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.
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.
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.
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
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.
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.
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
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 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.
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.
lukifer|1 year ago
eru|1 year ago
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.
---
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.
---
Arrow's theorem also doesn't apply when you allow bargaining, or people compensating each other.
---
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
compumetrika|1 year ago
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.
jfengel|1 year ago
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.
johnkpaul|1 year ago
That channel just released a video on the same topic.
https://youtu.be/v5ev-RAg7Xs?si=X1LY6Qc_s-HDqI3S
rtolsma|1 year ago
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
BriggyDwiggs42|1 year ago
contravariant|1 year ago
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.
dylwil3|1 year ago
vcdimension|1 year ago
rtolsma|1 year ago
unfamiliar|1 year ago
> 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.
pxeger1|1 year ago
j16sdiz|1 year ago
The paragraph you quoted introduce a generalized version, where voters can give continuous scores and have full spectrum of choice.
cfgauss2718|1 year ago
unknown|1 year ago
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