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
no, it has nothing to do with capturing preferences. it simply says that no ordinal social welfare function can simultaneously satisfy these criteria:
There is no dictator.
If every voter prefers A to B then so does the group.
The relative positions of A and B in the group ranking depend on their relative positions in the individual rankings, but do not depend on the individual rankings of any irrelevant alternative C.but technically it only applies to social welfare functions, not voting methods.
i had a chance to visit kenneth arrow at his home in palo alto circa 2015 and we had a nice little chat about this.
utterly false. https://www.rangevoting.org/PConsumer.html
Yes, but... https://politics.stackexchange.com/a/14245
that doesn't make sense. the result you get after bargaining would just _be_ one of the options.
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.
You are being governed with consent when the person who's elected is someone you are okay being governed by. And the person who wins an approval election is the person with that has the most people fine with being governed by them. Because approval voting doesn't ask people to rank candidates voting for someone you disapprove of only hurts you and voting for any subset of people you do approve of is sincere.
It's not some deep thing because it changes the target to something much easier. Finding the best candidate is hard, finding the candidate most people find acceptable is less so.
Approval voting gets more mathematically interesting when you assume people have preferences among the candidates they approve of and whether the best candidate gets elected but IRL you don't actually care about that anymore. You're fine electing someone who isn't the best.
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.
"They can only approve of one" is FPTP, the existing system. Everybody knows that sucks. The whole point of approval or score voting is to avoid that.
Right now if you favor candidate C but they have 5% of the vote and candidate A and B each have 45%, your preferred candidate has no chance and your vote can only change something in determining whether the winner is A or B, so you avoid voting for your preferred candidate.
With approval voting you vote for them and one of the major parties. Then people notice that third party candidates are immediately getting 30-40% of the vote because the people afraid of wasting their vote no longer have to refrain from voting for their preferred candidate. In some districts they even win. Which dissolves the two party system because people have to take third party candidate seriously and starting a new party has a real chance at succeeding rather than being an exercise in futility.
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.
That channel just released a video on the same topic.
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
> (we’ll take a slightly different approach).
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.
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.
> 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.
The paragraph you quoted introduce a generalized version, where voters can give continuous scores and have full spectrum of choice.