Topological Problems in Voting

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.

Re: Topological Problems in Voting

#5

Post by pxeger1 » Sat, Jun 15, 2024, 8:58 PM UTC

unfamiliar wrote: post #4

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…

I agree, it could do with a little more proofreading. Arrow’s theorem states that no voting state which ranks candidates can satisfy the the given conditions.

Re: Topological Problems in Voting

#6

Post by contravariant » Sat, Jun 15, 2024, 9:52 PM UTC

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.

Re: Topological Problems in Voting

#7

Post by lukifer » Sat, Jun 15, 2024, 10:46 PM UTC

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.)

Re: Topological Problems in Voting

#8

Post by dylwil3 » Sat, Jun 15, 2024, 11:05 PM UTC

contravariant wrote: post #6

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…

Yep, see Eckmann for a generalization and precise characterization: https://core.ac.uk/download/pdf/82385648.pdf

Re: Topological Problems in Voting

#9

Post by contravariant » Sat, Jun 15, 2024, 11:55 PM UTC

dylwil3 wrote: post #8

contravariant wrote: post #6

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…

Yep, see Eckmann for a generalization and precise characterization: https://core.ac.uk/download/pdf/82385648.pdf

Awesome, always nice to see my mathematical intuition still works. Also an interesting piece of mathematics.

My main takeaway was the following conclusion

> [E]xcept for the contractible case either no social choice function can exist on P, or if it exists for all n then unexpected properties turn up.

Re: Topological Problems in Voting

#10

Post by ykonstant » Sun, Jun 16, 2024, 12:54 AM UTC

dylwil3 wrote: post #8

contravariant wrote: post #6

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…

Yep, see Eckmann for a generalization and precise characterization: https://core.ac.uk/download/pdf/82385648.pdf

The notion of "space with mean" from that paper seems to be of independent interest; nice.