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Topological Problems in Voting
- deleted 2y ago[deleted]
- johnkpaul 2y agoHmm, 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 https://youtu.be/v5ev-RAg7Xs?si=X1LY6Qc_s-HDqI3S
- BriggyDwiggs42 2y agoI was thinking that too. Could be -plagiarized- inspired by the birds, since the flow of the article starts out the exact same way
- rtolsma 2y agoYes, 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
- TaylorAlexander 2y agoYeah you don’t want to get hbomberguy’d.
- unfamiliar 2y agoAm 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.
- pxeger1 2y agoI 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.
- j16sdiz 2y agoArrows theorem says it is impossible to have a system that always resolves (it is possible to have something work "sometimes" however. The paragraph you quoted introduce a generalized version, where voters can give continuous scores and have full spectrum of choice.
- contravariant 2y agoI'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.
- dylwil3 2y agoYep, see Eckmann for a generalization and precise characterization: https://core.ac.uk/download/pdf/82385648.pdf https://core.ac.uk/download/pdf/82385648.pdf
- contravariant 2y agoAwesome, 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.
- ykonstant 2y agoThe notion of "space with mean" from that paper seems to be of independent interest; nice.
- vcdimension 2y agoI 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.
- yjk 2y ago
- lukifer 2y agoArrow’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.)
- jfengel 2y agoI'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.
- wffurr 2y agoApproval voting with multi member districts.
- AnthonyMouse 2y agoThat'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.
- jfengel 2y agoList 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.
- cfgauss2718 2y agoOn 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.