Arrow’s Theorem is often invoked as a criticism of alternative voting systems (RCV, etc). And not while not wrong exactly, it seems textbook “perfect being the enemy of the good”. (It’s also one reason I prefer Approval Voting, which in addition to its benefit of simplicity, sidesteps Arrow by redefining the goal: not perfectly capturing preferences, but maximizing Consent of the Governed.)

Hmm, is this author related to the Physics for the Birds YouTube channel?

That channel just released a video on the same topic.

https://youtu.be/v5ev-RAg7Xs?si=X1LY6Qc_s-HDqI3S

I'm not quite sure why one would use a sphere, unless you were specifically trying to get a version of Arrow's theorem.

If anything it looks like it fails precisely because the space is not homologically trivial, but I'm a bit unsure how to make that precise. A similar set up with just [0,1]^n as preference space works perfectly fine just by averaging all the scores for each candidate.

I kind of sense that requiring a function X^k -> X to exist is somehow hard if X is not 'simple', but I'm not yet sure what the obstruction is.

Am I missing something or does the article fail to explain the point of Arrow’s Theorem? Is it satisfied for the discrete case, provably impossible, or what?

> While this applies to discrete rankings and voter preferences, one might wonder if it’s a unique property of its discrete nature in how candidates are only ranked by ordering. Unfortunately, a similarly flavored result holds even in the continuous setting! It seems there’s no getting around the fact that voting is pretty hard to get right.

I don’t follow any of this paragraph.

On a glance, the Chichilinsky theorem assumption of smoothness for the mapping between voter preferences And the vote result (the relation phi) seems burdensome. For example, many people might be effectively summarized as single issue voters - the topological consequences of a typical definition of differentiation (calculus) would seem unjustified. The exercise of exploring this world may be interesting, but I’m not convinced of its utility to politics.