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Topological Problems in Voting

ryantolsma.com

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Re: Topological Problems in Voting

#11

Post by jfengel »

lukifer wrote: post #7

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.)

I'm not convinced it can actually achieve that. There is still just one winner, just as now, and I'm not sure the people who picked them under duress will really feel they were listened to. (Or they can approve of only one, and almost certainly lose if it's not one of the two most popular parties.)

Still, I'm not averse to trying. Either it will help, or tactical voting will leave us more or less where we are now. If nothing else it's an opportunity to give the current deadlock a shove.

Re: Topological Problems in Voting

#12

Post by BriggyDwiggs42 »

johnkpaul wrote: post #3

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 was thinking that too. Could be -plagiarized- inspired by the birds, since the flow of the article starts out the exact same way

Re: Topological Problems in Voting

#13

Post by rtolsma »

johnkpaul wrote: post #3

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

Yes, I saw that! Inspired me to look at the original paper.

The video takes a slightly different approach from the paper and uses a retraction on the möbius strip to its boundary as a contradiction.

That particular argument doesn’t generalize as well in higher dimensions (in particular, the symmetric product won't always have a boundary to retract to), so I followed the original paper’s one instead. I'll add a link to that video as well

Re: Topological Problems in Voting

#14

Post by vcdimension »

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…

I thought about averaging the scores, which gives you a point inside the circle, and then projecting onto the circle with a ray from the centre, which is continuous everywhere apart from where the average is at the centre (e.g. for two voters this is when they have exactly opposite views). So if you have a continuous probability distribution on the domain the probability of undecidability has measure zero.

Re: Topological Problems in Voting

#15

Post by wffurr »

jfengel wrote: post #11
lukifer wrote: post #7

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.)

I'm not convinced it can actually achieve that. There is still just one winner, just as now, and I'm not sure the people who picked them under duress will really feel they were listened to. (Or they can approve of only one, and almost certainly lose if it's not one of the two most popular parties.) Still, I'm not averse to trying. Either it will help, or tactical voting will leave us more or less where we are now. If…

Approval voting with multi member districts.

Re: Topological Problems in Voting

#16

Post by eru »

lukifer wrote: post #7

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.)

Arrow's Theorem only applies to some voting systems and only in some situations.

Yes, the theorem doesn' apply to approval voting nor does it apply to score voting.

Arrow's theorem only applies to deterministic voting systems. So sortition (or other method based on random sampling) are not affected.

The theorem also doesn't apply to proportional representation systems. (Though they have their own problems, of course.)

Most RCV systems are very gameable with tactical voting. Though they aren't that useful, I guess.

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Arrow's theorem also doesn't guarantee that you will have problems. It just says that for some votings systems you can construct voting populations with preference that can't be captured well. It doesn't say whether these situations are likely to occur in practice.

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Arrow's theorem also doesn't apply when you allow bargaining, or people compensating each other.

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Of course, the problem with democracy in practice isn't so much that existing voting systems don't capture what voters want. Even first-past-the-post seems to be doing a reasonable job of that.

The problem is that voters want bad things like protectionism or war or price controls etc. See https://en.wikipedia.org/wiki/The_Myth_of_the_Rational_Voter

Re: Topological Problems in Voting

#17

Post by eru »

jfengel wrote: post #11
lukifer wrote: post #7

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.)

I'm not convinced it can actually achieve that. There is still just one winner, just as now, and I'm not sure the people who picked them under duress will really feel they were listened to. (Or they can approve of only one, and almost certainly lose if it's not one of the two most popular parties.) Still, I'm not averse to trying. Either it will help, or tactical voting will leave us more or less where we are now. If…

> There is still just one winner, just as now, and I'm not sure the people who picked them under duress will really feel they were listened to.

Huh, where is that duress coming from?

> (Or they can approve of only one, and almost certainly lose if it's not one of the two most popular parties.)

Huh, why can they approve of only one? The whole point is that you can approve more than one.

Re: Topological Problems in Voting

#18

Post by cfgauss2718 »

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.

Re: Topological Problems in Voting

#19

Post by rtolsma »

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…

Yea I think one reason to restrict to spheres is because the voting function takes as input the relative preferences (like in [0,1]^n how does all 0s differ from all 1s), which implies the vectors should be normalized

Re: Topological Problems in Voting

#20

Post by compumetrika »

lukifer wrote: post #7

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.)

Is there a proof or demostration somehwere about maximizing the consent of the governed?

Arrows theorem always has implied to me that the next step should be quantifuing some welfare measure for voters and then exploring which system maximized that welfare measure. "Consent of the governed" sounds like a welfare measure so I an intrigued.

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