, voting systems are inherently flawed. This is why two parties make sense. Between two choices (and no more than that), a vote can be fair. And how do we get down to only two choices? Political parties.