I find it funny how he immediately went to assuming that votes should be continuous because he needed some basis to found his topological problem solving, it is definitely NOT obvious that votes would be continuous, and the simplest way to do it, the popularity vote does not at all work that way
For the sake of the argument I’ll go down to simpler terms, imagine our voting function in which all people are weighted equally, now imagine an election between two movies called 1 and two, now imagine that the x axis is based on the percentage voting for movie two, it starts at 1 and stays completely level until 50%, then it sharply increases to two discontinuously, that can also be extended to more dimensions and the discontinuity stays, that is why the continuity assumption is really quite flawed, I seriously wonder what prompted you to make such an assumption
Not sure how continuity makes sense as a requirement. At a certain point, there has to be a discrete winner, and so there MUST be a point at which flipping one vote will change who/which set of candidates wins, and that just makes sense.
Our systems implement a discrete winner, but they don't have to. Actually, the discontinuity causes issues in our election systems (let's just take the US one for now, assuming you're American): A third-party candidate in a swing state could side with one of the major parties negotiating outsized influence in return. Outsized in relation to the "few" votes they bring to the "allied" major party. You could share power on a fractional basis or have the length of the presidency be based on the difference in votes. It's less obvious to me where continuity would fail there, but the theorem says it does.
That sounds more like an issue with treating votes as anything other than equal, by design. Parties break that system, because the parties are the things doing the real voting, right? @@TrueMilli
I don't think this is the problem, though. Because in the video he's mapping everyone's opinions on a circle. The last step would be "snapping" the circle to the closest axis to get the mode. The continuity requirement is in the sense that your opinions can be continuous (to a certain extent, of course), so if you change your opinion slightly, the resulting circle should also change by a small amount. It's not about the final result, which of course has to be discrete, but the step before that.
@@TrueMilli Welll the problem still sort of lies in the fact that is the outcome teaters around 50%, it doesnt matter who gets the edge, the one with the slight lead will get all the power. (i.e. 49.5% red reps and 50.5% blue reps is a vastly different outcome then the other way around.) Because passing policy is a discrete function, power assignment must be disconnected.
The discrete number of votes corresponds to the number of dimensions of the strip, not the continuity of the strip (the video showed two votes mapping onto a two-dimensional mobius strip, three votes would be a three-dimensional space, etc)
@@ajbiffl4695 You're right. I mistook that. It is what everyone else said. The space of all preference-pairings of all voters is dense (assuming ppl have preferences for things that can be quantified by numbers in R) but the election outcome is discrete. So the function cant be continuous. I corrected my comment
@@infamedepatates2502 Well no, because 'discrete' already implies a topology. Once we apply a different topology to the discrete space, it may cease being discrete.
This is the problem. My feeling is that this applies to votes that are of a different nature. Say a jury votes for how long a culprit has to stay in jail and how much money they need to pay. Then everyone chooses a pair of real values. This is a point in a 2D plane. Reducing it to a circle equates to choosing only the ratio between money and jail time (without choosing any of the two absolute amounts). Then this theorem applies. But this situation is very different from elections...
If you think about it, the idea of fundamental rights exists precisely because the magnitude of certain opinions matters. If a constitutional system forbids slavery, it doesn't matter if 90% vote to enslave the 10%. That's justified by the idea that the magnitude of the suffering of the enslaved outweighs the common will.
10:00 "I personally think that the most likely condition to fail is continuity". Theoretically, I think you're right: pragmatically, I think the most likely condition to fail will be that the ballots are of the form "Pick a point on a circle".
@@caofan5190 The mathematical result is about a function that takes continuously-valued parameters. This corresponds to, say, questions on a questionnaire where you could answer 0% to 100%, and you can use an infinite sequence of digits after the decimal point in your answer to precisely specify your value. Ballots are usually of the form, rank these options from most-preferred to least-preferred, or, select your most preferred option, or other types - but they're all essentially discrete-valued, categorically different from continuously-valued. And that's the way it should be - continuously-valued ballots would be horrendous. If, in the real world, we can't even get the data collection to match the premise of the theorem, the result about the data processing is moot. At least, in the form shown here.
Yes! When I first saw your Möbius strip synth I knew you'd be able to do something really cool with my Synth-a-Sette. Thanks for making with MicroKits!
@@churchboy4609 Ratchet Theory or Ratchet Effect feels more like what Ivan is referring to. Where the duopoly induced by first past the post elections leads to conservative drift as liberal politicians use the excuse of having to chase imagined "centrist" or "moderate" votes to maintain conservative policy positions that just happen to also reinforce their upper class family economically.
At the part with the two "ch" sounds in a row I noticed you pronounce "ch", "dj" "sh", "zh" in a completely different way than most people. I think your "sh" is a voiceless velar lateral fricative, with "zh" being a voiced velar lateral fricative, "ch" a voiceless velar lateral affricate and "dj" a voiced velar lateral affricate. I think they all have audio files on Wikipedia. For the common pronunciation of these you can swap "velar lateral" with "postalveolar" or "postalveolar sibilant", those also have audio. Cheers
I came here expecting a discussion of the topographical analysis of voting. With some conclusion being along the lines of "put your polling center at sea level or lower to get the best turnout for your district. Nobody wants to go uphill to vote."
If you take a real vector space and quotient by scaling by a positive constant, you don't just get a sphere. There is also the vector 0 that can't be scaled to any other vector. This is the utility function that represents indifference between all the options. It seems like the theorem cheats by not allowing this point as a possible output (it's easy to accommodate it as an input, just ignore any indifferent voter). In particular, in the two voter case you could say that the overall utility is indifferent precisely when the two voters are diametrically opposed. This has the effect of cutting out the centre line from the Möbius strip, whereupon the rest of it does indeed retract onto the boundary.
I think in order to have a mapping that covers the exact-opposites case, then you have to throw out the anonymity condition - someone's vote will have to matter less, just so that you can get back to the circle
You are misunderstanding what we are doing here. We are not "quotienting the vector space by scaling by a positive constant". And there is no vector 0, because there's no indifference curve whose oriented normal vector is the vector 0. And the vector 0 would not represent "indifference between all the options". That vector already exists in the construction shown in the video, and its the unit vector forming a 45° angle with the positive horizontal axis
@@UnCavi I don't think that's right. The unit vector represents the direction of the gradient of the utility function. If someone has no preference the utility function is flat, and the gradient is zero. There are no indifference curves, it's an indifference plane.
@@ajbiffl4695 The example shown in the video, where the indifference curves are diagonal straight lines with slope -1, represents a situation where the voter preference doesn’t lean towards either option (fries, burger). Take for example the line with equation y + x = 1: moving along this line, we get points where we either maximally support “fries” and give 0 points to “burger”, or we maximally support “burger” and give 0 points to “fries”, or every combination in between, with equal weight: this is exactly what it means to be indifferent to each candidate. And the normal unit vector to these indifference lines is a 45° vector on the unit circle What you’re talking about with your “indifference plane” would be a constant function with 0 gradient at every point, and while yes, this would also correspond to some kind of indifference form the voter perspective, we are not interested in including this scenario in our description , because it doesn’t add anything useful as there is already another way to express indifference with a non-zero gradient. Also, a constant function would just be “not voting”, so we don’t count them. And if one wants to give their vote equally to each candidate, giving a “null” preference, they can choose the unit vector 45° to the horizontal
I saw a Google Tech Talk about this years ago and somewhat understood it. They also went into details about which voting systems broke which fairness principles. But I have to say your topological explanation makes more sense and I think the reasoning will stick with me longer. Alas, I still can't properly explain it to people at parties. Unless I carry around props.
I actually think the most likely condition to fail is 'ordinal voting'. If you have only two voters with exactly equal and opposite preferences, the optimal election result would be no change at all, rather than a change in a completely orthogonal direction.
But "no change at all" has a length of zero. It is not part of the preference space. So instead, the tiniest differences get magnified. When 50 people vote left and 50 people vote right, then 1 vote for backwards would win the election.
in reality it requires AT LEAST 2 against 1 to ENFORCE the will of the (social) "majority" onto the (asocial) "minority" with an obvious outcome (so the minority accepts defeat without "testing" the resilience of the majority). Voting systems that are based on simple majorities (50%+1) are not natural and socially unsustainable.. systems based on 2:1, 3:1, 4:1 etc. pp. on will result in more natural outcomes and obviously ever lower common goals being enforced ON ALL - which is a feature and NOT a bug.
This sounds right. The problem is that there is no "obvious" compromise between opposed preferences in the space of unit vectors. A usual way to make sure that compromises exist is to assume that the choice space is convex. But the circle is not even contractible. What the theorem is saying seems to be just that democracy requires that compromises are available. It doesn't sound like an amazing insight, but it's reasonable. It says nothing about the viability of democracy over decisions that do admit compromises, though (in that case Arrow's theorem and its variants apply and provide the powerful and nontrivial insight that the idea of democracy is not a clear guide if there at least two dimensions of disagreement).
what u people forget is that 'democracy' is the "peaceful" path by which opposing goals are being solved, which normally - in wilderness - are being solved via force, via physically overcoming the opposing party. This means in a 1:1 scenario democracy CAN NOT solve the problem, because it is NOT CLEAR who will "win" this 'conflict'. Once we get to 2:1 or better it is much more clearer which 'opinion' will be inferior if the rubber hits the road.. which is the way by which MINORITIES knuckle under and stay 'peaceful'. Democracies are minority suppressing mechanisms that allow for peaceful solutions, if there is a 2:1 or better majority behind the preference. Any democracy that allows for less than 2:1 preferences to become the rule for all WILL sooner or later fail as it is CATERING to a non-capable 'minority' against the 'majority'.
"obviously voting for the 99% Hitler is better than the 100% Hitler, there's absolutely nothing else to do other than to hope 99% Hitler dies in next 4 years and we vote for someone less than 100% Hitler in 2028" -Liberal westerner
I was so distracted by the anthem played on a two-axis keyboard that I had to replay the intro multiple times! Such a great execution of an odd concept!
I think the mistake is that in reality, you cannot vote against a party (except by only voting for the other ones). At 4:14, there is only one party, but you can still vote against. In reality, having only one party would have only one voting option. To fix this, we should only have a positive vote, creating a quarter circle (in 2D). A quarter circle doesn't wrap around, so I don't think realistic voting is a möbius strip.
Fascinating application of Mathematics, which I like to see. I would say on the topic of the continuity supposition, though, that I don't think we would intuit voting functions to be continuous. Results in a election are in their actual expression always form a finite space. For example in a FPTP electorate the space of results will be exactly the candidates that run, and the result could move from one candidate to another by the change of one vote. The same applies just with more discontinuities in say a MMP election for a legislature (e.g. Germany, NZ, etc..) because of the nature of what the result is.
oh shoot Nicky Case also made a small interactive thingy about voting ages ago. I'm with Nicky, scoring each candidate looks cool and more representative than ranking, which in turn is more representative than just pick one.
Yep. The assumption that voting is about ordinal preferences (rankings) is probably the biggest blunder in social choice theory, making it basically useless in practice.
@@1ucasvbOrdinal preferences at least have a clear meaning. The meaning of a scalar seems, unclear? Like, how do I determine what an “honest vote” would be so that I can vote honestly?
But scoring just encourages voters to give all the candidates they can tolerate the maximum score and give all the candidates they hate the minimum score. Even if the system somehow accounts for this, loads of people will do it anyway.
@@sir_fuzzball1486 well yeah it is possible for it to devolve into that, but that would also be better than saying yes to one guy and that's it. with score you can say yes and no to multiple people. you can say varying amounts of yes or no. i think having something that is more representative its better.
Continuity makes sense, Voting is not continuous in many cases. If we have a bunch of primary schoolers voting for pineapple or no pineapple on pizza, then it is possible for one vote to take us from 0% pineapple to 100% pineapple a discontinuity if there ever was one. (say you have 11 kids and 5 vote for No Pineapple and 6 vote for pineapple, resulting in a pizza that has pineapple if one kid changed their vote we would shift from 5-6 to 6-5 for No pineapple) Continuity can only ever make sense in voting schemes where blended results are possible. which does not generally represent most votes
I think it's perfectly fine to model opinions as multi-dimensional vectors, it just doesn't make any sense to expect that mapping those opinions onto a finite list could ever be continuous. In fact, given the fact that the results space is not densely populated, it boggles me to understand why anyone would think voting could *ever* be continuous, regardless of other factors.
You misunderstood the video. We are not mapping the input opinions onto a finite list: the codomain of the voting function is the unit circle, and the output is a unit preference vector, just as any vote cast is.
@UnCavi nah that's a fine understanding of the video, because the assumption is, in and of itself, moronic. The codomain might be continuous, but the image is a set of discrete choices and that's all that matters as far as real world application is concerned. Expecting a voting system to be continuous is in and of itself absurd and not a valid criticism of the system
@@zachariahkindle8926 "the image is a set of discrete choices' Can you clarify this? We don't ever fix the specific election function, so we can't know what the image (of the whole domain, so the "range") is. For all we know the function could be surjective, the range coincides with the codomain, the unit circle
Assuming continuity of results is so flawed it's hilarious. "The result of the vote is that we want 11% of candidate A and 89% candidate B" "Sir, that's... not..." "MAKE IT WORK"
@@brunopargaproportional representation is nice, but in the above case, you would still elect candidate B, even if the average voter disagrees with him 11% of the time. To preserve continuity, the system would not only accurately measure the levels of average voter's approval on every issue, but also allocate authority to each candidate proportional to the popularity of their position in that space. It's practically impossible. Every voter just tries to map their own views onto a party or candidate and hopes that their election promises resemble the effects of their administration.
The presentation is unclear. This function doesn't map voter preferences to a particular candidate. It attempts to map voter preferences (seen as vectors) to a single global preference (also a vector). The winner of the election would be whatever candidate maximises the global preference. For instance the dimensions here might be "welfare state vs. low taxes" and "religious freedom vs. theocracy". Each voter has a preference for these things, like say a person may strongly want a welfare state and be indifferent to religion. We would attempt to takes the preferences of every voter and get a global preference vector that tells you "the country as a whole likes theocracy a lot and secondarily values low taxes a little bit" or something. Continuity just means that if you make a slight change to voter preferences, global preference should change little.
As I understand it, continuity will always break if the decision is discrete (you EITHER choose one president or the other and that could depend on a single vote). The way to patch this is for outcomes to be distributed according to votes (e.g. a parliamentary system where you have representatives proportional to amount of votes. In the perfectly continuous case, each party has "exactly the amount of power" proportional to their votes). And I guess this breaks at "one level up" where if one party has majority representation (regardless of if majority is 50% or 2/3rds) then they become dictatorial and also the votes end up being discontinuous (a single vote pushes the party's power above the majority threshold). Please correct me if I'm on the wrong track :)
Something interesting/funny to consider is what if we assign votes or power according to some probability mass function such that the expected value is proportional with respect to the voting distribution? For example, suppose we have a direct elections and candidate A receives 51% of the vote and B the remaining votes. We'd have some random number generator choose candidate A 51% of the time and B 49%. We'd expect that power would be distributed proportional to the voting preferences of the public in aggregate over many elections. The only problem is if some candidate receives 0.1% of the vote and happens to catch a lucky break (haha).
The problem with this is that when passing policies, the ammount of power a party can have is a discrete space: you either have a majority or you do not. So even if seats are assigned perfectly continuous, power is not.
Well this kind of fails because people dont care (or at least shouldnt) about whos in parliament but rather which policies they enforce. And here this doesnt fulfill if all pi=p then p is the result because youre limited to what the parties offer. Lets say party one wants A,B,C party two (not A),B,C party 3 A,(not B),(not C) but the whole population wants neither A nor B nor C. Then you still dont get the result
@@HopUpOutDaBed single outliers dont break the rule. It's only a break of the rule if all no exceptions vote some way and it has a different outcome. Also neither does removing a single outlier because it's not a preselected person. You could permutate voters and still get the same result
As a mathematics student, i can tell you most of the voting systems in the world suffer from a limited image first before you even get to the impossibility of functions to satisfy fairness axioms.
Saw the veritasium video in my feed. Didn't watch it but went straight to this video's comments to see if they "borrowed" the same ideas.
5 หลายเดือนก่อน +98
The continuity cannot be met: The output will always switch from option A to B directly, discontinuous. You could have 1000 votes on A, and 1000 votes on B, and then just adding 1 vote is enough to select either A or B - the last vote will always have a lot more weight. To solve that your system would need to be able to output 49.5% A + 50.5% B. Some voting systems do this by assigning seats, but because the output is still an integer number and not a full real number, it would still not be continuous.
As long as it's a cardinal voting system, even if there were a continuous variable as output you still get discontinuities because of the boundary of the closed space (like 0 and 360 degrees, if looking at points around a circle)
@@viliml2763 in a mapping from the reals^2 to the reals, where you take mod 360 of your inputs, the output is discontinuous (unless you use trig functions, which sort of do the mod 360 anyway, and then you’ll have singularities where you have to break anonymity to get an output)
You misunderstood the video. The output space is not a discrete list of options, it’s the unit circle, and every possible output of the election function is a unit vector on that circle, just as every vote cast is
I love this channel ... of all the ways I've seen someone explain why "every voting system is wrong" this one is DEFINATELY the most unique and personally satisfying
Loved this, and it comes at a very appropriate time too! We just had elections here in the EU, to elect the new European Parliament, and one thing that stuck with me (and that it isn't being talked about nearly as much as I'd like to) is the general dissatisfaction of people towards voting in general. In my own country (Italy) only about 49.5% of all the voters actually showed up and voted, which makes it the first election in our entire history to have less than 50% attendance. But still, both winners and losers are either celebrating their victory or lamenting their defeat. The most voted party got about 29% and is now celebrating a "landslide victory", but given the overall attendance, that only represents about 13% of the overall population (figures are rounded, I don't have the actual results at hand right now, I'm going by memory). Hardly a victory at all, in my opinion... So yeah, what's the deal with people who don't show up to the ballots? As you said in the video, this looks like a continuity fail of the function, because we literally have missing data from the inputs. How do we deal with them? Should they just be ignored? Or should we somehow take into account the fact that a portion of the voters chose not to vote? Is there even a way to account for this? Either way, it's a huge problem, and the political system seems indifferent to it, which is infuriating to me.
The flip side doesn't sound better. If you ever have the chance to, talk to a Brazilian about mandatory voting. It sounds like there's a lot of "here's a beer, vote for us" on the day of voting to sway the disinterested voting block.
@@RiffZifnab Yeah, both extremes suck. But I don't blame the people. I blame the politics. The most common motivations for non voting are "no one represents me", and "my vote doesn't matter, everyone sucks anyway", and i've seen people fighting because "if you don't show up, it's your fault, you're allowing others to decide for you". But in a system that is increasingly more corrupt, on all sides of the political spectrum, it's easy to see why people would lose confidence in the idea of voting to begin with. Some systems handle the problem by just ignoring their votes, so that an overall 13% with 49% attendance "magically" becomes a 29%, or, as you said, they resort to corruption and bribing to get everyone to vote. But no one actually cares about tackling the real problem, that is that people are becomin increasingly disinterested about politics as a whole.
@@Merione In the Netherlands it is allowed to leave the ballot blank to show you don't agree with any party. While this is great in theory, our EU vote turnout was 46.2%, so... Maybe people just aren't aware this is a thing? Or are they actually disinterested instead of just frustrated?
@@mitsync Here in Italy leaving the ballot blank is allowed as well, but (as far as I know, and feel free to correct me if I'm wrong), blank ballots are still not counted in the overall results. They only count towards the attendance, but the vote percentages themselves remain unaffected. They simply get ignored and the percentages are calculated based only on the valid votes
I *_LOVE_* unexpected uses of the Möbius strip (and other Mathematical super-stars). I have actually deduced one such use, myself; and was super surprised and super happy, at that point: You can represent a Tetrachromatic or a Pentachromatic Spectrum, with a Möbius strip; because they’re ”Doubly Cyclical”, and the Primary and Tertiary Colours (as well, as the Secondary and Quaternary Colours, in the case of the Pentachromatic Spectrum) coincide / occupy the same positions, on them; and thus, to truly show everything that can be shown*, you need to traverse the loop twice, with a slightly different ”point of view” / focus; and what’s the shape you traverse twice, and view from a slightly different ”point of view”? Why, that’s a Möbius strip. * I said: ”all that can be shown”; because you, actually can’t show all the Chromatic colours (14 for the Tetrachromatic, and 30 for the Pentachromatic Spectrum), on these Spectra. I have actually proven Mathematically that, from an n-Chromatic Spectrum (with n Primary Colours), you always need to omit (2^n - 2) - (n² - n) Chromatic Colours (the so-called: ”Aspectral Colours”); and this formula works, for any positive integer n. For example: For n = 3: (2^n - 2) - (n² - n) = (2³ - 2) - (3² - 3) = (8 - 2) - (9 - 3) = 6 - 6 = 0. We don’t need to omit any Chromatic Colours, from our Trichromatic Spectrum. For n = 4: (2^n - 2) - (n² - n) = (2^4 - 2) - (4² - 4) = (16 - 2) - (16 - 4) = 14 - 12 = 2. We need to omit 2 Secondary Colours, from a Tetrachromatic Spectrum. For n = 5: (2^n - 2) - (n² - n) = (2^5 - 2) - (5² - 5) = (32 - 2) - (25 - 5) = 30 - 20 = 10. We need to omit 5 Secondary Colours and 5 Tertiary Colours, from a Pentachromatic Spectrum, totalling at 10 Chromatic Colours.
Maybe I misunderstood the "continuity" part, but if we have a discrete output (let's say outcome A or outcome B), then by definition we can't have continuity, because there isn't a "transitional space" between A and B. So at the 50-50 point we have a jump from A to B. Or am I missing something?
I think the idea is that there’s a (continuous) value function that spits out some overall measure of preference, and then another map that takes this to a discrete result. Small changes in inputs might happen to make this final result jump (by pushing the output past some threshold) but shouldn’t make the value function jump around. In most voting systems though, the inputs themselves would be discrete, and so continuity goes out the window.
You CAN make a binary outcome continuous by introducing probability. For example, if A gets 70% of votes and B gets 30%, then we flip a 70-30 coin to decide which option wins in the end. And honestly I can't see which one of the three rules described in the video this system violates
@@haoyu53 It breaks anonymity. For anonymity you have to ensure that every permutation of a given set of inputs leads to the same result. With your function you can't even ensure that the very same set of inputs, without being non-trivially permutated, always leads to the same result. Or in other words: anonymity (as defined in the video) implies that the function must be deterministic.
I think this is the first time I really regret not being able to give multiple votes through the like button :) What an excellent video! Funny but also interesting, I'm so glad to have stumbled upon this channel!
Honestly, the whole idea feel very flawed because it starts with a non-sensical requirement: Continuousness; if 51 people vote A and 50 vote B, then A should win, but if one person changes their vote from A to B then B should win, which is the smallest possible change in input having the biggest possible change in output; even in a more complicated system, there is always going to be some way to arrange things such that there is a sudden change in result from a minor change in input. As such it feels less "no voting system is perfect" and more "if you add an impossible requirement the whole thing becomes impossible" which feel on par with discovering the Pigeonhole Principle.
I wanna say that you are going to be a great inspiration for all if us, but honestly you already are. I can see your content among the best visual, matter-of-fact, humorous, enjoyable, mesmerizing, relaxing, well-researched, well-put-together, did I say relaxing? content on youtube I know. I hope you realize this, that your effort is what sparks everyone's interest in science and tells the world why to should study science and engineering beyond exams and school and money at all, and I know that it did for me. I hope to be someone with enough talent and skill to similarly contribute my part in return to this world of craziness one day, but until then - kudos my friend, keep up the good work and never stop! Edit : Trusting my phone's keyboard on a long comment wasn't really the best idea.
Lets consider an example: A normalized sum. Under this system, all opinion vectors are added together, and then the result is summed, this sum is normalized back to one, and then the sum is normalized back to length one, returning an opinion vector. Intuitively, this seems to follow most of the requested properties. Unanimity is maintained, because the sum of two parallel vectors is always parallel to those vectors. Anonymity is maintained, due to the commutative property of addition. Even continuity appears like it should be maintained, because vector addition is a continuous function. However, continuity is not actually maintained. This is because while addition is continuous, normalization is not. For any continuous range of inputs, the output is continuous if the inputs do not contain the zero vector. However, while the votes were nonzero, the sum is not necessarily. This means that the outcome is discontinous whenever the vote crosses a tie. Intuitively this makes some sense. How can we fix this? We could consider not normalizing the outcome to 1, and instead use a continuous function to shrink the outcome (ex: sigmoid function or dividing by voter count). This would make the government responsive to how *much* people agree on something. With some political questions this makes sense and can be done, with others it doesn't make sense.
@@UnCavi I know you didn’t read my comment in full, so I’ll summarize my findings. 1: In a trivial system, discontinuities only occur at ties. 2: Discontinuities can be eliminated by expanding the outcome space. (Doing so changes the geometry and breaks the video’s proof.)
@@UnCavi The video wasn’t about ordinal voting either. It was about voting on a 1D circular loop of preferences, which is a weird and specific method of voting. Analogous to a higher dimensional version of a binary voting choice perhaps, but not the same. Other methods don’t necessarily have such problems. Obviously ordinal voting systems are discontinuous, because ordinal numbers aren’t continuous. (Except in the case of a single number? Depends on your definition of continuous.)
@@haph2087 Yes, the video is about ordinal voting. He explains how choosing a point on the unit circle corresponds to an ordinal preference from 2:20 to 4:20. In particular, from 3:07 to 3:43, where at 3:43 he explicitly states that an ordinal preference corresponds to a vector on the unit circle.
Yes, but you could imagine a spectrum of politician. And your voting system would be pick the politician who's at the median or average vector. Or some other "average".
@@poutineausyropderable7108 in this situation would a 'centrist' politician win with 0 votes if the vote was evenly split between the most 'right' and most 'left'. Not necessarily a bad outcome in my opinion, but a funny one.
@@poutineausyropderable7108 Does not matter it would still be discrete lest you mean having an infinite variety of politicians representing the floating point number.
Dude the ISSUE is that it’s inherently discontinuous. What happens to all the kids who didn’t vote for the movie that won and maybe even hate that movie? They shut up and play along to keep the peace OR they have a temper tantrum and get shunned. The very issue with our voting system is the same. It doesn’t let AMERICA win. It lets slightly over half of America win. Sometimes LESS than half.
I'm glad to see someone acknowledging the limitations inherent in voting. I see people making very intelligent videos but yet blindly falling into the painful trap of berating voting systems that don't "weigh all votes equally."
The US's election system does violate one of the conditions, like any other, but the electoral college has nothing to do with it. What it actually violates is continuity, by using a first-past-the-post system..
The "continuity" criteria just seems ill-defined. A lot of people are arguing that it's fundamentally impossible for the simple reason that "one candidate has to win"; while I interpreted it in a way that makes it something that even first-past-the-post satisfies. I think you've made the mistake of taking something that says something generally bad about voting systems and jumped to "that must be a reason to stop using first-past-the-post", when in this case the criteria in question just wasn't useful and didn't say anything about FPTP in particular.
@@Tzizenorec nah I think you misunderstood. I was not necessarily arguing against first past the post, but simply stating what rule it actually violates. True, the way it's described in the video is misleading, but if you look at the theorem this video is based on, Arrow's Impossibility Theorem, that has an actual rigorous usable definition for it, then it applies.
@@David_Box You made it sound like first-past-the-post violates that rule while other voting systems do not. But the whole point of Arrow's Impossibility Theorem (and Gibbard's better impossibility theorem) is that there is no voting system that _doesn't_ violate one of the rules. So once you move to the actual rigorous usable definition... that's where you wind up.
Nice video! Here's how to make formal the idea that the boundary B of a Mobius strip M "wrapping around twice" makes it impossible for it to be a retract of M. B is homeomorphic to the circle, and this homeomorphism induces an isomorphism Z->Z on the fundamental groups, which we can take to be the identity by an appropriate choice of generator. Now, if B were a retract of M, this homeomorphism would extend to a continuous map M -> S^1 (by composing with the retraction). But since the identity in pi1(B) becomes twice the identity in pi1(M) (more formally, the inclusion B -> M induces a multiplication by 2 on the fundamental groups), the induced map pi1(M) -> pi1(S^1) will send 2 to 1, which is impossible (any homomorphism Z->Z maps 2 to something even).
At least psychology tries to prove the assumptions they make about how people behave (take that, Economics!). Anyway, I would imagine that continuity is the obvious failure given that voting seems to be discrete by nature. I would also recommend looking into the public choice branch of economics if you haven't already. Your idea for a 2-note synth would make a killer VST plugin!
Real world eonomists don't treat utility as cardinal, but ordinal. Dunno why he implies the opposite. There is no magnitude information because it's not interpersonally comparable. Person A says they love hot dogs more than Person B does, while B says they like them more... there's no way to prove either of them right, and the field doesn't pretend it can be done.
Psychology has also never proved anything, seeing as how most papers have never been replicated. Actors with free will can't be explained by equations, even with fancy metric spaces, gauge theory, or any other kind of "wiggle room" math.
Thanks for the wonderful video! Just a quick correction for a topological statement in the video. The fact that you're using in this video is that "the boundary of the Mobius strip is not a retract of the Mobius strip." I think you made a mistake of saying 'deformation retract' instead of 'retract.' Thank you.
If you understood the first 15 seconds which are about US presidential elections being held every 4 years, etc., which I have no doubt you could follow without difficulty, you've understood more than 2.3% of the video! 😀
Collecting votes as a 'order these things' is just a really bad way to collect preferences in the first place. I think it was CPG grey who did a video about a voting system where you get to vote yes/no per candidate, which leads to generally more satisfaction of the group as a whole. But a much better approach would be where voters rank each option on a slide scale (or ommit). So you get to represent if you're explicitly pro or against with an option, and how much.
Yes. The idea that the only meaningful information we can aggregate are rankings is ludicrous. This idea is based on an idealistic and individualistic notion of collective decision making that defeats the purpose of it. It's the same issue pointed out in Sen's Liberal Paradox. Ordinal preferences are meaningless if your preferences affect other people's preferences, and that's the entire point of voting.
Liked the video! Thanks for uploading it. I had one thing in my mind, though... The argument holds for 2 voters, but the induction for n was not clear to me. I imagine I would figure it out on my own if I tried, but I also imagine it would add to the video format if it was shown. Keep up the good work!
Lot of comments about the continuity condition being broken when there's a discrete output, and that makes sense - there has to be a discontinuity at the boundary between options. BUT in theory, the theorem says that even for a continuous output, something in there breaks. Which is odd, because it seems like for the case of continuous inputs (votes are a real in [0,1]), if the output is just the mean, it should meet all criteria. Take the case of two voters, x and y: 1) Continuity: (x+y)/2 is continuous 2) Anonymity: (x+y)/2 = (y+x)/2 3) Unanimity: (x+x)/2 = x I think I'm misunderstanding something about the setup. Maybe it HAS to be a discrete output? But then why a continuous input (space of a circle?). And the proof doesn't seem to depend on the exact nature of the output? Maybe has something to do with the input space needing to be closed, i.e. the reals mod 1 instead of [0,1], but how does that change anything? ========================================== EDIT because people seem to be misunderstanding: First, @MaiZhang had it correct: with the reals mod 1, the mean is not continuous for inputs at 0 or 1 (which are the same point), thus this example fails the continuity condition. Second, remember that the vote-input space in this example is a circle, where each point is representing a unit vector in 2-space (this unit vector being the direction of the gradient of the utility function, i.e. the direction that the person's preferred policies "point" in). So the preference space is indeed one-dimensional (a closed loop, and as we saw, is the boundary of a mobius strip), and two combined votes create a 2d space of possible vote pairs (the surface of the mobius strip). But it's not just choosing reals in some interval, which is maybe confusing. It's probably less confusing to say voters choose the angle their unit vector makes with the +x axis -- picking from [0,2pi] (or [0,360] if you're a degrees enjoyer). Which brings me to my next point, which is that the "mean" is a BAD solution for voting. If one person picks 350 degrees, and one person picks 10 degrees, the mean of those two is 180 degrees - almost exactly opposite the direction the voters wanted to travel in. I think people are picturing choosing two points in cartesian space, and then the mean takes you to halfway between those points, which is maybe a good way to compromise - but that's not what's happening.
Correct me if I'm wrong. In the video it is said, that unanimity means, the point in the preference space sits on the boundary of the mobius strip. Therefore for a function to satisfy all the requirements it needs to be a continuous mapping from the strip onto its edge. As they said that's not possible. So even if we take the mean of all votes, the outcome still sits on the boundary of the mobius strip. So the problem persists and we cant find a continuous function from the strip onto its edge which would be required to meet unanimity. Not entirely sure though
The output has to come from the list of inputs, I think. So even if you say voters can choose a real number from [0,1], the output of the voting function has to coincide with one of the numbers being chosen. Thus, the average is not a valid voting function. Or in other words, the output doesn't just have to life in the space of possible choices, but in the subset of choices being made.
If you're taking the mean, then votes close to the mean will have less influence over the result than outlier votes far away from the mean. Could this count as breaking the continuity requirement?
Yes the input space needs to wrap around (like you said, reals mod 1), which is what caused the torus topology. Then the average function breaks continuity at 0 (or 1, which is the same point)
The function you describe is not based on cardinal voting where you put a score on each option and we conclude your direction of preference which will make a spherical input space.
3:36 "once a person votes the strength of the opinion becomes irrelevant" Here is the problem with this: When you have multiple choices to vote for, you can also vote for inbetweens. So it becomes more continuous in a multiple party system. Or in other words: If 1000 kids vote between 100 movies, it will be more fair. A lot of kids will also decide on something that is not only best for them, but best to watch in the hole group, even more compromise.
I was confused the whole time about why you wanted continuity. with only 2 candidates and 99 voters, the outcome changes as soon as one gets 50 votes. there are only 2 outcomes for 50 different inputs. maybe I don't understand what you mean by "continuous"?
The space where the output of the election function lives isn’t a discrete list of options, it’s the unit circle. Every result of an election is a unit vector on that circle
For a discrete list of possible outputs it's obvious that the voting function can't be continous. With 2 candidates you only have 1 issue (candidate A or candidate B, as in 4:08). The preference space would only contain 2 discrete points, not a continuous circle. The video shows that this problem can't be solved by making the inputs continous (e.g. allowing a mix of 50% A and -50% B in the input and output). But I think this is obvious, because people are still free to have discrete preferences.
I just got this suggested by YT algo. I'm at 0:00, looking at a thumbnail. As if it was not enough that video title is "The Topological Problem with Voting", like, come on, the channel name is one of the greatest channel names ever: "Physics for the Birds". Friends, if this is not a nerd heaven then I don't know what is.
I can't believe you mentioned Mobius strips without taking a strip of paper, giving it a twist, and taping the ends together, then running your finger along it to demonstrate that it _only has one side._
The American elections aren't very democratic or fair. Let's say there are 10 people voting for where to eat. 3 of them vote for steak house. The remaining 7 are vegetarians. 2 of them vote for pizza , 2 vote for salad, 2 vote for tofu and 1 for icecream. In American voting system all of them would go to steak house, even only minirity of 3 eat meat. In a multi party system they would go to the food court and the 70% majority would have veggie options, but the carnivore minority would also be catered for.
Thank you mr Bird, now I understand that I use a cardinal value system with a threshold, and if that threshold is not satisfied, I'm not voting for that candidate, regardless of rank.
for example, 3 people voting for 2 options, the vote it's 2:1 and one of the 2 that voted the same change the vote, then the outcome is totally different
The video indeed seems to have used a criterion that just... shouldn't ever be satisfied anyway. But, there are other things that could be described as "continuous" that _do_ make sense. For example, let's say candidate A has 1000 votes and candidate B has 900 votes... "continuous" could mean that if one more person votes, then the gap between A and B has to change by no more than 1 vote (either A is winning by 101 votes, or A is winning by 99 votes, or something in between; A can't suddenly be winning by 50 votes, or losing by 200 votes, because one person voted).
Excellent explanation! I've heard this result mentioned many times, but I never imagined that the proof involved the topology of a Mobius strip - that's pretty wild! As far as fair elections go, perhaps ordinal voting needs to be replaced with a form of cardinal voting, or else having runoff elections, like they do in Europe. I can think of many problems with the voting system in the USA that have nothing to do with this result, by the way, but more to do with human nature, greed and corruption in particular!
You can mathematically prove that the perfect voting system cannot exist but I'm pretty sure you can also prove that the US presidential elections follow the absolute worst system short of a dictatorship.
“The US system except that when tallying electoral votes, the tallest candidate among those that got at least one electoral vote, gets 4 extra electoral votes”
@@drdca8263 dunno. Giving outsized weight to voters from Wyoming or giving outsized weight to candidates with long femurs sounds about equally random to me ;)
@@drdca8263 okay, before we go on I have to make sure that we're both on the same page: this _is_ still a joke, right? Sorry if I'm stating the obvious but on the internet, you never know.
Thank you, Combrade, for this great contribution to our motherland's prosper. You will not be lose to history, forever in our heart despide your now inevitable tragic demise. Your credible account will trule assist in the downfall of the Americans. Best of luck to you, Cimbrade!
When discreetness and sinusoidal collapse seems imminent, I go to the handy dandy Fourier transform. It’s about the peaks, limits, amplitudes, and shifts in direction around a mean.
For the two-voter two-topics example, it just means, if both of them have opposite "preferences" (on these topics), there is no compromise which is also a "preference". Not very surprising.
glad that all the comments are pointing out the discrete nature of voting functions. you're right that it's incorrect to map the opinion space as an n-dimensional vector, because not all opinions are possible. you'd have a circle with holes in it
I had to check the paper for the critical missing definition; this is ONLY valid for ordinal choice. There's an easy counter-example otherwise - the average of a set of vectors satisfies all three properties. It's when you demand that the output to the function is a vector of norm 1 that there's a discontinuity. The vector average function's discontinuity is at the zero vector (every choice is equally preferred)
Very funny and interesting. I think that continuity hypothesis has to be discarded, here the reason: - the function argument is discrete, only a definite number of choices are allowed, and the output, too. - little change in one of the voter's opinion can change the results: in a majoritarian system, when there is a tie, changing the vote of one person will change the result. In a proportional system, changing a few votes will change the tresholds and the distribution of seats.
In the voting literature, most people explain Chichilnisky's theorem using the example of where to have a party. Imagine you and I live along a circular road, and each of us wants to have the party closer to home. A natural way to decide where the party should go is to pick the halfway point between our two homes (or, if we live together, to just have it there). But this rule leads to a discontinuous jump. Suppose I live at 12 o'clock and you live at 6:01, just slightly past the bottom of the circle going clockwise. Then the party will be way over on the left side of the circle (near 9 o'clock), since that's halfway between us. But if you were to move just slightly counter-clockwise -- say, to 5:59 -- the party would have to move way over to the right (near 3 o'clock). An arbitrarily tiny nudge to your preferences can lead to a big leap in our collectively preferred location. And this point generalizes to other anonymous rules that respect unanimity. Interestingly, there is a topological condition that will get us out of the problem. You can have an anonymous, continuous rule that respects unanimity so long as the space of preferences is contractible -- which in this case means a line rather than a circle. (BTW great video.)
I mean, we can see pretty clearly there's a lack of continuity in most voting systems we employ. Either option A wins or option B wins. There's no middleground, which means no continuity.
Regarding the discrete nature of preferences in practical applications, note that you can consider as valid outputs any convex combinations of preferences. For example, if Trump and Biden are the possible preferences, the combination 0.15 Biden + 0.85 Trump Explains how much you prefer one to the other. This is used in may contexts, e.g. in neural networks to allow categorical variables to be treated as continous variables. The categorical result is then extracted by taking the variable with higher coefficient, with a chosen convention when draws occur.
What's interesting is that a lot of people in the comments are focusing on continuity whereas to me the obvious one is unaminity. The biggest simplification for the model is disreagrding how strongly people feel about an issue. Perhaps that's the one that SHOULD get much better utility in a voting system. Marked preference seems to account for this by electing the person most people deem acceptable but it has a few problems: 1. Reliance on vote counting machines (even when processing paper ballots) or accepting it could take several days for the votes to be counted and verified. 2. It can feel somewhat unfair because the person who received the most first preference votes can still loose to second preference if they don't clear the high majority needed. Same can go for 3rd preference, etc. 3. Still leans into party politics at best and tribalism at worst. Whereas most people these days care about issue politics, i.e. how are you going to fix problems X, Y, & Z? I'd like to see what topology something like Quadratic Voting maps onto and what conditions that could satisfy.
When your favorite tool is a topological hammer, every problem starts to look like a higher dimensional nail.
without loss of generality, since a nail is a genus 0 surface but in the video voting was hammered into a genus 1 shape
Good comment
A hypernail
I find it funny how he immediately went to assuming that votes should be continuous because he needed some basis to found his topological problem solving, it is definitely NOT obvious that votes would be continuous, and the simplest way to do it, the popularity vote does not at all work that way
For the sake of the argument I’ll go down to simpler terms, imagine our voting function in which all people are weighted equally, now imagine an election between two movies called 1 and two, now imagine that the x axis is based on the percentage voting for movie two, it starts at 1 and stays completely level until 50%, then it sharply increases to two discontinuously, that can also be extended to more dimensions and the discontinuity stays, that is why the continuity assumption is really quite flawed, I seriously wonder what prompted you to make such an assumption
Not sure how continuity makes sense as a requirement. At a certain point, there has to be a discrete winner, and so there MUST be a point at which flipping one vote will change who/which set of candidates wins, and that just makes sense.
Our systems implement a discrete winner, but they don't have to.
Actually, the discontinuity causes issues in our election systems (let's just take the US one for now, assuming you're American): A third-party candidate in a swing state could side with one of the major parties negotiating outsized influence in return. Outsized in relation to the "few" votes they bring to the "allied" major party.
You could share power on a fractional basis or have the length of the presidency be based on the difference in votes.
It's less obvious to me where continuity would fail there, but the theorem says it does.
Mode is not continuous which tripped me up as soon as he introduced continuity as a criteria
That sounds more like an issue with treating votes as anything other than equal, by design. Parties break that system, because the parties are the things doing the real voting, right? @@TrueMilli
I don't think this is the problem, though. Because in the video he's mapping everyone's opinions on a circle. The last step would be "snapping" the circle to the closest axis to get the mode.
The continuity requirement is in the sense that your opinions can be continuous (to a certain extent, of course), so if you change your opinion slightly, the resulting circle should also change by a small amount. It's not about the final result, which of course has to be discrete, but the step before that.
@@TrueMilli Welll the problem still sort of lies in the fact that is the outcome teaters around 50%, it doesnt matter who gets the edge, the one with the slight lead will get all the power. (i.e. 49.5% red reps and 50.5% blue reps is a vastly different outcome then the other way around.) Because passing policy is a discrete function, power assignment must be disconnected.
I reckon continuity is broken since the elction outcome is discrete and not a dense space. That means the function cant be continuous
The discrete number of votes corresponds to the number of dimensions of the strip, not the continuity of the strip (the video showed two votes mapping onto a two-dimensional mobius strip, three votes would be a three-dimensional space, etc)
@@ajbiffl4695 You're right. I mistook that. It is what everyone else said. The space of all preference-pairings of all voters is dense (assuming ppl have preferences for things that can be quantified by numbers in R) but the election outcome is discrete. So the function cant be continuous. I corrected my comment
A function to a discrete space can be continuous if we define the correct topology on the image space
@@infamedepatates2502 Well no, because 'discrete' already implies a topology. Once we apply a different topology to the discrete space, it may cease being discrete.
This is the problem. My feeling is that this applies to votes that are of a different nature.
Say a jury votes for how long a culprit has to stay in jail and how much money they need to pay. Then everyone chooses a pair of real values. This is a point in a 2D plane. Reducing it to a circle equates to choosing only the ratio between money and jail time (without choosing any of the two absolute amounts).
Then this theorem applies. But this situation is very different from elections...
"We don't care about the magnitude of someone's opinion" goes so hard
At least they care about its direction. Quality over quantity.
If you think about it, the idea of fundamental rights exists precisely because the magnitude of certain opinions matters. If a constitutional system forbids slavery, it doesn't matter if 90% vote to enslave the 10%. That's justified by the idea that the magnitude of the suffering of the enslaved outweighs the common will.
@@WWLinkMasterX What if only one person is enslaved? Seems more like a limit on individual suffering rather than on an aggregate.
@@Transyst I see we've arrived at Omelas.
@@elliotgott2993 Thanks for the reference, wishlisted. And it's from Le Guin, nice.
10:00 "I personally think that the most likely condition to fail is continuity". Theoretically, I think you're right: pragmatically, I think the most likely condition to fail will be that the ballots are of the form "Pick a point on a circle".
Isn't that the same thing?
@@caofan5190
Nope. You could pick from a continuous flower.
Look up R(t)*e^it
And you'll see nice shapes.
Like 3B1B fourrier's drawing.
the most likely situation is fraud :^)
Well, what's wrong with that?
@@caofan5190 The mathematical result is about a function that takes continuously-valued parameters. This corresponds to, say, questions on a questionnaire where you could answer 0% to 100%, and you can use an infinite sequence of digits after the decimal point in your answer to precisely specify your value. Ballots are usually of the form, rank these options from most-preferred to least-preferred, or, select your most preferred option, or other types - but they're all essentially discrete-valued, categorically different from continuously-valued. And that's the way it should be - continuously-valued ballots would be horrendous.
If, in the real world, we can't even get the data collection to match the premise of the theorem, the result about the data processing is moot. At least, in the form shown here.
I like how you delivered "here's a big twist" without fanfare and carried on as if it wasn't amusing or planned.
I would be honored to have my opinions mapped to a topological space
you biden or trump?
@@grimaffiliations3671 no i'm michael
is this the Krusty Krab?
@@robeagleR NO, THIS IS PATRICK
Yes! When I first saw your Möbius strip synth I knew you'd be able to do something really cool with my Synth-a-Sette. Thanks for making with MicroKits!
I find it ironic that video about voting in *two* party system emerges to be about a shape with *one* side.
Kinds like horseshoe theorem?
He presented it in 2 parties for simplicity since it's easy to abstract to n parties (but not easy to make an n dimensional mobius strip)
@@lordzekrom2 he presented it in 2 parties because US elections are soon :)
@@churchboy4609More like Monoparty. You get one group and several different flavors of controlled opposition. Just like in real life!
@@churchboy4609 Ratchet Theory or Ratchet Effect feels more like what Ivan is referring to. Where the duopoly induced by first past the post elections leads to conservative drift as liberal politicians use the excuse of having to chase imagined "centrist" or "moderate" votes to maintain conservative policy positions that just happen to also reinforce their upper class family economically.
At the part with the two "ch" sounds in a row I noticed you pronounce "ch", "dj" "sh", "zh" in a completely different way than most people. I think your "sh" is a voiceless velar lateral fricative, with "zh" being a voiced velar lateral fricative, "ch" a voiceless velar lateral affricate and "dj" a voiced velar lateral affricate. I think they all have audio files on Wikipedia. For the common pronunciation of these you can swap "velar lateral" with "postalveolar" or "postalveolar sibilant", those also have audio. Cheers
I used to have this exact pronunciation until I "corrected" it and can confirm, velar lateral.
I love to stumble across comments referencing linguistics in the wild.
A phonetics comment in a maths video? I love this
yay, people have accents and dialects.
And yes, it's absolutely possible for people to pronounce "god", "guard" and "gourd" the exact same way.
This video is kinda unique
It's 1 minute old and the Video is 10 minutes long
Nice profile pic
Good pfp
oi, tudo otimo, e vc?
Does uniqueness have magnitude?
I'd like to offer another topological perspective:
Let's say each voter is choosing between n options, where 1
I came here expecting a discussion of the topographical analysis of voting. With some conclusion being along the lines of "put your polling center at sea level or lower to get the best turnout for your district. Nobody wants to go uphill to vote."
"Back in my day, we went to vote uphill, both ways."
😆
@@Transyst underrated comment
If you take a real vector space and quotient by scaling by a positive constant, you don't just get a sphere. There is also the vector 0 that can't be scaled to any other vector. This is the utility function that represents indifference between all the options.
It seems like the theorem cheats by not allowing this point as a possible output (it's easy to accommodate it as an input, just ignore any indifferent voter). In particular, in the two voter case you could say that the overall utility is indifferent precisely when the two voters are diametrically opposed. This has the effect of cutting out the centre line from the Möbius strip, whereupon the rest of it does indeed retract onto the boundary.
I think in order to have a mapping that covers the exact-opposites case, then you have to throw out the anonymity condition - someone's vote will have to matter less, just so that you can get back to the circle
@@ajbiffl4695how so?
You are misunderstanding what we are doing here. We are not "quotienting the vector space by scaling by a positive constant". And there is no vector 0, because there's no indifference curve whose oriented normal vector is the vector 0.
And the vector 0 would not represent "indifference between all the options". That vector already exists in the construction shown in the video, and its the unit vector forming a 45° angle with the positive horizontal axis
@@UnCavi I don't think that's right. The unit vector represents the direction of the gradient of the utility function. If someone has no preference the utility function is flat, and the gradient is zero. There are no indifference curves, it's an indifference plane.
@@ajbiffl4695 The example shown in the video, where the indifference curves are diagonal straight lines with slope -1, represents a situation where the voter preference doesn’t lean towards either option (fries, burger). Take for example the line with equation y + x = 1: moving along this line, we get points where we either maximally support “fries” and give 0 points to “burger”, or we maximally support “burger” and give 0 points to “fries”, or every combination in between, with equal weight: this is exactly what it means to be indifferent to each candidate. And the normal unit vector to these indifference lines is a 45° vector on the unit circle
What you’re talking about with your “indifference plane” would be a constant function with 0 gradient at every point, and while yes, this would also correspond to some kind of indifference form the voter perspective, we are not interested in including this scenario in our description , because it doesn’t add anything useful as there is already another way to express indifference with a non-zero gradient. Also, a constant function would just be “not voting”, so we don’t count them. And if one wants to give their vote equally to each candidate, giving a “null” preference, they can choose the unit vector 45° to the horizontal
I saw a Google Tech Talk about this years ago and somewhat understood it. They also went into details about which voting systems broke which fairness principles. But I have to say your topological explanation makes more sense and I think the reasoning will stick with me longer. Alas, I still can't properly explain it to people at parties. Unless I carry around props.
The solution, ofc, is to carry a Möbius strip with you at all times!
@@davidshi451 What's your EDC mobius strip? (:
Check out the Mr Beat (not Beast) video about voting systems.
Never thought I'd see a video on the physics of voting
Well, math, but same difference
Neither did I, but here we are😂😂
you watched a whole video about topology?
try and read about the "Voter Model", its variants and computational social sciences in general :D
I actually think the most likely condition to fail is 'ordinal voting'. If you have only two voters with exactly equal and opposite preferences, the optimal election result would be no change at all, rather than a change in a completely orthogonal direction.
"Then I guess there's nothing for dinner and we just die of stupid, honey"
But "no change at all" has a length of zero. It is not part of the preference space. So instead, the tiniest differences get magnified. When 50 people vote left and 50 people vote right, then 1 vote for backwards would win the election.
in reality it requires AT LEAST 2 against 1 to ENFORCE the will of the (social) "majority" onto the (asocial) "minority" with an obvious outcome (so the minority accepts defeat without "testing" the resilience of the majority).
Voting systems that are based on simple majorities (50%+1) are not natural and socially unsustainable.. systems based on 2:1, 3:1, 4:1 etc. pp. on will result in more natural outcomes and obviously ever lower common goals being enforced ON ALL - which is a feature and NOT a bug.
This sounds right. The problem is that there is no "obvious" compromise between opposed preferences in the space of unit vectors. A usual way to make sure that compromises exist is to assume that the choice space is convex. But the circle is not even contractible. What the theorem is saying seems to be just that democracy requires that compromises are available. It doesn't sound like an amazing insight, but it's reasonable. It says nothing about the viability of democracy over decisions that do admit compromises, though (in that case Arrow's theorem and its variants apply and provide the powerful and nontrivial insight that the idea of democracy is not a clear guide if there at least two dimensions of disagreement).
what u people forget is that 'democracy' is the "peaceful" path by which opposing goals are being solved, which normally - in wilderness - are being solved via force, via physically overcoming the opposing party.
This means in a 1:1 scenario democracy CAN NOT solve the problem, because it is NOT CLEAR who will "win" this 'conflict'.
Once we get to 2:1 or better it is much more clearer which 'opinion' will be inferior if the rubber hits the road.. which is the way by which MINORITIES knuckle under and stay 'peaceful'.
Democracies are minority suppressing mechanisms that allow for peaceful solutions, if there is a 2:1 or better majority behind the preference.
Any democracy that allows for less than 2:1 preferences to become the rule for all WILL sooner or later fail as it is CATERING to a non-capable 'minority' against the 'majority'.
My initial reaction was "Mobius strip and voting? Oh yeah, two sides that are actually one..."
"obviously voting for the 99% Hitler is better than the 100% Hitler, there's absolutely nothing else to do other than to hope 99% Hitler dies in next 4 years and we vote for someone less than 100% Hitler in 2028"
-Liberal westerner
I was so distracted by the anthem played on a two-axis keyboard that I had to replay the intro multiple times! Such a great execution of an odd concept!
2:04 "it is one of the most unexpected uses of a mobius strip, and here's the big twist" XD
I think the mistake is that in reality, you cannot vote against a party (except by only voting for the other ones). At 4:14, there is only one party, but you can still vote against. In reality, having only one party would have only one voting option.
To fix this, we should only have a positive vote, creating a quarter circle (in 2D). A quarter circle doesn't wrap around, so I don't think realistic voting is a möbius strip.
Fascinating application of Mathematics, which I like to see. I would say on the topic of the continuity supposition, though, that I don't think we would intuit voting functions to be continuous. Results in a election are in their actual expression always form a finite space. For example in a FPTP electorate the space of results will be exactly the candidates that run, and the result could move from one candidate to another by the change of one vote. The same applies just with more discontinuities in say a MMP election for a legislature (e.g. Germany, NZ, etc..) because of the nature of what the result is.
Perfect timing! I just watched every video on this channel a couple days ago.
oh shoot Nicky Case also made a small interactive thingy about voting ages ago.
I'm with Nicky, scoring each candidate looks cool and more representative than ranking, which in turn is more representative than just pick one.
Yep. The assumption that voting is about ordinal preferences (rankings) is probably the biggest blunder in social choice theory, making it basically useless in practice.
@@1ucasvbOrdinal preferences at least have a clear meaning. The meaning of a scalar seems, unclear?
Like, how do I determine what an “honest vote” would be so that I can vote honestly?
But scoring just encourages voters to give all the candidates they can tolerate the maximum score and give all the candidates they hate the minimum score. Even if the system somehow accounts for this, loads of people will do it anyway.
@@sir_fuzzball1486 if the scores are either -1, 0, or 1, then that seems fine?
@@sir_fuzzball1486 well yeah it is possible for it to devolve into that, but that would also be better than saying yes to one guy and that's it.
with score you can say yes and no to multiple people.
you can say varying amounts of yes or no.
i think having something that is more representative its better.
as a maths major, I'm disappointed at the number of people who don't treat my opinion as an n dimensional vector
Continuity makes sense, Voting is not continuous in many cases. If we have a bunch of primary schoolers voting for pineapple or no pineapple on pizza, then it is possible for one vote to take us from 0% pineapple to 100% pineapple a discontinuity if there ever was one. (say you have 11 kids and 5 vote for No Pineapple and 6 vote for pineapple, resulting in a pizza that has pineapple if one kid changed their vote we would shift from 5-6 to 6-5 for No pineapple)
Continuity can only ever make sense in voting schemes where blended results are possible. which does not generally represent most votes
I still think we should have multi-member districts for duverger's law, and approval based voting (for the largest overlap of expectations to results)
I think it's perfectly fine to model opinions as multi-dimensional vectors, it just doesn't make any sense to expect that mapping those opinions onto a finite list could ever be continuous. In fact, given the fact that the results space is not densely populated, it boggles me to understand why anyone would think voting could *ever* be continuous, regardless of other factors.
You misunderstood the video. We are not mapping the input opinions onto a finite list: the codomain of the voting function is the unit circle, and the output is a unit preference vector, just as any vote cast is.
@UnCavi nah that's a fine understanding of the video, because the assumption is, in and of itself, moronic. The codomain might be continuous, but the image is a set of discrete choices and that's all that matters as far as real world application is concerned. Expecting a voting system to be continuous is in and of itself absurd and not a valid criticism of the system
@@zachariahkindle8926 "the image is a set of discrete choices'
Can you clarify this? We don't ever fix the specific election function, so we can't know what the image (of the whole domain, so the "range") is. For all we know the function could be surjective, the range coincides with the codomain, the unit circle
Assuming continuity of results is so flawed it's hilarious.
"The result of the vote is that we want 11% of candidate A and 89% candidate B"
"Sir, that's... not..."
"MAKE IT WORK"
You mean, proportional representation?
@@brunopargain an vote with a single winner?
@@brunopargaproportional representation is nice, but in the above case, you would still elect candidate B, even if the average voter disagrees with him 11% of the time. To preserve continuity, the system would not only accurately measure the levels of average voter's approval on every issue, but also allocate authority to each candidate proportional to the popularity of their position in that space.
It's practically impossible. Every voter just tries to map their own views onto a party or candidate and hopes that their election promises resemble the effects of their administration.
@@brunoparga even that deals with discrete people
The presentation is unclear. This function doesn't map voter preferences to a particular candidate. It attempts to map voter preferences (seen as vectors) to a single global preference (also a vector). The winner of the election would be whatever candidate maximises the global preference.
For instance the dimensions here might be "welfare state vs. low taxes" and "religious freedom vs. theocracy". Each voter has a preference for these things, like say a person may strongly want a welfare state and be indifferent to religion. We would attempt to takes the preferences of every voter and get a global preference vector that tells you "the country as a whole likes theocracy a lot and secondarily values low taxes a little bit" or something. Continuity just means that if you make a slight change to voter preferences, global preference should change little.
In your example the voting was continuous, but we have discrete options, I think this changes the problem
As I understand it, continuity will always break if the decision is discrete (you EITHER choose one president or the other and that could depend on a single vote). The way to patch this is for outcomes to be distributed according to votes (e.g. a parliamentary system where you have representatives proportional to amount of votes. In the perfectly continuous case, each party has "exactly the amount of power" proportional to their votes).
And I guess this breaks at "one level up" where if one party has majority representation (regardless of if majority is 50% or 2/3rds) then they become dictatorial and also the votes end up being discontinuous (a single vote pushes the party's power above the majority threshold). Please correct me if I'm on the wrong track :)
Something interesting/funny to consider is what if we assign votes or power according to some probability mass function such that the expected value is proportional with respect to the voting distribution?
For example, suppose we have a direct elections and candidate A receives 51% of the vote and B the remaining votes. We'd have some random number generator choose candidate A 51% of the time and B 49%. We'd expect that power would be distributed proportional to the voting preferences of the public in aggregate over many elections.
The only problem is if some candidate receives 0.1% of the vote and happens to catch a lucky break (haha).
The problem with this is that when passing policies, the ammount of power a party can have is a discrete space: you either have a majority or you do not. So even if seats are assigned perfectly continuous, power is not.
Well this kind of fails because people dont care (or at least shouldnt) about whos in parliament but rather which policies they enforce. And here this doesnt fulfill if all pi=p then p is the result because youre limited to what the parties offer. Lets say party one wants A,B,C party two (not A),B,C party 3 A,(not B),(not C) but the whole population wants neither A nor B nor C. Then you still dont get the result
@@HopUpOutDaBed single outliers dont break the rule. It's only a break of the rule if all no exceptions vote some way and it has a different outcome.
Also neither does removing a single outlier because it's not a preselected person. You could permutate voters and still get the same result
@@HopUpOutDaBed i didnt bring a scenario where it doesn't work i only repeated the necessary condition for a rule break which is never fulfilled
It's hard to find good political video that is unbiased.
Thanks for that ❤
I have literally no idea what you’re talking about but I absolutely love this video.
As a mathematics student, i can tell you most of the voting systems in the world suffer from a limited image first before you even get to the impossibility of functions to satisfy fairness axioms.
Would be helpful to have some kind of units or descriptions for the axis. I lost track of what is happening around half mark of that video.
Maybe I need to watch your other videos to get some context.
Saw the veritasium video in my feed. Didn't watch it but went straight to this video's comments to see if they "borrowed" the same ideas.
The continuity cannot be met: The output will always switch from option A to B directly, discontinuous. You could have 1000 votes on A, and 1000 votes on B, and then just adding 1 vote is enough to select either A or B - the last vote will always have a lot more weight. To solve that your system would need to be able to output 49.5% A + 50.5% B. Some voting systems do this by assigning seats, but because the output is still an integer number and not a full real number, it would still not be continuous.
As long as it's a cardinal voting system, even if there were a continuous variable as output you still get discontinuities because of the boundary of the closed space (like 0 and 360 degrees, if looking at points around a circle)
But imagine a continuous set of politician and the voting system is nearest neighbor of the median vector.
@@ajbiffl4695 0 and 360 are the same thing as angles so that still counts as continuous
@@viliml2763 in a mapping from the reals^2 to the reals, where you take mod 360 of your inputs, the output is discontinuous (unless you use trig functions, which sort of do the mod 360 anyway, and then you’ll have singularities where you have to break anonymity to get an output)
You misunderstood the video. The output space is not a discrete list of options, it’s the unit circle, and every possible output of the election function is a unit vector on that circle, just as every vote cast is
I love this channel ... of all the ways I've seen someone explain why "every voting system is wrong" this one is DEFINATELY the most unique and personally satisfying
Loved this, and it comes at a very appropriate time too! We just had elections here in the EU, to elect the new European Parliament, and one thing that stuck with me (and that it isn't being talked about nearly as much as I'd like to) is the general dissatisfaction of people towards voting in general. In my own country (Italy) only about 49.5% of all the voters actually showed up and voted, which makes it the first election in our entire history to have less than 50% attendance. But still, both winners and losers are either celebrating their victory or lamenting their defeat. The most voted party got about 29% and is now celebrating a "landslide victory", but given the overall attendance, that only represents about 13% of the overall population (figures are rounded, I don't have the actual results at hand right now, I'm going by memory). Hardly a victory at all, in my opinion... So yeah, what's the deal with people who don't show up to the ballots? As you said in the video, this looks like a continuity fail of the function, because we literally have missing data from the inputs. How do we deal with them? Should they just be ignored? Or should we somehow take into account the fact that a portion of the voters chose not to vote? Is there even a way to account for this? Either way, it's a huge problem, and the political system seems indifferent to it, which is infuriating to me.
The flip side doesn't sound better. If you ever have the chance to, talk to a Brazilian about mandatory voting. It sounds like there's a lot of "here's a beer, vote for us" on the day of voting to sway the disinterested voting block.
@@RiffZifnab Yeah, both extremes suck. But I don't blame the people. I blame the politics. The most common motivations for non voting are "no one represents me", and "my vote doesn't matter, everyone sucks anyway", and i've seen people fighting because "if you don't show up, it's your fault, you're allowing others to decide for you". But in a system that is increasingly more corrupt, on all sides of the political spectrum, it's easy to see why people would lose confidence in the idea of voting to begin with. Some systems handle the problem by just ignoring their votes, so that an overall 13% with 49% attendance "magically" becomes a 29%, or, as you said, they resort to corruption and bribing to get everyone to vote. But no one actually cares about tackling the real problem, that is that people are becomin increasingly disinterested about politics as a whole.
Democracy is a inherently flawed system
@@Merione In the Netherlands it is allowed to leave the ballot blank to show you don't agree with any party. While this is great in theory, our EU vote turnout was 46.2%, so...
Maybe people just aren't aware this is a thing? Or are they actually disinterested instead of just frustrated?
@@mitsync Here in Italy leaving the ballot blank is allowed as well, but (as far as I know, and feel free to correct me if I'm wrong), blank ballots are still not counted in the overall results. They only count towards the attendance, but the vote percentages themselves remain unaffected. They simply get ignored and the percentages are calculated based only on the valid votes
I *_LOVE_* unexpected uses of the Möbius strip (and other Mathematical super-stars). I have actually deduced one such use, myself; and was super surprised and super happy, at that point: You can represent a Tetrachromatic or a Pentachromatic Spectrum, with a Möbius strip; because they’re ”Doubly Cyclical”, and the Primary and Tertiary Colours (as well, as the Secondary and Quaternary Colours, in the case of the Pentachromatic Spectrum) coincide / occupy the same positions, on them; and thus, to truly show everything that can be shown*, you need to traverse the loop twice, with a slightly different ”point of view” / focus; and what’s the shape you traverse twice, and view from a slightly different ”point of view”? Why, that’s a Möbius strip.
* I said: ”all that can be shown”; because you, actually can’t show all the Chromatic colours (14 for the Tetrachromatic, and 30 for the Pentachromatic Spectrum), on these Spectra. I have actually proven Mathematically that, from an n-Chromatic Spectrum (with n Primary Colours), you always need to omit (2^n - 2) - (n² - n) Chromatic Colours (the so-called: ”Aspectral Colours”); and this formula works, for any positive integer n. For example:
For n = 3:
(2^n - 2) - (n² - n) = (2³ - 2) - (3² - 3) =
(8 - 2) - (9 - 3) = 6 - 6 = 0.
We don’t need to omit any Chromatic Colours, from our Trichromatic Spectrum.
For n = 4:
(2^n - 2) - (n² - n) = (2^4 - 2) - (4² - 4) =
(16 - 2) - (16 - 4) = 14 - 12 = 2.
We need to omit 2 Secondary Colours, from a Tetrachromatic Spectrum.
For n = 5:
(2^n - 2) - (n² - n) = (2^5 - 2) - (5² - 5) =
(32 - 2) - (25 - 5) = 30 - 20 = 10.
We need to omit 5 Secondary Colours and 5 Tertiary Colours, from a Pentachromatic Spectrum, totalling at 10 Chromatic Colours.
Maybe I misunderstood the "continuity" part, but if we have a discrete output (let's say outcome A or outcome B), then by definition we can't have continuity, because there isn't a "transitional space" between A and B. So at the 50-50 point we have a jump from A to B. Or am I missing something?
I think the idea is that there’s a (continuous) value function that spits out some overall measure of preference, and then another map that takes this to a discrete result. Small changes in inputs might happen to make this final result jump (by pushing the output past some threshold) but shouldn’t make the value function jump around.
In most voting systems though, the inputs themselves would be discrete, and so continuity goes out the window.
You CAN make a binary outcome continuous by introducing probability. For example, if A gets 70% of votes and B gets 30%, then we flip a 70-30 coin to decide which option wins in the end. And honestly I can't see which one of the three rules described in the video this system violates
@@maxchemtov3482the discreteness of the input is negligible when you have infinite voters, right?
@@haoyu53 It breaks anonymity. For anonymity you have to ensure that every permutation of a given set of inputs leads to the same result. With your function you can't even ensure that the very same set of inputs, without being non-trivially permutated, always leads to the same result.
Or in other words: anonymity (as defined in the video) implies that the function must be deterministic.
@@lonestarr1490 The result of the function is a unit vector, not a single winner candidate
I think this is the first time I really regret not being able to give multiple votes through the like button :)
What an excellent video! Funny but also interesting, I'm so glad to have stumbled upon this channel!
....w-what?
Thanks for making me knoe why after every vote, I feel somewhat unsatisfied withe the result no matter what the result/my-vote combination was.
“ and here’s the big twist“ 👏👏👏
I wrote an essay on this in my 2nd year of uni and forgot all of it since, good stuff
Honestly, the whole idea feel very flawed because it starts with a non-sensical requirement: Continuousness; if 51 people vote A and 50 vote B, then A should win, but if one person changes their vote from A to B then B should win, which is the smallest possible change in input having the biggest possible change in output; even in a more complicated system, there is always going to be some way to arrange things such that there is a sudden change in result from a minor change in input. As such it feels less "no voting system is perfect" and more "if you add an impossible requirement the whole thing becomes impossible" which feel on par with discovering the Pigeonhole Principle.
I wanna say that you are going to be a great inspiration for all if us, but honestly you already are. I can see your content among the best visual, matter-of-fact, humorous, enjoyable, mesmerizing, relaxing, well-researched, well-put-together, did I say relaxing? content on youtube I know. I hope you realize this, that your effort is what sparks everyone's interest in science and tells the world why to should study science and engineering beyond exams and school and money at all, and I know that it did for me. I hope to be someone with enough talent and skill to similarly contribute my part in return to this world of craziness one day, but until then - kudos my friend, keep up the good work and never stop!
Edit : Trusting my phone's keyboard on a long comment wasn't really the best idea.
Lets consider an example:
A normalized sum.
Under this system, all opinion vectors are added together, and then the result is summed, this sum is normalized back to one, and then the sum is normalized back to length one, returning an opinion vector.
Intuitively, this seems to follow most of the requested properties. Unanimity is maintained, because the sum of two parallel vectors is always parallel to those vectors. Anonymity is maintained, due to the commutative property of addition. Even continuity appears like it should be maintained, because vector addition is a continuous function.
However, continuity is not actually maintained. This is because while addition is continuous, normalization is not. For any continuous range of inputs, the output is continuous if the inputs do not contain the zero vector. However, while the votes were nonzero, the sum is not necessarily. This means that the outcome is discontinous whenever the vote crosses a tie.
Intuitively this makes some sense.
How can we fix this? We could consider not normalizing the outcome to 1, and instead use a continuous function to shrink the outcome (ex: sigmoid function or dividing by voter count). This would make the government responsive to how *much* people agree on something. With some political questions this makes sense and can be done, with others it doesn't make sense.
Whatever way you think you found for fixing continuity can't work, because of the conclusion of the theorem shown in the video
@@UnCavi I know you didn’t read my comment in full, so I’ll summarize my findings.
1: In a trivial system, discontinuities only occur at ties.
2: Discontinuities can be eliminated by expanding the outcome space. (Doing so changes the geometry and breaks the video’s proof.)
@@haph2087 I've read your comment.
Changing the output space means that we're not talking about ordinal voting anymore
@@UnCavi The video wasn’t about ordinal voting either. It was about voting on a 1D circular loop of preferences, which is a weird and specific method of voting. Analogous to a higher dimensional version of a binary voting choice perhaps, but not the same.
Other methods don’t necessarily have such problems.
Obviously ordinal voting systems are discontinuous, because ordinal numbers aren’t continuous. (Except in the case of a single number? Depends on your definition of continuous.)
@@haph2087 Yes, the video is about ordinal voting. He explains how choosing a point on the unit circle corresponds to an ordinal preference from 2:20 to 4:20. In particular, from 3:07 to 3:43, where at 3:43 he explicitly states that an ordinal preference corresponds to a vector on the unit circle.
Your last sentence got me in the feels as I did my bsc thesis on the axelrod model, which does exactly that ;_;
I see that Veritasium just copied this videos exact subject. A bit weird.
Yeah, weird indeed. At least they changed the content this time!
Amazing, what an amazing video! Had zero idea about how topology could be related to voting theory!! Thanks for this fantastic explanation.
I mean, if more than half of the kids vote on a movie then it is definitive, so that's a point of discontinuity. It's not as arbitrary as you say.
Yes, but you could imagine a spectrum of politician.
And your voting system would be pick the politician who's at the median or average vector. Or some other "average".
@@poutineausyropderable7108 in this situation would a 'centrist' politician win with 0 votes if the vote was evenly split between the most 'right' and most 'left'.
Not necessarily a bad outcome in my opinion, but a funny one.
@@Rb61926exactly, you went from satisfying the majority of the voters to satisfying no one
@@poutineausyropderable7108 Does not matter it would still be discrete lest you mean having an infinite variety of politicians representing the floating point number.
Dude the ISSUE is that it’s inherently discontinuous.
What happens to all the kids who didn’t vote for the movie that won and maybe even hate that movie? They shut up and play along to keep the peace OR they have a temper tantrum and get shunned.
The very issue with our voting system is the same. It doesn’t let AMERICA win. It lets slightly over half of America win. Sometimes LESS than half.
Me who understood 0% of the video and can't figure out what the hell this has to do with voting
I'm glad to see someone acknowledging the limitations inherent in voting. I see people making very intelligent videos but yet blindly falling into the painful trap of berating voting systems that don't "weigh all votes equally."
The US's election system does violate one of the conditions, like any other, but the electoral college has nothing to do with it. What it actually violates is continuity, by using a first-past-the-post system..
The "continuity" criteria just seems ill-defined. A lot of people are arguing that it's fundamentally impossible for the simple reason that "one candidate has to win"; while I interpreted it in a way that makes it something that even first-past-the-post satisfies.
I think you've made the mistake of taking something that says something generally bad about voting systems and jumped to "that must be a reason to stop using first-past-the-post", when in this case the criteria in question just wasn't useful and didn't say anything about FPTP in particular.
@@Tzizenorec nah I think you misunderstood. I was not necessarily arguing against first past the post, but simply stating what rule it actually violates. True, the way it's described in the video is misleading, but if you look at the theorem this video is based on, Arrow's Impossibility Theorem, that has an actual rigorous usable definition for it, then it applies.
@@David_Box You made it sound like first-past-the-post violates that rule while other voting systems do not. But the whole point of Arrow's Impossibility Theorem (and Gibbard's better impossibility theorem) is that there is no voting system that _doesn't_ violate one of the rules.
So once you move to the actual rigorous usable definition... that's where you wind up.
@@Tzizenorec yeah I know that all systems violate at least one rule, hence why the original comment said "like any other"
Nice video! Here's how to make formal the idea that the boundary B of a Mobius strip M "wrapping around twice" makes it impossible for it to be a retract of M. B is homeomorphic to the circle, and this homeomorphism induces an isomorphism Z->Z on the fundamental groups, which we can take to be the identity by an appropriate choice of generator. Now, if B were a retract of M, this homeomorphism would extend to a continuous map M -> S^1 (by composing with the retraction). But since the identity in pi1(B) becomes twice the identity in pi1(M) (more formally, the inclusion B -> M induces a multiplication by 2 on the fundamental groups), the induced map pi1(M) -> pi1(S^1) will send 2 to 1, which is impossible (any homomorphism Z->Z maps 2 to something even).
At least psychology tries to prove the assumptions they make about how people behave (take that, Economics!). Anyway, I would imagine that continuity is the obvious failure given that voting seems to be discrete by nature. I would also recommend looking into the public choice branch of economics if you haven't already. Your idea for a 2-note synth would make a killer VST plugin!
Real world eonomists don't treat utility as cardinal, but ordinal. Dunno why he implies the opposite. There is no magnitude information because it's not interpersonally comparable.
Person A says they love hot dogs more than Person B does, while B says they like them more... there's no way to prove either of them right, and the field doesn't pretend it can be done.
Psychology has also never proved anything, seeing as how most papers have never been replicated. Actors with free will can't be explained by equations, even with fancy metric spaces, gauge theory, or any other kind of "wiggle room" math.
Thanks for the wonderful video! Just a quick correction for a topological statement in the video. The fact that you're using in this video is that "the boundary of the Mobius strip is not a retract of the Mobius strip." I think you made a mistake of saying 'deformation retract' instead of 'retract.' Thank you.
It's rare that I find a youtube video I understand absolutely 0% of lol
If you understood the first 15 seconds which are about US presidential elections being held every 4 years, etc., which I have no doubt you could follow without difficulty, you've understood more than 2.3% of the video! 😀
@@jensraab2902 not more math😞
@@wisconsinengines 😂
Same lol
Amazing topological explanation and visualizations. Great job!!! 👏
Collecting votes as a 'order these things' is just a really bad way to collect preferences in the first place. I think it was CPG grey who did a video about a voting system where you get to vote yes/no per candidate, which leads to generally more satisfaction of the group as a whole. But a much better approach would be where voters rank each option on a slide scale (or ommit). So you get to represent if you're explicitly pro or against with an option, and how much.
Yes. The idea that the only meaningful information we can aggregate are rankings is ludicrous. This idea is based on an idealistic and individualistic notion of collective decision making that defeats the purpose of it. It's the same issue pointed out in Sen's Liberal Paradox. Ordinal preferences are meaningless if your preferences affect other people's preferences, and that's the entire point of voting.
Still better than first past the post
Liked the video! Thanks for uploading it.
I had one thing in my mind, though... The argument holds for 2 voters, but the induction for n was not clear to me. I imagine I would figure it out on my own if I tried, but I also imagine it would add to the video format if it was shown.
Keep up the good work!
Lot of comments about the continuity condition being broken when there's a discrete output, and that makes sense - there has to be a discontinuity at the boundary between options.
BUT in theory, the theorem says that even for a continuous output, something in there breaks.
Which is odd, because it seems like for the case of continuous inputs (votes are a real in [0,1]), if the output is just the mean, it should meet all criteria. Take the case of two voters, x and y:
1) Continuity: (x+y)/2 is continuous
2) Anonymity: (x+y)/2 = (y+x)/2
3) Unanimity: (x+x)/2 = x
I think I'm misunderstanding something about the setup. Maybe it HAS to be a discrete output? But then why a continuous input (space of a circle?). And the proof doesn't seem to depend on the exact nature of the output?
Maybe has something to do with the input space needing to be closed, i.e. the reals mod 1 instead of [0,1], but how does that change anything?
==========================================
EDIT because people seem to be misunderstanding:
First, @MaiZhang had it correct: with the reals mod 1, the mean is not continuous for inputs at 0 or 1 (which are the same point), thus this example fails the continuity condition.
Second, remember that the vote-input space in this example is a circle, where each point is representing a unit vector in 2-space (this unit vector being the direction of the gradient of the utility function, i.e. the direction that the person's preferred policies "point" in). So the preference space is indeed one-dimensional (a closed loop, and as we saw, is the boundary of a mobius strip), and two combined votes create a 2d space of possible vote pairs (the surface of the mobius strip). But it's not just choosing reals in some interval, which is maybe confusing. It's probably less confusing to say voters choose the angle their unit vector makes with the +x axis -- picking from [0,2pi] (or [0,360] if you're a degrees enjoyer).
Which brings me to my next point, which is that the "mean" is a BAD solution for voting. If one person picks 350 degrees, and one person picks 10 degrees, the mean of those two is 180 degrees - almost exactly opposite the direction the voters wanted to travel in. I think people are picturing choosing two points in cartesian space, and then the mean takes you to halfway between those points, which is maybe a good way to compromise - but that's not what's happening.
Correct me if I'm wrong. In the video it is said, that unanimity means, the point in the preference space sits on the boundary of the mobius strip. Therefore for a function to satisfy all the requirements it needs to be a continuous mapping from the strip onto its edge. As they said that's not possible. So even if we take the mean of all votes, the outcome still sits on the boundary of the mobius strip. So the problem persists and we cant find a continuous function from the strip onto its edge which would be required to meet unanimity. Not entirely sure though
The output has to come from the list of inputs, I think. So even if you say voters can choose a real number from [0,1], the output of the voting function has to coincide with one of the numbers being chosen. Thus, the average is not a valid voting function.
Or in other words, the output doesn't just have to life in the space of possible choices, but in the subset of choices being made.
If you're taking the mean, then votes close to the mean will have less influence over the result than outlier votes far away from the mean. Could this count as breaking the continuity requirement?
Yes the input space needs to wrap around (like you said, reals mod 1), which is what caused the torus topology. Then the average function breaks continuity at 0 (or 1, which is the same point)
The function you describe is not based on cardinal voting where you put a score on each option and we conclude your direction of preference which will make a spherical input space.
3:36 "once a person votes the strength of the opinion becomes irrelevant"
Here is the problem with this: When you have multiple choices to vote for, you can also vote for inbetweens. So it becomes more continuous in a multiple party system. Or in other words: If 1000 kids vote between 100 movies, it will be more fair. A lot of kids will also decide on something that is not only best for them, but best to watch in the hole group, even more compromise.
I was confused the whole time about why you wanted continuity.
with only 2 candidates and 99 voters, the outcome changes as soon as one gets 50 votes. there are only 2 outcomes for 50 different inputs.
maybe I don't understand what you mean by "continuous"?
The space where the output of the election function lives isn’t a discrete list of options, it’s the unit circle. Every result of an election is a unit vector on that circle
@@UnCavi those are the inputs, though
For a discrete list of possible outputs it's obvious that the voting function can't be continous. With 2 candidates you only have 1 issue (candidate A or candidate B, as in 4:08). The preference space would only contain 2 discrete points, not a continuous circle. The video shows that this problem can't be solved by making the inputs continous (e.g. allowing a mix of 50% A and -50% B in the input and output). But I think this is obvious, because people are still free to have discrete preferences.
I just got this suggested by YT algo. I'm at 0:00, looking at a thumbnail. As if it was not enough that video title is "The Topological Problem with Voting", like, come on, the channel name is one of the greatest channel names ever: "Physics for the Birds".
Friends, if this is not a nerd heaven then I don't know what is.
im so early!! this channel is amazing and im so happy there’s now more content !!
I can't believe you mentioned Mobius strips without taking a strip of paper, giving it a twist, and taping the ends together, then running your finger along it to demonstrate that it _only has one side._
Im waay too dumb to understand this :(
I suddenly fell on my knees as soon as this video started. I don't know why.
The American elections aren't very democratic or fair. Let's say there are 10 people voting for where to eat. 3 of them vote for steak house. The remaining 7 are vegetarians. 2 of them vote for pizza , 2 vote for salad, 2 vote for tofu and 1 for icecream. In American voting system all of them would go to steak house, even only minirity of 3 eat meat. In a multi party system they would go to the food court and the 70% majority would have veggie options, but the carnivore minority would also be catered for.
You stole this analogy from CGP gray. Atleast give him credit.
Thank you mr Bird, now I understand that I use a cardinal value system with a threshold, and if that threshold is not satisfied, I'm not voting for that candidate, regardless of rank.
when u talked about continuity, I thought how a vote system could even be continuous?
for example, 3 people voting for 2 options, the vote it's 2:1 and one of the 2 that voted the same change the vote, then the outcome is totally different
The video indeed seems to have used a criterion that just... shouldn't ever be satisfied anyway.
But, there are other things that could be described as "continuous" that _do_ make sense. For example, let's say candidate A has 1000 votes and candidate B has 900 votes... "continuous" could mean that if one more person votes, then the gap between A and B has to change by no more than 1 vote (either A is winning by 101 votes, or A is winning by 99 votes, or something in between; A can't suddenly be winning by 50 votes, or losing by 200 votes, because one person voted).
Excellent explanation! I've heard this result mentioned many times, but I never imagined that the proof involved the topology of a Mobius strip - that's pretty wild! As far as fair elections go, perhaps ordinal voting needs to be replaced with a form of cardinal voting, or else having runoff elections, like they do in Europe. I can think of many problems with the voting system in the USA that have nothing to do with this result, by the way, but more to do with human nature, greed and corruption in particular!
You can mathematically prove that the perfect voting system cannot exist but I'm pretty sure you can also prove that the US presidential elections follow the absolute worst system short of a dictatorship.
“The US system except that when tallying electoral votes, the tallest candidate among those that got at least one electoral vote, gets 4 extra electoral votes”
@@drdca8263 dunno. Giving outsized weight to voters from Wyoming or giving outsized weight to candidates with long femurs sounds about equally random to me ;)
@@unvergebeneid but doing both?
@@drdca8263 okay, before we go on I have to make sure that we're both on the same page: this _is_ still a joke, right? Sorry if I'm stating the obvious but on the internet, you never know.
@@unvergebeneid I knew you were joking, but, for some reason, responded to it as if it were not a joke. I don’t know why I do this.
That fact that I understood all of the terminology and logic here makes me happy
Thank you, Combrade, for this great contribution to our motherland's prosper. You will not be lose to history, forever in our heart despide your now inevitable tragic demise. Your credible account will trule assist in the downfall of the Americans. Best of luck to you, Cimbrade!
That musical instrument at the beginning is really dope!
When discreetness and sinusoidal collapse seems imminent, I go to the handy dandy Fourier transform. It’s about the peaks, limits, amplitudes, and shifts in direction around a mean.
For the two-voter two-topics example, it just means, if both of them have opposite "preferences" (on these topics), there is no compromise which is also a "preference". Not very surprising.
I'm so happy I found your channel today! This video was wonderful.
glad that all the comments are pointing out the discrete nature of voting functions. you're right that it's incorrect to map the opinion space as an n-dimensional vector, because not all opinions are possible. you'd have a circle with holes in it
This is eccelent work and I feel gratefull towards this beirdo for theaching me not only mathematics but also logiics and civics
this reminds me of the rectangle on any arbitrary closed curve. as the set of all pairs of unordered points on a closed curve is a möbius strip.
I had to check the paper for the critical missing definition; this is ONLY valid for ordinal choice. There's an easy counter-example otherwise - the average of a set of vectors satisfies all three properties. It's when you demand that the output to the function is a vector of norm 1 that there's a discontinuity. The vector average function's discontinuity is at the zero vector (every choice is equally preferred)
Very funny and interesting. I think that continuity hypothesis has to be discarded, here the reason:
- the function argument is discrete, only a definite number of choices are allowed, and the output, too.
- little change in one of the voter's opinion can change the results: in a majoritarian system, when there is a tie, changing the vote of one person will change the result. In a proportional system, changing a few votes will change the tresholds and the distribution of seats.
Interesting video. Thanks. (PS a circle is not a 1-D sphere; it's 2-D.)
Interesting, thank you for citing the sources!
What a freaking bomb-ass intro: 0:00 - 0:10
Immediately rapt
I vote your video over Veritasiums for algorithm promotion
Today I found a diamond... namely THIS CHANNEL 😍
This is nearly identical to 3b1b's inscribed rectangle proof. Neat!
Any video using Viridis gets my vote!
In the voting literature, most people explain Chichilnisky's theorem using the example of where to have a party.
Imagine you and I live along a circular road, and each of us wants to have the party closer to home. A natural way to decide where the party should go is to pick the halfway point between our two homes (or, if we live together, to just have it there). But this rule leads to a discontinuous jump. Suppose I live at 12 o'clock and you live at 6:01, just slightly past the bottom of the circle going clockwise. Then the party will be way over on the left side of the circle (near 9 o'clock), since that's halfway between us. But if you were to move just slightly counter-clockwise -- say, to 5:59 -- the party would have to move way over to the right (near 3 o'clock). An arbitrarily tiny nudge to your preferences can lead to a big leap in our collectively preferred location. And this point generalizes to other anonymous rules that respect unanimity.
Interestingly, there is a topological condition that will get us out of the problem. You can have an anonymous, continuous rule that respects unanimity so long as the space of preferences is contractible -- which in this case means a line rather than a circle.
(BTW great video.)
I mean, we can see pretty clearly there's a lack of continuity in most voting systems we employ. Either option A wins or option B wins. There's no middleground, which means no continuity.
this video has changed something in me. i dont know what, or if for the better, but something.
Regarding the discrete nature of preferences in practical applications, note that you can consider as valid outputs any convex combinations of preferences. For example, if Trump and Biden are the possible preferences, the combination
0.15 Biden + 0.85 Trump
Explains how much you prefer one to the other. This is used in may contexts, e.g. in neural networks to allow categorical variables to be treated as continous variables. The categorical result is then extracted by taking the variable with higher coefficient, with a chosen convention when draws occur.
Thinking of D. Kahneman's "Noise", an n-dim opinion-vector very often seems to be worth achieving ;) Cheers
What's interesting is that a lot of people in the comments are focusing on continuity whereas to me the obvious one is unaminity.
The biggest simplification for the model is disreagrding how strongly people feel about an issue. Perhaps that's the one that SHOULD get much better utility in a voting system.
Marked preference seems to account for this by electing the person most people deem acceptable but it has a few problems:
1. Reliance on vote counting machines (even when processing paper ballots) or accepting it could take several days for the votes to be counted and verified.
2. It can feel somewhat unfair because the person who received the most first preference votes can still loose to second preference if they don't clear the high majority needed. Same can go for 3rd preference, etc.
3. Still leans into party politics at best and tribalism at worst. Whereas most people these days care about issue politics, i.e. how are you going to fix problems X, Y, & Z?
I'd like to see what topology something like Quadratic Voting maps onto and what conditions that could satisfy.