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Topological Problems in Voting

ryantolsma.com

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Re: Topological Problems in Voting

#31

Post by jfengel »

AnthonyMouse wrote: post #28
edflsafoiewq wrote: post #21

Earlier quoted context omitted.

I think the idea is that if there are two popular candidates A and B, one of whom is almost certain to win, a voter feels forced to approve of whichever of A or B they prefer even if they don't really want either one, just to hedge against the other winning, exactly as in FPTP. Or they can approve of only the candidates they really want, but they will likely lose. Approval only seems to be able to break this gridlock…

The problem with FPTP is that as soon as you have more than two parties, the two most similar parties split the vote among their common constituency and give the win to the least similar party. As a result any candidate who wants a chance at winning has to run on the ticket of the major party they most agree with, or else they split the vote with them and lose. Hence two party system. With a cardinal voting system, s…

That's not necessarily all that different from now. We have a two stage system. In the primary people with broadly similar platforms run against each other. The "third parties" are factions within the two major ones.

Those options exist, and it's a multi way election. Primaries receive far less attention but they are where the real work of democracy is done.

I believe people are hoping they can vote for a radical candidate and a mainstream candidate, on the off chance people will love the radical candidate if they just get on the general ballot. I'm not convinced that will ever happen, and such people will be not just disappointed, but continue to be convinced the system is rigged against them.

Re: Topological Problems in Voting

#32

Post by Spivak »

compumetrika wrote: post #20
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.

I don't really think you need too much to prove it yourself.

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.

Re: Topological Problems in Voting

#33

Post by jfengel »

AnthonyMouse wrote: post #29
wffurr wrote: post #15

Earlier quoted context omitted.

Approval voting with multi member districts.

That's likely to reduce diverse representation vs. single-member districts. If there are e.g. 8 seats a party could run 8 identical candidates and they'd all get the highest approval ratings for the combined district if one of them would, and other parties wouldn't get any.

List voting might work as an alternative to single member districts. You vote for your favorite party, and they are allocated a proportion of the total seats.

You lose the ability to know your local candidate, but how many people really do these days? It's what we set up in Iraq, but we don't do it ourselves.

It doesn't solve the problem that there is still exactly one chief executive. You can try making that a committee but that has other downsides.

Re: Topological Problems in Voting

#34

Post by dataflow »

eru wrote: post #16
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.…

> Arrow's Theorem only applies to some voting systems

Yes, but... https://politics.stackexchange.com/a/14245

Re: Topological Problems in Voting

#35

Post by contravariant »

rtolsma wrote: post #19
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

As it turns out choosing a simplex instead doesn't change things much from the hypercube. I think the arithmetic mean also still works. In stark contrast to the sphere.

Re: Topological Problems in Voting

#36

Post by yjk »

vcdimension wrote: post #14
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.

In addition, I would argue undecidability is a feature, not a bug. I'm not sure why any other answer would be desired in that case.

Re: Topological Problems in Voting

#37

Post by contravariant »

yjk wrote: post #36
vcdimension wrote: post #14

Earlier quoted context omitted.

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.

In addition, I would argue undecidability is a feature, not a bug. I'm not sure why any other answer would be desired in that case.

It's not the undecidability that is a problem, it's the discontinuity. Undecidable answers are manageable, random answers however are very annoying to deal with.

Re: Topological Problems in Voting

#38

Post by ClayShentrup »

eru wrote: post #16
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.…

well, yes. https://www.rangevoting.org/ArrowThm

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.

Re: Topological Problems in Voting

#39

Post by ClayShentrup »

eru wrote: post #16
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.…

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

that doesn't make sense. the result you get after bargaining would just _be_ one of the options.

Re: Topological Problems in Voting

#40

Post by ClayShentrup »

eru wrote: post #16
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.…

> 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.

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.
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