Is Democracy Impossible? (Arrow's Theorem)

[@AnotherPanther]

After watching this I understood less than before

[@magnus123DF]

Another Penguin I preferred to not watch the whole video...

[@kyrilcouda]

Because it comments on a really specific way of majority voting, which is not used anywhere anyways. Read my comment and you will (maybe) see what is the problem and what is the solution.

[@vnen]

When you get to the point that everyone prefers Squirtle over Charmander but the pivot makes Charmander best than Squirtle, I got lost. What I don't understand is that if you look at each pair, you see that the majorities prefer Charmander over Bulbasaur, Bulbasaur over Squirtle, and Squirtle over Charmander, thus creating an infinite loop of preferences. How can this still keep the Independence of Irrelevant Alternatives? It seems that the Unanimity is the biggest problem since it makes the order you read the votes important, when it shouldn't be. If you just looked at the end result you get into an inconclusive state unless you have other factors to decide the winner.

[@ganondorfchampin]

The problem is that you are used to majority rules voting systems and are assuming this voting system must follow that. But this isn't saying the majority rules, instead it's enforcing different criteria. Those criteria are unanimity (while majority rules enforces unanimity, unanimity only refers to the case where EVERYONE, INCLUDING THE PIVOT agrees) and independence of irrelevant alternatives. Unamity was never violated because Mister F disagreed, so there was no universal consensus that Squirtle is better than Charmander. The reason the second property is held is because it's in regard to the final decision, not individual preferences. The only reason Mr. F could make change the group preference for Charmander over Bulbasaur is because he was defined as being the pivot. The fact they CANNOT change the preference is based on the transitive property: Charmander is ranked above Bulbasaur, and Bulbasaur is ranked above Squirtle, thus Charmander must be ranked above Squirtle as long as nobody changes how they rank Bulbasaur. What the independence of irrelevant alternatives actually requires is that this hierarchy stays unchanged, because nobody changed their relative preference for Bulbasaur. Also, order doesn't matter. The dictator is the dictator based on being the dictator, not on position. Any voting system with unamity must have a pivot, but the pivot doesn't have to be determined by order. For example, in majority rules the pivot is anyone who votes for a someone that is one point ahead of being tied. Getting rid of unamity is a REALLY bad idea, probably even worse than a dictatorship, as it means that even if literally everyone doesn't want an option it still ends up happening. At least in a dictatorship at least one person gets what they want. In fact, getting rid of pivots is very bad, as the only way get rid of pivots entirely is to ensure that the outcome is always the same regardless of how ANYONE votes. The actual problem is the independence of irrelevant alternatives, as any sane voting system has the other two properties.

[@AstralS7orm]

@@ganondorfchampin What do you mean by sane? Exploitable? See, if the pivot is unpredictable as in it is mathematically hard to calculate if any given voter is a pivot, the voting system is not possible by any defined "Mr. F" to exploit systematically. The resistance to dictatorship (either direct or by tactics) is exactly the property of unpredictability of a pivot and not lack of any pivots! Most IIA systems satisfy this property, so unless you're a voting oracle, are dictatorship proof. The one way to break IIA satisfying voting system is to allow for multiple rounds in small voter population and using strong predictive tools. Maybe. That said, this is used to break democracy with other systems already. For example, Majority Judgement (about the most resistant system that's based on scoring) in small scales can be broken because it relies on median preferences and medians are reasonably easily computable - but not for larger constituencies. Most voting systems in fact will fail "dictatorship/corruption" (predictability of pivot) in small scales. Ones using averages or direct counts are even easier to break. So, while the proof is sufficient that this situation *can exist* it says nothing on probability of any given F being the "dictator/corruptible". Which is much more important. You cannot bribe someone if you don't know who it is, nor you can break the voting system more by messing with constituencies if the voting is reasonably consistent. (Full consistency is not strictly necessary - only predictable, exploitable inconsistency has to be avoided.) Nor you can mess up the voting by introducing fake candidates. Heck, by most of those measures except majority conditions sortition is better... You probably want more properties like low Bayesian regret and high Bayesian gain for a really fully acceptable voting ordinance. In this regard, range voting always wins over other commonly applied systems (IRV, plurality, Borda count) regardless of tactics... and it satisfies IIA. Presumably in Majority Judgement hard to exploit variant that keeps most of majority conditions.

[@ganondorfchampin]

AstralStorm That’s not how it works, at all. You can’t exploit the pivot, in many situations knowing you would have been the pivot makes no difference as in that case changing the way you vote wouldn’t change the outcome, it just changes who the pivot ends up being. The point is, ALL voting systems are exploitable, and all ranked voting systems have a particular weakness. However, having the weakness is still better than having a dictatorship or not making decision when literally EVERYONE agrees on it, that’s what I mean by sane.

[@ganondorfchampin]

AstralStorm Also, Mr. F is not the dictator because he is the pivot. He is the dictator because he is the dictator. The main thing you’re not understanding is that for a system to respect both unanimity and irrelevance of independent alternatives the system MUST be a dictatorship. In a dictatorship the dictator is the pivot, but if there is no dictator then the pivot isn’t a dictator.

[@regalrayquaza7609]

It was a confusing choice to use Pokémon when in the games the trainers receive different Pokémon, but here they are deciding on which one the group as a whole prefers.

[@chrstfer2452]

I've been looking for a channel like this for so long. This video is amazing (great diction and clear explanation in particular, also good graphics), and the rest of the channel looks similarly epic. Thank you.

[@arrowstheorem1881]

I highly recommend such videos before a good sleep

[@miguelfonseca1104]

what always confused me is the specialness of the dictator here. Why does it matter if one voter's choice happens to mimic the election result? if all votes are counted at the same time, why is there being one random dictator matter? "dictator's voice is all that matters" how exactly? because someone in the voting system can be the tie breaker? this reminds me of the common equivocation you often see in intro to philosophy/logic classes of confusing logical implication with causation.

[@MustSeto]

The proof doesn't just show that one person will happen to have the same rank order as the results, the proof shows that the method only ever cared about a specific voter's ballot, you don't even have to look at anyone else's ballot. You can change their votes however you like, but the final result will still be a copy of the dictator's ballot, or else the method doesn't pass IIA and Unanimity (or doesn't use ranks, as the next video explains) PBS Infinite Series has another video going over Arrow's Theorem which I found it slightly easier to follow. Apparently it can still give the impression that the dictator is just someone who's ballot happens to match the results, but remember that at 9:04 in that video, the ballots can be anything. No matter what those are, if you use the same set of test ballots from earlier, the final results will always be a copy of voter2's ballot. kzbin.info/www/bejne/d5m5g2qde7J-hMk

[@IceNein763]

There will only be a certain person who is (functionally) a dictator if that person has knowledge of all of the other voters preference, and the preferences are balanced in such a way that one persons decision will sway the vote one way or the other. In reality, if there was anonymous polling in the example you gave, there would be a "dictator" but nobody, including the dictator would be able to determine who that dictator was. Your examples conveniently use a simple five person vote (so that it is impossible to have a tie) and in addition persons 1&2 voted the same, and people 4&5 voted the same. Other times you'd have a situation where persons 1&5, 2&4 voted the same on choice A, and 1&2, 4&5 voted the same on choice B, always leaving both sides balanced to enable person 3 to be the dictator. In an actual election scenario, there will be hundreds of thousands of voters, and the only case where somebody will actually be a dictator (instead of a dictator who happens to agree with the majority) is in the one case where the votes are even except for the one person. Essentially what I understand of Arrow's impossibility theorum is that it's an interesting parlor trick to demonstrate a mathematical fact that there is technically one person who gets to decide, despite the fact that nobody, including that person knows it.

[@IceNein763]

Also, I'd like to point out that in a case where you are choosing between two options, and you have 400,001 voters, 200,000 voting one way, and 200,001 voting the other, the dictator is equally any of the 200,001 people.

[@UndefinedBehavior]

I think you may be confusing the "pivot" with the "dictator." The person who is the pivot of A over B is the person that will cause the result to change to A over B once their vote is counted in some arbitrary ordering of the voters. This will pretty much always happen in all voting schemas, and we just use their existence so that we can infer how the vote will look in a specific situation. The power of the "independence of irrelevant alternatives" property is that we can then use the results from one vote to infer how the result must look in another vote. The dictator of A over B is someone who makes A beat B if they say that, and it doesn't matter what the rest of the voters say. It is incredibly problematic if any voting schema has a dictatorship as that means that one person's vote counts more than another person's (in fact everyone else's vote doesn't matter at all). We started by looking at the pivot, and then realized that they will also end up being a dictator. The moment we got into a state where the result reflected the pivot's vote even when everyone else disagreed meant that that person will always dictate the result because of the independence of irrelevant alternatives. It may be true that we don't know ahead of time who the dictator will be. For example, your voting system might be "pick one person at random and use their ballot as the result." So yes, in this case the dictator won't know that they are the dictator, but this voting system is still clearly unfair and unrepresentative of the voters at large.

[@IceNein763]

Undefined Behavior It will only ever be "pick one person at random and use their ballot" in an exceedingly narrow situation where there is N people voting for A (N-1) people voting for B, and 2N/3 people voting for A over B, 2N/3 people voting for B over C and 2N/3 people voting for C over A. This might be a persuasive reason not to use that system where there is a small number of voters, and their individual votes are a matter of public record, and voters can base what they want to vote for given their knowledge of other people's votes.

[@UndefinedBehavior]

No worries! The proof can be pretty confusing and the questions are really important. A couple side notes: we are allowing voters to give their full ranking of their preferences of all choices, not just cast a single vote for their favorite. Also, we don't know that the majority will win (in fact we show that we are in fact not using majority rule, but rather a dictatorship). I show the pivot as the middle voter simply for convenience, but we technically don't know who the pivot will be. The dictator is not someone that casts the deciding vote (or resolves a 50/50 split), the dictator is someone who forces the result to be one way no matter what anyone else says. If you look at what's happening at 5:40, the result is that Charmander beats Squirtle, but Mr. F is actually the only person that has that on his ballot whereas every other person says the opposite. Later in the proof, we find out that Mr. F doesn't just dictate that one specific choice, but all of the choices. This means that even if all other voters collude together, it doesn't matter; the result will always be exactly what Mr. F says and nothing anyone else does can change that.

[@UndefinedBehavior]

+Clint Davis I'm a little confused as to what you're saying. What the theorem shows is that the only way to satisfy both unanimity and independence of irrelevant alternatives is a dictatorship. You could make your dictatorship more random by saying "pick a random person and only use their ballot for the result," but ultimately the system must be some form of "one person rules the entire vote, and nothing anyone else says can change the result." Again, a dictator is not someone that breaks a tie. A dictatorship is a voting system where we ignore all but one person's vote. Even if everybody changes their vote to act against the dictator, the dictator still rules. The reason we do not have a dictatorship in our elections is because we break independence of irrelevant alternatives in our current voting systems.

[@Wouldntyouliketoknow2]

At 5:48 I don't understand why charmander wins.. if 3 points for first pick, 2 points for second pick and 1 point for third pick, he would score 9. Squirtle who you say comes in bottom, scores 11. When we count votes we typically assign value to order of preference right?

[@MustSeto]

There are a ton of rank-based method, out there, and no, the most popular ones generally don't give points based on ranks like that (the most popular are IRV and various Condorcet methods, the main family of methods I know of that give points based on ranks are "Borda Count" related methods) Plus for this theorem the exact method is kept very non-specific. We only assume it has a couple broad properties, namely that it passes IIA and Unanimity. At least until the end when we realize that there's only one specific method that can have all these properties: a dictatorship.

[@maxfriis]

I have not thought about Arrow's impossibility theorem as an explanation for strategic voting. It's interesting combined with another theorem called the revelation principle theorem. It states that any mechanism can always be replicated by a truth telling mechanism where agents reveal there true preferences.

[@pancreasdragonheart9765]

I always picked Charmander, which resulted in Gary picking Squirtle and me catching a grass pokemon, which is the first pokemon one could catch. Easiest way to beat early game :D

[@heimdall1973]

The way I would probably interpret preferential vote fits unanimity but violates independence. But is it the best way? It's one of the following: (1) Each possible result (ordering of all candidates) gets a score. For every pair candidates add the number of voters who put those two candidates in the same order as the result. (For example, a possible result is 1 Bulbasaur, 2 Charmander, 3 Squirtle, and the score of that result is number of voters that prefer B to C + number of voters that prefer B to S + number of voters that prefer C to S.) The ordering with the highest score is the actual result or election. (2) Every candidate gets a score. To get the score for candidate A, we add the number of voters who prefered A to X for every other candidate X. (For example, Bulbasaur's score is number of voters that prefer B to C + number of voters that prefer B to S.) The candidate with the highest score wins. The obvious difficulty is if 1/3 votes BCS, 1/3 CSB and 1/3 SBC. Both my systems will then cause a draw. But is there a system that can resolve this? Is there a system for 2 candidates that can resolve 50/50 votes? Suppose 35% voters say BCS, 34% CSB and 31% SBC. We want each two candidates to be ordered according to majority. But majority (69%) prefer C than S, majority (65%) prefer S than B and majority (66%) prefer B than C. You can't satisfy all majorities, but you can satisfy any 2, and system (1) will satisfy the 2 bigger majorities (C before S and B before C), resulting in BCS with score 66% + 35% + 69% = 170% (which is 1700 if there are 1000 voters). Other scores: CSB 168%, SBC 162%, CBS 138%, BSC 132%, SCB 130%. The winner is B. The highest possible score for 3 candidates is 300% (1000% for 4 candidates), achieved if every voter submits the same order. Note that the top three orderings have similar scores because of closeness of votes. System (2) will give C the winning score 69% (for CS) + 34% (for CB) = 103%. Other scores: B 101%, S 96%. The highest possible score with 3 candidates is 200% (300% with 4 candidates), achieved if every voter puts the same candidate on top. Again close scores for the same reason. The problem with my systems is, who should then be the winner, B (according to (1)) or C (according to (2))? System (2) is actually equivalent to scoring each candidate by looking at system (1)'s scores and adding up scores of those orders that put the candidate on top. With Pokemon election, B's alternative score would be the sum of scores for BCS and BSC, and so on. So although ordering BCS has the highest score, CSB is not far behind, and the low score of BSC lets B down in comparison with C. There are other systems: (3) Eliminating the candidate with the fewest votes would mean S with 31% is out, and the preference means that those 31% votes are added to B's votes, who then wins with 66%. (4) First past the post would just see that B got 35% of votes, C 34% and S 31%, so B wins. What about if we change the votes: 35% say BCS, 31% CSB, 34% SBC? Now both my systems will favour B. Ordering BCS will have the winning score of 170% in system (1) and B will have the winning score of 104% in system (2). System (4) will also simply favour B. But system (3), which is supposed to be fairer than first past the post (4), will eliminate C with 31% who voted CSB, adding those votes to S, making S the winner with 65%. So which system is fairer?

[@patrickwienhoft7987]

IIA seems nonsensical in the real world. Say there are only 2 groups of voters. One love Charmander, finds Bulbasaur ok but hates Squirtle. The other group loves Bulbasaur, finds Squirtle ok but hates Charmander. A reasonable solution would be to elect Bulbasaur. He beats Squirtle unanimously and is loved by 50% of the population, just like Charmander, but unlike charmander also tolerated by the other 50%. Charmander and Squirtle are hated by 50% of the population, but at least Charmander is loved by the other 50%, hence he should be 2nd. So our result is B>C>S. In a different world, where our groups vote C>S>B and B>C>S (S moved up 1 in group 1 and down 1 in group 2), C>S unanimously and C and B are loved by 50% each, but now C is accepted by the other 50% and B is hated by the other 50%. Hence our result should be C>B>S. Changing only S's position changed the position of C and B, however the reasons for that are sound. I think there are two fundamental problems with this interpretation of IIA: 1. It does not consider by how much one candidate wins over another on a ballot. It acts as if winning against another choice on a ballot as first vs. last is worth just as much as winning as first vs. second or as second last against last. 2. There isn't really a thing like "changing the position of one candidate" in a ranking system. Changing one position will always affect at least one other position as people can't share a position. Because of that, an illustration of IIA should at least include 4 parties and show that swapping two does not affect the order of the other 2.

[@drewcole5375]

By far the best explanation for this theorem, thank you so much!

[@babycrab1661]

I have a trouble understanding why can we switch charmender and squirtle to argue Mr.F must be the pivot for both two cases? We only know that Mr.F dictates charmender over squirtle, not squirtle over charmender.

[@UndefinedBehavior]

Yup, you got the most hand-waivey part of my presentation of the proof. Essentially, there is nothing special about my specific use of Charmander, Bulbasaur, and Squirtle, and you can redo the entire proof by replacing Charmander with Squirtle and Squirtle with Charmander. The difference when doing the proof a second time is that we know a little extra information about how Mr. F influences the election, and we can combine it to realize that Mr. F will be the same pivot in the second proof. You continue to redo this whole process for every pair, and he will be the dictator with every pair, meaning he will dictate the entire election.

[@babycrab1661]

Thanks for your reply. I understand what you said quite well, but I think I didn't state my question clearly. (C=Charmander, S=Squirtle) In the second part of the proof, we want to figure out who is the pivot for C beating S if we've already known Mr.F is the dictator for C beating S. Then by argument, we find that the pivot for C beating S must either be him or before him and the pivot for S beating C must either be him or after him. Finally, you argue that if we repeat the argument C and S switched, we'll see that the pivot for S beating C cannot after him and the pivot for C beating S cannot before him. I don't understand the final step since I believe that the first argument is based on our current knowledge of Mr.F dictating C over S. However, we haven't decided that Mr.F dictating S over C at the moment. Maybe I got something wrong (Mr.F dictating S over C can be also decided or the repeat of the argument can also be done), but could you explain the detail of the proof to me? Thanks a lot!

[@UndefinedBehavior]

We start with Mr. F as a pivot for C over B. This leads to Mr. F being a dictator of C over S. We then see that, by the nature of being a dictator, the pivot for C over S will be before (or equal) the pivot of C over B, and the pivot for S over C will be after (or equal) the pivot of C over B. We then repeat the entire proof with C and S switched. I didn't go into detail of doing it all again because of tedium, but essentially we will go through the process of using the pivot of S over B to discover a dictator of S over C. We then use the knowledge of the existence of this dictator to show that the pivot of S over C must come before (or equal) the pivot of C over S. We now see that the pivot of S/C and the pivot of C/S both come before and after each other, meaning that they must be equal. This means that the pivot of S/C = pivot of C/S = pivot C/B. We've now chained together a ton of pivots, and if we continue to repeat this whole argument again for different pairs, we will see that all of the pivots will link together. Here's the Wiki link to the argument, but they get nice formatting on their math which might help (although they also hand waive at the part you are confused about): en.wikipedia.org/wiki/Arrow's_impossibility_theorem#Part_three:_There_can_be_at_most_one_dictator

[@babycrab1661]

Your explanation is so wonderful! I think I have grasped your point. Thanks for your patience. I have to say using animation to explain such a puzzling proof is quite a good way. Hope Undefined Behavior will release more series.

[@ganondorfchampin]

I didn't even realize what the problem was I watched the video again, as I didn't realize dictatorship/pivots didn't go both ways, meaning the S/C pivot isn't necessarily the same as the C/S pivot. I think you're main problem is that you used the name Mister F for the second dictator, when the big conclusion is the two dictators are one and the same. A little bit of scratching showed the two dictators had to be one and the same as otherwise one of the two pivots wouldn't exist, but that wasn't quite clear from the video, which I think is due to the fact you used the name twice it wasn't clear you were doing the entire process of finding a dictator again. What it looked like you were saying is because because the pivot be both before and after it must be Mister F, basically just a rehash of the last segment of the video, which comes across as rather trivial without understanding that the two pivots are different.

[@Dracolith1]

What you describe is not a dictatorship, OR a dictatorship is not bad --- does not have the negative attributes popularly associated with dictatorships. During a real election (1) Nobody can change their vote after the polls have closed or after they cast it, (2) The ballot is secret, and (3) The votes are not counted in a specific order. So even if there is a pivot, there is no way of predicting what vote it will be.

[@kellel5610]

Dracolith1 a pivot could be like picking a random person on the street vote and they decide who is president whitch is horrible

[@MustSeto]

The dictator in arrow's theorem is the only person the other method ever cares about. They cannot be defeated even if everyone could hypothetically change their vote afterwards. Of course this does mean the method would have to have some way of identifying a dictator and throwing out everyone else's vote, so if a method behaves like you say, then it cannot be a dictatorship, and therefore cannot pass IIA and Unanimity (if it's also a rank-based method).

[@toastbrot97]

Am i missing something or couldn't one get a fair result by just averaging all votes? If at 5:47 we add all the placements by each person and devide the result by the amount of people that have voted we end up with: Squirtle: 1+1+3+2+2 = 9 9 / 5 = 1.8 Bulbasaur: 3+3+2+1+1 = 10 10 / 5 = 2 Charmander: 2+2+1+3+3 = 11 11 / 5 = 2.2 Therefore squirtle being 1st, bulbasaur 2nd and charmander 3rd. Seams fair and square to me.

[@Tulbih]

Very uncertain about this. The problem might be, that in economics you use ordinal and not cardinal utility. Ordinal utility only ranks utility levels but doesn´t account for "distance" between different utility levels. Therefore you lose information that you would need to meaningfully average across utility levels.

[@MustSeto]

This is The Borda Count, it fails Independence of Irrelevant Alternatives

[@mesplin3]

A random dictator could be interesting. Insane, but interesting. (Pick a voter at random and consider their vote. Whoever they vote for is the winner.)

[@MustSeto]

Just FYI, that would fail IIA, and wouldn't be a dictatorship in the same way as in the video

[@mesplin3]

@@MustSeto darn

[@ganondorfchampin]

A random dictator does not in fact fail IIA, as far as Arrow’s impossible theorem is concerned it is in fact a dictatorship. Anyway, I think such a system is actually not as insane as it’s generally thought to be, considering our existing systems give us insane results like Trump. In particular, random dictatorship completely eliminates strategic voting as people cannot get a better outcome then by honestly ranking their candidates. Counterintuitively, it also makes people’s choices matter more than most systems, as someone voting always increases the chance that they will get the outcome they want, often by more than other systems.

[@MustSeto]

@@ganondorfchampin I think it sort of depends on how dictatorship, IIA, random and maybe even what a "rank-based method" are defined. For example take an election like: 1x A > B 1x B > A If you run this exact same election twice but pick a dictator at random each time, then no relative orderings will change, but sometimes A can win and sometimes B can win, which should be IIA failure. You can sort of finagle it by saying something like "the dictator is always the first ballot submitted", in which case A would always win here. But then if you "actually" ran the election multiple times, couldn't sometimes the B>A voter submit theirs first? Then B would win. But of course, then the "voters" are not the actual humans, but just the abstract positions in a list, and THOSE _have_ changed their relative A/B orders even though the humans didn't, so IIA is still not considered violated. But this feels strange. You could probably finagle things differently but I think effects like this would always exist. Part of this is what Arrow considers a rank-based method. I proved Arrow's Theorem in AGDA for fun, and one thing I noticed is that using the normal definitions, I couldn't use a random method if I wanted to. If I wanted a random method, I'd need rank-based method that, in addition to an ordered list of ranked ballots, also needed something to represent a random state (this extra data would already mean that you couldn't apply Arrow's Theorem to it as-is any more). Then whether or not a random dictatorship passes or fails IIA depends on whether you allow the state to vary across elections. If you do, then I think you can prove it fails IIA. If you don't, then I think you can prove it passes. "Strategy" is also kind of weird with random dictator/random ballot examples like this. It's not strategy in the traditional sense, this is true, but if write-ins are allowed, then arguably a lot of people's best option is to just vote for themselves. Their vote doesn't contribute anything at all unless it gets picked, in which case the others can't bring it down, so there's no need to cooperate. I wonder what would happen if we had a Score election, but instead of electing the score with the highest total/average, we elect a candidate "randomly", except weighted by their totals/averages?

[@matthewvicendese1896]

The other things that also make a democracy to work are inequitable economic structures leading to some people having more possibilities to push their agenda and a political/media class that wants this economic model to continue because they are all richly rewarded by it.

[@zix2421]

8:55 who “we”?

[@JazzyWaffles]

This honestly just seems like some mathematical parlor trick combined with careful wordplay rather than a legitimate thing. Which... it probably is? Less of anything to actually worry about and more of a weird quirk of the system that, if you define your terms carefully, can SOUND scary when it actually isn't.

[@MustSeto]

Not really? I've noticed a lot of people who see explanations like these thinks it only proves "the will be one voter who's ballot happens to match the final rankings by coincidence", maybe because the others "cancel out" or something, but that if the others had voted differently the "dictator" would be someone else. Unfortunately, that's not true, Arrow's theorem proves that, if a rank-based method passes IIA and Unanimity, etc., only one vote _ever_ matters at all. Even if everyone else changes their vote, or votes against the dictator, the dictator still decides the election alone Of course real rank-based methods do fail IIA, and Arrow's theorem doesn't say anything about how or how often methods fail. And as another of UB's video point out, rating-based methods aren't covered by arrow's theorem and are probably better anyways

[@CapitanAmerica96]

Following my microeconomic's class, this has been of great help. His explanation perfectly conforms to Arrow's theorem and gives a simple nonmathematical explanation to it.

[@edwardlau892]

The F in Mr. F stands for Favored. Favored by the Arrow's theorem as the pivotal minority.

[@theultimatereductionist7592]

We went through the proof of Arrow's Theorem in a graduate mathematics course taught by a Dr Fred Roberts at Rutgers University in 1992. I never liked that anyone thought that the 4 particular axioms or conditions that regarding what the group ranking should or should not satisfy were special or necessary. I mean, one could/can make up infinitely many conceivable restrictions/conditions. 1 or 2 of them made sense: e.g. a no dictator rule, for example. But, I see nothing wrong with a Borda count, for example, even if it means the top ranked candidate in the group function does not win a plurality of top rankings by voters. So what? The Borda count takes into account ALL the feelings of the voters, for example. So, sensationalistic headlines such as "is democracy impossible?" really annoy me, because they're total bullshit. No. Arrow's Theorem just says that this particular set of arbitrarily chosen conditions are not all simultaneously achievable. NOT that "democracy is impossible".

[@MustSeto]

>"I see nothing wrong with a Borda count, for example, even if it means the top ranked candidate in the group function does not win a plurality of top rankings by voters." The top ranked group winner winning a plurality of top rankings isn't a requirement for Arrow's theorem, Borda fails Arrow's theorem basically because it's vulnerable to the spoiler effect (but it actually has lots of problems, so whatever) Also as this video "sequel" shows a method that does pass all of Arrow's explicit requirements, by failing an implicit/unstated requirement: being based on ranks, which inherently ignores some of the feelings of the voters

[@mrowlbert]

Thank you! I've seen the Arrow's Theorem exists therefore democracy is impossible conclusion way too much. Ugh

[@jakedewey3686]

The most confusing part of this video is how nondescript the "dictator's" role in the vote is. What Arrow's Theorem states regarding dictators is that for a given set of votes, there should be no single voter such that changing that voter's choices causes the result of the vote to change to mirror that change.

[@MustSeto]

It might not be completely clear from the video, but the dictator in arrow's theorem is the only voter that can possibly matter; even if the other voters could hypothetically change their vote, the results would still just be a copy of the dictator's vote, unless the dictator changes, in which case the results would be a copy of the same dictator's new vote.

[@damienhughes4559]

I love the overall explanation but as a non-Pokemon guy, I quickly get lost by the references to Squirtle, Charmander, and Bulbasaur. Any chance you could make the same video sometime in the future except using ice cream flavors (vanilla, chocolate, and strawberry) instead?

[@minwoorhee4451]

transitivity is not mentioned, however, this is an awesome video. Thank you!

[@ganondorfchampin]

It's implied by the structure of the problem, which implies that a ranking of group preferences can be determined by the voting system.

[@xsaberfaye]

Yeah Arrow's definition of dictator is too convenient to be realistic. A dictatorship shouldn't be a voting system where "there exists one person in any situation whose preferences will match the final result, assuming all other people do not change their vote." Instead, a dictatorship should be a voting system where "there exists one person in any situation whose preferences will match the final result, EVEN IF all other people change their votes." Sorry Arrow, your theorem is correct, it is also interesting, but it has a much more limited use than it tries to imply. In fact, it's just outright misleading at this point. The definition of dictator has become so far removed from common understanding of the word.

[@MustSeto]

The dictator in Arrow's theorem _is_ "one person in any situation whose preferences will match the final result, EVEN IF all other people change their votes". It might not be clear from this video's explanation though.

[@wolfpackjew]

Can someone explain to me why a pivot would exist in the first place? Once the votes have been cast it's not like anyone can go back and switch their vote. There could be the one vote that gets a candidate into a majority...but even that exists in a vacuum unless you ignore every vote that comes after it.

[@MustSeto]

It might help to look at other proofs as well. This one made more sense to me: kzbin.info/www/bejne/d5m5g2qde7J-hMk It can apparently still give the impression the dictator is just a random person though, but this isn't the case. The dictator is the only voter which ever matters, and the results will be a copy of the dictator's ballot even if everyone else is able to change their vote in any way they like, even if they try to vote against the dictator using whatever coordinated strategy they can think of. That is, the election they show at 8;24 can be anything, but the results must always be a copy of voter2's ballot.

[@TheMindCrushGroup]

5:50 I don't understand. Isn't this an "unsolvable problem"? The way I see it: -Cases in which Squirrel>Charmander: 4 (S beats C) -Cases in which Bulbasur>Squirrel: 3 (B beats S) - Cases in which Charmander>Bulbasur: 3 (C beats B) Therefore: Squirrel beats Charmander, Charmander beats Bulbasur, and Bulbasur beats Squirrel... Who won? I would argue that Squirrel would win, since S>someone in 4 cases, while B>someone in 3 cases. But something tells me this not accurate either. Help!

[@nguyenquangminh4814]

you deserve as many subscribers as CGP Grey dude. Please don't quit making noice ass videos

[@banditski]

I'm sure this is great for people who already know pokemon, but to a middle aged guy like me, I got lost with the names pretty quickly. I watched some other videos about voting systems that used animals (lions, tigers, gorillas, owls, etc.) in the same way you did and it was a lot easier to follow because I already know what picture matched the name "gorilla" the narrator is talking about. I don't have to waste time trying to tie them back together and then miss the point being made.

[@Eccentrick218]

So basically, the centrists decide who wins? I see no problem in that

[@MustSeto]

Nope, the dictator picks who wins, the other voters don't matter even if they change their votes. So if the dictator picks an extremist, then the extremist will win, even if everyone else votes against them.

[@Eccentrick218]

@@MustSeto I'll argue that in a small group, dictator is singular. But in a large group, the dictator is is plural, meaning the goal is to win over people in the middle. Bernie and Trump found a loophole to this, but winning over extremists that turnout high in the name of reform, therefore ignoring the centrists for another untapped 'dictator' group. But the 'dictator' seems inevitable in democracies, and isn't necessarily indicative of anything bad. "Key voting blocks" is just another word for dictators presented in this video.

[@MustSeto]

@@Eccentrick218 I'm not sure what you mean, but the dictator is always one person (singular?) in any rank-based method that passes IIA and Unanimity, no matter how large the group is. Of course dictatorships are obviously not inevitable, since FPTP/IRV/etc are not dictatorial in the formal sense used in Arrow's theorem

[@MateusSFigueiredo]

The scenario in 5:15 bugs me. Given those votes, Charmander would have been eliminated in a Instant-Runoff voting system, since only 1 person ranked it first.

[@f_f_f_8142]

Instant-Runoff voting does violate the independence of irrelevant alternatives.

[@redundantideas]

I think we moved into the twilight zone at this point, where we moved away from how voting actually works, rather we just applied certain princples of the video. Personally I think the argument that one person determines the outcome is misleading, because you still require the other 50% to enable the outcome (all of them are equally "dictators".) More importantly, if less than 50% agree with you, then you are denied your choice by more than one person; all of the "excess" voters have made that decision against you. You need a near even preference distribution to have a "dictator" scenario, and it would need to be even across candidates, which virtually never happens. This brings me to another point: How can Independence of irrelevant alternatives actually exist when there are more than two candidates? Surely in a genuine democracy, all candidates must complete with one another, and therefore the result must be influenced by each candidate? Condorcet seems to be the only method I've read that minimises (but does not remove) this issue, but it is relatively complex and potentially untrustworthy by the general public.

[@joy_gantic]

@@f_f_f_8142 Proportional Vote and Alternative Vote do not though

[@MustSeto]

@@joy_gantic What do you mean by "Alternative Vote"? As a name, I've only heard that as another name for Instant Runoff.

[@SidHancock3]

This Anti-Bulbasaur rhetoric is unacceptable. In the new Bulbasaur regime, which my vote will singly bring about, you shall be the first to face the public vine whipping.

[@AndrewCMyers]

Very nice explanation! I don't quite buy the conclusions, though. You jump from the claim that favorite betrayal is possible to a pretty sweeping claim that all ranked voting systems are intrinsically unacceptable. I think a stronger argument is needed that losing IIA is so terrible, given that Condorcet methods seem to actually work pretty well in practice.

[@Claijohn]

Mr. F becomes the deciding vote only if the competition is neck to neck. If there is a majority then I don't see how Mr. F's vote changes the course.

[@MustSeto]

Dictatorships are not majoritarian, and neither is any rank-based method that passes Unanimity and IIA

[@randomguy-tg7ok]

I don't quite understand what's happened at 5:38 - in the example, the relative ranking of Squirtle to Charmander has changed, and that hasn't been reflected. The ultimate result should put Charmander last, with the 'winner' depending on what system is used (Squirtle in a points-based system, Bulbasaur in STV). Perhaps the problem here is that the magnitude of relative rankings isn't being respected? In the above result, Bulbasaur comes above Charmander despite being preferred by only two of five because said two have a much stronger preference than the other three. Is that something that violates the premise of the theorem, or?

[@MustSeto]

By IIA, B > S. Also by IIA, C > B. But Rankings are transitive, so since C > B and B > S, C > S as well. The problem is that it passes IIA. Which together with Unanimity means the method has a dictator. It's not that the magnitude of relative ranking isn't being respected, it's that only one voter is considered at all, no mater how anyone else votes.

[@osvaldo701]

Love it! Great video and explanation!

[@reversedragon3]

The independence of alternatives may be the single biggest problem here. "Mr. F" seems to actually arise from an incorrect assumption that voters all act independently rather than that voters are necessarily interacting through their votes and assembling into populations, and there is generally either a best or least-bad population for everybody to assemble into. The vote tables on screen immediately suggest a possible voting system where the three ranks get a different weight and we consider the total scores of each candidate to reflect the overall approval for each possible voter population across the real population. I'm not sure how good this system actually is but it seems like it has more realistic assumptions

[@MustSeto]

You sound like you're either describing the Borda Count or one of its relatives (all of which fail IIA) or something like Score voting (which is not rank-based, and so is not covered by Arrow's theorem)

[@UN-Seki]

@@MustSeto why would it have to be a rank-based system necessarily to be a democracy?

[@MustSeto]

@@UN-Seki Methods don't have to be rank-based to be democratic, but Arrow's Theorem only applies to rank-based methods. So a rating-based method could be democratic, but Arrow's theorem by itself can't tell you anything about its properties one way or another

[@UN-Seki]

@@MustSeto I see, I don't think this was pointed out in the video at all though.

[@MustSeto]

@@UN-Seki It's mentioned at the end: 9:49 He also has a sequel video on Range/Score Voting: kzbin.info/www/bejne/m2Sqd3pmqL18n8k

[@iwersonsch5131]

What happens if you limit independence of irrelevant alternatives to the winner of the election?

[@hecko-yes]

But who is the "dictator"? Is it everyone who voted Charmander over Squirtle? If so, it's still the group's preference.

[@hengistcz1940]

THis video is awesome man. I am doing seminar work on the subject and I am not native english speaker so I find Arrows books unreadable with kind of language he uses but this explanations with pokemon made me understand everything. Kunos to you for your work. I will definetly reference you in it. Although profesor may give me a bad eye for it. :D

[@Xectopus]

ok but would rank choiced vote be more democratic if there was more than 3 candidates?

[@dougwebb7799]

Thank you! Need to re-watch as it goes pretty fast. What is the main difference between Arrows theorem and the Satterthwaite theorem? Score Voting and Approval Voting FTW.

[@ganondorfchampin]

Satterthwaite theorem is the collary to Arrows theorem, even though it's a direct conclusion from it what they actually say is quite different.

[@procheese-tw6813]

the conclusion amazed me

[@EyeoftheAbyss]

Really should have used something other than these odd names, a simple ABC would have sufficed, or more easily associated names with the pictures. It especially becomes a challenge to keep names, places, logic, and pictures associated correctly around 5:20

[@0004332]

Ye the whole Pokemon reference really irritated me.

[@yan-qu5zm]

Armin Kraemer Lol k

[@ganondorfchampin]

There is no such thing as there is no universal language. The intended audience of this video is expected to be familiar with pokemon, and concrete examples work better for most people than abstractions.

[@rahulmalik1083]

For people who have watched Pokemon , this is better than A B C but I get your point

[@nielsfrederiksen6636]

Or you could just.. you know... do preportional representation and not a first-past-the-post system

[@MustSeto]

first-past-the-post generally refers to a specific single-winner method, there are a lot of other single-winner methods, but the rank-based ones are generally still covered by this theorem (although rating-based ones can evade it, even while remaining single-winner) And finding a good single-winner methods will always be necessary, not just because some positions currently only have a single seat, but also because after you pick you representatives, those representatives will then need some way to vote on laws/policies, which are often inherently single-winner

[@omaolligain]

Would love it if you'd do Sen's Liberal Paradox with pokemon too.

[@olesdrow6711]

Why do we call Pivotal as a dictator here, if anyone after a certain point could be a pivotal and in anonymous voting you can't tell if you are the pivit or not before the results?

[@MustSeto]

Because it turns out they're not just pivotal, they're a true dictator. It's true that anonymous methods can't be dictatorships, but we don't assume anonymity. If you ran another election with the _exact_ same method, the results will always be a copy of the same voter's ballot, no mater how anyone else votes. Even if everyone knows how the method works, and knows how the dictator is going to vote, the other voters would not even be able to cooperate to change the final results even a little bit. If this proof is confusing to you, you can find another proof using another style by PBS Infinite Series. Personally I like that one better.

[@Nidomy]

When there is three things to choose from, is it not at least a third of the votes?

[@Castle3179]

How do people know if they're the pivot? They can't. Nobody can be a pivot unless they know they are?

[@MustSeto]

They could know they're the pivot if they were involved in the design of the "method" or told ahead of time would be the dictator

[@khere4787]

Possible Solution: Have an even number of voters?

[@anselmschueler]

No

[@khere4787]

@@anselmschueler Ok

[@christiancatalan2943]

Thank you for more clearer explanation, for visual learners

[@konradpietras8030]

I don't agree with what is said in 4:50. What if pivot person is variable and it changes when we change the order of voting. For example third person may be pivot when he changes order from bulbasaur, charmander, squirtle to charmander, squirtle, bulbasaur but when he changes to charmander, bulbasaur, squirtle then nothing changes in final order and the pivot person will be fourth or fifth person.

[@MustSeto]

The pivot is only defined by one change in a specific scenario. Once we've found that we don't immediately assume they are the pivot for another even slightly different scenario (though we eventually derive it). Instead we just use Unanimity and IIA. To try to define the pivot another way, this time using the specific scenario in the video, they're the first voter such that if everyone before them votes C>S>B and everyone after them says B>C>S, their switching between B>C>S and C>B>S switches the results between B>C>S and C>B>S. (We don't then assume they'll remain "pivotal" in the same way if the ballots were tweaked slightly _pre se._ We just immediately, blindly start applying IIA, only using the pivot scenarios as a starting point. We don't assume the pivot is preserved after we start doing that.) Proving that such a pivotal situation must exist at all in this concrete scenario is mostly what the first part of the proof was doing, but the positions of Bulbasaur and Charmander were only _relative,_ and they kept the _relative_ position of Squirtle in the votes and results abstract/unknown. But eventually they strategically assume Squirtle was in positions such that unanimity ensured they would finish last anyways, letting us use a more concrete example where everyone's rankings in the votes and results are absolute.

[@leondost3575]

please correct me if i'm wrong, but i think there is a mistake from 2:24 onwards. as i understand it, and as UB describes it at first, independence of irrelevant alternatives considers a scenario where one option (or all but the two being compared) are removed from the vote. i don't think this is the same as assuming they stay in the race but change positions. for example, i wouldn't say removing donald trump from a presidential election(by the system shown) is the same as moving him up or down on anyones ballot paper.

[@seahlilynn]

Really love the way you have presented this. It's entertaining :) and made me learn something new. Nice and clear voice too! Thanks for the amazing content!

[@DrewryPope]

but ranked choice optimizes what i care about which is who i don't want in?

[@youtubeuniversity3638]

I'd disagree on what dictatorship qualifies as. If anyone can be the decider, then as far as I'm concerned, the title is inapplicable just as postulate. It's effectively the same as saying "Because it is possible to pick a winner, it's a dictatorship."

[@MustSeto]

Arrow's dictator is always the same dictator regardless of how anyone else votes. Only the dictator can ever be the decider.

[@adamyamadakelly7242]

I don't agree with the dictatorship criteria. Just because there is a pivot doesn't mean that person is a dictator, mostly because they do not know they are the pivot and whether they are a pivot or not depends on who everyone else voted for.

[@MustSeto]

It might not be clear, but this is not correct. The dictator is always the same dictator no matter how other people vote.

[@prithwishguha309]

I don't understand, you said even after moving squirttle above of charmender, charmender still is the winner; but how? doesn't it break the unanimity?

[@MustSeto]

No, Unanimity is only used at a couple points. Unanimity was in involved in proving what the result of the election had to be before squirtle was moved, but only to set up the situation. From there IIA took over to prove that the results couldn't change even if squirtle was moved in that way

[@jstan55]

@@MustSeto but why doesn't IIA mean there is a preference for squirtal beating charmander?

[@MustSeto]

@@jstan55 At that point in the proof we had already established that Charmander > Bulbasaur and Bulbasaur > Squirtle. Then the non-pivots move Squirtle, but this doesn't change anyone's relative order between Squirtle and Bulbasaur, so Bulbasaur > Squirtle must stay the same by IIA. But it also doesn't change anyone's relative order of Chamander and Squirtle, so Charmander > Bulbasaur must also stay the same by IIA. And if Charmander > Bulbasaur and Bulbasaur > Squirtle are both true, the only way to put them together is Charmander > Bulbasaur > Squirtle. Which means Charmander must (still) be above Squirtle as well. I think this is partly an artifact that the final results must be a simple ranking list. If you allowed a nontransitive final "ranking", then you might get some other weird effects or cycles (Charmander > Bulbasaur > Squirtle > Charmander or something), but I'm not sure these specific criteria would be coherent anymore. In fact you could probably argue that's what Arrow's theorem really suggests: simple rankings for the inputs and output isn't enough. As the next video mentions, using scores/ratings doesn't exactly have this problem

[@jstan55]

@@MustSeto I'm pretty sure Arrow calls the result a paradox (1950: 329) because it does not accord with the law of transitivity. It can be proven simply, it just hasn't been done here. As you increase the nuance of the inputs the result becomes more possible, not less... I think I might be over thinking it though!

[@vishesh0512]

I am not very comfortable with this idea of the pivot being a 'person' and not a 'number' (the passing vote %age or something), because then the order in which you are putting people rigs the elections. What I am saying is that if you could prove that Mr. F is the dictator for everything, then this scenario: A A B B A B B A A B C C C C C has the results B > A > C (cause the middle guy is the dictator for everything) While: A A A B B B B B A A C C C C C has the result A > B > C (cause the middle guy is still the dictator), even though both the situations must have the same outcome.

[@MustSeto]

> [...] even though both the situations must have the same outcome. Nothing says those must have the same outcome. For example, they could obviously have different outcomes in a dictatorship. So maybe it just means it's a dictatorship?

[@ganondorfchampin]

A better definition of a pivot is a player who, in certain election outcomes, could have changed the result by changing their vote. That way it has nothing to do with voting order. There could potentially be many pivots, it’s just with any unanimous system at least one pivot must exist so group preference can switch from one option to another. If you also have IIA, then the only pivot is the dictator.

[@Nole2701]

6:57 why must the pivot for C/S be before Mr. F? i dont get it. From what I see, nobody before Mr. F can change the fact that S/C if everyone after and including Mr. F votes S/C. It must be the result of one of the people after Mr. F or himself changing their mind to C/S that changes the outcome. If everyone before Mr. F really wants C/S, it can only come true once Mr. F or someone after him gives in to the mob and becomes the pivot as a result. So from my standpoint it looks like the pivot for C/S would be Mr. F or anyone after him. not before Mr. F like you said. please help explain anybody.

[@UndefinedBehavior]

It's a bit confusing and I went through this part a little fast. So at this point in the proof, we know that Mr. F is a dictator for C/S. This means that once Mr. F votes for C/S, C/S must be represented in the result. If we want to find the pivot for C/S, we start with everyone voting S/C. By unanimity, S/C wins. By the time we get to Mr. F, when we flip his vote, C/S is now forced (because he's the dictator). A priori, it's possible that one of the votes before him flipped this result; the only restriction is that after Mr. F, no one else can flip the vote. This means that no one after Mr. F can be the pivot because the result has already flipped by the time it gets to them.

[@Nole2701]

sorry im still lost. 6:40 and 7:10 look exactly the same but just with the pokemon roles reversed. Well, im still 15 so am I missing any information that might have been taught before I reach this topic?

[@UndefinedBehavior]

I think you've got the main idea, it's just a little reversed. It is definitely confusing with all the different moving parts. When I say Mr. F is a dictator for C/S, what I mean is that if he puts C above S, C will definitely be above S in the result. However, at this point, if he has S over C, there isn't any implication about the result. In other words, C can be over S in the result even if he doesn't explicitly vote that way, but if he does, then C must definitely be above S. Once we flip his vote to have C over S, then the result is forced to be C over S (by the definition of him being dictator of C/S). What I'm describing from 6:40 to 7:10 is this process. Because the result switches from S over C to C over S sometime up to or including Mr. F, no one after Mr. F can be the pivot for C/S because the result will already have switched.

[@Nole2701]

so basically you are saying that Mr. F doesnt have to the one to change the result. It can be someone before him. Which means that the one before Mr. F who changes the result would be the "new Mr. F"

[@UndefinedBehavior]

Yup, that's exactly right. It turns out that if we do this a few times, we'll discover that the "new Mr. F" is the same as the original Mr. F.

[@addisonwu9324]

How can this channel only get so few views

[@PvblivsAelivs]

I tend to prefer "Condorcet" methods. (It should be noted that the system we currently have is not majority, but rather plurality. In most "practical" elections, there will be a Condorcet winner and no incentive for strategic voting. Arrow's Impossibility Theorem is the result of preference cycles A > B, B > C, C > A.

[@ganondorfchampin]

I think practice has shown that assumption is ridiculous, usually there is no Condorcet winner as a result of the population being so polarized on various issues.

[@PvblivsAelivs]

You think wrong. A polarized population makes a Condorcet winner even more likely.

[@ganondorfchampin]

John Undefined It would be if polarization was deep, but in practice it’s not. What polarization actually consists of in reality is a bunch of different factions forming alliances to push their popularity as close to 50% of population as possible, but if you were to break them back apart again there is numerous differences. Thus whenever a third party enters, they tend to act as a disruptor, preventing Condorcet winners.

[@ChrisInTheNorth]

I was at a disadvantage in this video because I know nothing about Pokemon! Of course that shouldn't matter.. they are just names of candidates.. but it did leave me trying to keep up. Its still interesting stuff though

[@tjcraft7233]

Tell me if I'm wrong but there is no such a person as mr. F right? Everybody in the majority gets to be able to decide the outcome of the vote ... no???

[@rahulmalik1083]

Even I think like that , let's say they count the vote randomly , let the sequence of counting votes be 1 , 2 , 3 2 , 3 , 1 3 ,1 , 2 There is possibility of vote counting from any of these possibilities , then in each case some other person is the "Pivot " or let's say the "dictator" . If anyone can be a dictator without his knowledge of being a dictator , it's not dictatorship , it's totally "majority" . Arrows impossibility theorum seems bs , but that's just me .

[@MustSeto]

Ranked methods that pass IIA and Unanimity are outright dictatorships, where the dictator is always the same and is the only one who ever matters, and can never be overruled even if everyone else changes their votes and/or votes against them. Of course practical methods all either fail IIA or are rating-based, so they don't have a dictator.

[@Elocess]

If a random person chosen from a population dictates the voting result without knowing he/she had such power, is that really more unfair than democracy?

[@MustSeto]

Arrow's theorem doesn't say it's a random person who doesn't know they had the power, it says the method must have a way to single out one person as a dictator, even across elections, and will only ever care about them, even if everyone else votes against them

[@Santiago_valdivieso852]

Before the problem of the dictatorship we have a problem of construction of the social welfare function (in particular, we have a not well defined one) : Charmander is prefered to Bulbasur, Bulbasor is prefered to Squirtle BUT Squirtle is prefered to Charmander. So we have a dicatorship ONLY IF we decide in what order should we count who beats who. If we dont do that, we just can't define netiher the social function nor the winner but in Arrow's Theorem it is possible to end the problem there (the violation of the first condition). PD: Thanks for your videos. Now im making my thesis thanks to the ideas this content gave me.

[@EHuelsz]

The solution is simple, taking the independence of irrelevant alternatives as a fair property, is a mistake, because those alternatives do matter at the end. A way you can make a fair election, taking into account people preferences, could be just by giving 3 points to the candidate they choose 1st, 2 points to the 2nd one, and 1 point to the 3th, so the candidate with more points win, in that way you can still keep the unanimity property and get a possible and real democracy.

[@JDSileo]

That also doesn't pass the test

[@Ptotheextreme]

Why choose just one system? Why not have a mixture to please as many as possible?

[@tirzakharisma3566]

I will do a presentation for this, but I don't want to do 10 minute explanation like this since they don't have to know this deep. How to do that, I wonder...

[@ZealousAnubis]

So cool, using pokemon icons to explain these things!

[@jamowrer]

omg i just lost my last brain cell

[@paulgemperlein626]

Wouldn't ranking candidates individually (0 to 10 where multiple candidates can have the same rank) completely avoid this problem? Sounds like you're saying, at least in part, that ranking is arbitrary and preference for each candidate is not neccisarily seperated evenly by 1st, 2nd, 3rd, etc ranking. As an example, a lot of people would have been forced to say 1st Bernie, 2nd Hillary, 3rd Trump if the US had this system but really it should be Bernie: 10 out of 10, Hillary: 8 out of 10, Trump: 1 out of 10

[@UndefinedBehavior]

What you're describing is range/score voting (here's our video: kzbin.info/www/bejne/m2Sqd3pmqL18n8k), and yes, it totally solves this problem.

[@paulgemperlein626]

Oh awesome!

[@brycejohansen7114]

I don't think it's a dictatorship if no one knows if they are the dictator, the fact that it's random unknown factor makes it fair. I'd personally have two voting systems. The first is pick the pokemon you don't want... if it's a tie, you have a winner If not, you have a second vote on what pokemon from the remaining two you want.

[@MustSeto]

It's not random, the dictator is always the same dictator no matter how other people vote.

[@brycejohansen7114]

@@MustSeto That's simply not true, if everyone else's vote was switched to identical yet opposing the dictator then it doesn't matter what the dictator voted... you're not always going to have a swing vote situation. If anything as the voting pool gets bigger, you'd need *multiple* swing voters that can push the vote in either direction rather than a single voter.

[@MustSeto]

@@brycejohansen7114 It might not be obvious from this video's explanation, but yes, the dictator is the only vote which ever matters, the others are fundamentally ignored and never considered by the method, even if they change, even if they all oppose the dictator. The dictator isn't simply a swing voter, they just look like one momentarially due to the way the proof is constructed, until it's proven they were actually always a dictator. Mr. F changing their vote causes the final results to change, right? That could either be because they were simply a swing voter, and their vote was just enough to push them over the edge---OR it could be that the votes changed because the method was only ever considering Mr. F, and this is just when Mr. F change their vote. The rest of the proof is checking which of these was the case, and it ends up being that if the method passes Unanimity and IIA, it can only be because Mr. F was an outright dictator.

[@brycejohansen7114]

@@MustSeto And if they get removed from the poll? Naturally the Dictator would shift to another person.

[@MustSeto]

@@brycejohansen7114 Well, the same thing that happens to any voting method that gets 0 votes. The method only really counts the dictator's vote anyways, so even if there are other "votes", it's the same as there being 0

[@kyrilcouda]

To all you idiots who think that this proves somehow that democracy is not possible, just stop right there and read: This video describes a scenario, where a relation is taken into account. Relation, that is an ordering defined in a really abstract way. This whole paradox is possible because we are taking pairwise orderings based on majority for those pairs, while ignoring the overall relation of the pair ro the rest of the candidates for each voter. This voting system is inherently flawed, as described in the video, but that only comes from the pure relational aspect forced by the UNANIMITY requirement. Lets take into account another aspect. The "extent to which a voter likes the candidate". For example, if I strongly prefer candidate A over B, C, and D, but do not care in any way about B, C and D, I should be able to express that. In the system above, I cant express that, since there is a forced ordering with fixed weight. Weighted majority voting: One, very simple, yet powerful idea, is to let the voters to rank every candidate with a real number in and then sum rankings for each candidate. This is a method used in machine learning for majority voting since ensemble methods became a thing and it works flawlessly. Not only you can capture the ordering as you have in the example above, but you can also rank multiple candidates the same amount (without forced ordering). The sum of ranks then does not take into account the majority pairwise ordering, but the complete weight of the candidate across all of them. UNANIMITY here does not make sense... because the ordering itself is not really important, the only important thing is the PREFERENCE WEIGHT, which is taken into account perfectly here and captures the overall voters preferences in teh best way possible. I want to see an example which would disqualify the idea of weighted majority voting. So far everyone I was talking to was waving hands and screaming Arrow theorem to me, yet no one really understood it themself.

[@MustSeto]

This video has a follow-up which already addresses that. Look for "How We Should Vote (Range Voting)". Your idea sounds exactly like Range Voting (more commonly called Score Voting) and yes, as the video mentions it evades Arrow's Theorem by virtue of not being based on ranks, but on ratings

[@RipleySawzen]

Unanimity and Independence of "irrelevant" alternatives are both stupid though. I don't want either. Unanimity is stupid because consider the following situation: You have 4 candidates. Three of the candidates have 1/3 of the top vote each. The remaining candidate gets ALL of the 2nd votes. And everyone HATES their 3rd and 4th choices. It would make sense for the compromise candidate to win. No one is happy, but no one is miserable either. As for Independence of "irrelevant" alternatives, I don't consider them to ever be irrelevant. The fact that their removal can change the results proves it. And I don't even see it being a problem anyway. It's obvious that it only comes up during a very tight election that's essentially a tie.

[@MustSeto]

In your unanimity example, no candidate is unanimously preferred to any other. So a method can pick the compromise candidate everyone put 2nd without violating unanimity. I'm not sure what you're trying to say about independence of irrelevant alternatives. The theory is that ideally, removing a candidate should not be able to change the results if they weren't the winner. It's not a priori obvious that this is impossible, and indeed, there arguably are methods where this doesn't happen.

[@td9250]

Maybe this would've been more relevant if the votes were magic and changing in the ballot.

[@jetpackkblues]

that arrested development reference tho

[@scottjuhnke6825]

Go read Aristotle's Politics. His critique of Democracy explains why Democracy is the worst form of Tyranny.

[@Auroral_Anomaly]

Why don’t we just I don’t know, vote once and not rank them.

[@MustSeto]

Like in Plurality? Plurality votes can be losslessly encoded as ranks, and fails IIA. You could score them instead though. Scores can't be losslessly encoded as ranks, so Arrow's theorem doesn't apply.

[@Auroral_Anomaly]

@@MustSeto Just see who ever has the most votes, IDK.

[@MustSeto]

@@Auroral_Anomaly That's a great method, if you allow voters to vote for more than one candidate (Approval). Otherwise, if you only let them vote for one candidate, it's terrible (Plurality).

[@Auroral_Anomaly]

@@MustSeto Ok, ig.

[@viniciuskfm]

How's charmander winning and sqirtle losing in 5:40? I get there's independence of irrelevant alternatives, but that's not the case Charmander shouldn't even be first in that scenario, let alone be agead of squirtle

[@MustSeto]

Why shouldn't they be first? I mean, we never assumed the method _didn't_ have a dictator. And if F is a dictator for the method, then obviously the method would put Charmander first there. The application of Unanimity and IIA just helps us find who they might be. The process described in this video can be thought of as the process of diagnosing an unknown method to find who any dictator might be.

[@hounamao7140]

5:50, why is charmender the winner ? He didn’t get the majority. Didn’t we decide on the majority rule ? Unanimity is only useful to rank the rest

[@MustSeto]

No, we didn't decide on Majority Rule. And it turns out no rank-based method that passes IIA and Unanimity can be majoritarian, because it must be a dictatorship, and dictatorships are not majoritarian.

[@mrowlbert]

My poor little brain can't handle this lol

[@exoendo]

im high right now and I can't tell if this is really fucking obvious shit that doesn't even need to be explained, or some deep theorem

[@gogigaga1677]

GREAT EXPLANATION THANKS

[@whatno5090]

Idea: Perform a matching algorithm using each peoples choices to match to them the best selection. Then, do majority vote on that (if possible). If not, repeat and make them change until a single choice is selected. The reason this doesnt violate the theorem is because we have to count the vote multiple times (if majority is selected then it isnt a violating edge case)

[@TheTrueAdept]

No this theorem has cullinaries which goes into alternative systems of voting... and they go into how alternatives will go back to the problem of the original impossibility theorem.

[@reubena7854]

surely you can implement condorcet in range voting?

[@MustSeto]

In range voting the winner has the highest score, so their score would win in head-to-head matchups against any other candidate's score. In this sense they are always Condorcet winners, _and_ there is always a winner (barring exact ties). This isn't usually how Condorcet winners are defined though. But I don't think Condorcet winners in the normal sense are strictly desirable.

[@panda4247]

5:35 isn't this like rock paper scissors? I mean: if we ignore Squirtle, we can see that Charmander beats Bulbasaur. if we ignore Charmander, we can see that Bulbasaur beats Squirtle. if we ignore Bulbasaur, we can see that Squirtle beats Charmander. Therefore, the ranking Ch>Bu>Sq that you present, exists only because you looked at the first 2 things only. You can't say that Rock beats Scissor, and Scissor beat Paper, therefore Rock should beat Paper. When you have e.g. 10 candidates, you have 45 pairs to compare, but you only need 9 (put 9 > signs) to make some ranking like A>C>G>B>E>F>H>J>D>I So, in this system, the true "dictator" is not the one single voter, but the person choosing which pairs to compare.

[@ganondorfchampin]

No, because it's not based on majority, but unanimity. Squirtle does not beat Charmander because Mr F says no. If there actually was unanimity, it would be impossible for a strange loop to exist as people can't give strange loops in their ballots, and everyone would be agreeing with the Mr. F anyway. What you noticed is that Condorcet winners might not exist, but even though the final rank must be transitive it doesn't matter that no Condorcet winner exists as the voting process doesn't have to be democratic. We know the relationships between anything and bulbasaur had to stay unchanged because nobody changed their preferences for bulbasaur, that's why we only look at them. It has nothing to do with order or choosing pairs, it has to do with the fact we are assuming independence of irrelevant alternatives.

[@kanewilliams3613]

great video, you explained a hard concept very simply thanks for making it :)

[@20acivi08]

5:34 doesn't make sense to me. why is charmander no 1? should be squirtle to me charmander 3:2 bulbasaur charmander 1:4 squirtle bulbasaur 3:2 squirtle everybody wins 1 and loses 1, but the most "single" wins has squirtle... dunno which system you took to count. even if you give 3 points for every 1st place, 2 points for every 2nd, and 1 point for every 3rd place, squirtle would win. moreover charmander is the only one that has just one 1st place, while the other two are on first place twice.

[@MustSeto]

The method used to count was one which passes IIA and Unanimity. So charmander wins because it's forced by a few assumptions made earlier in the video: 1. The method gives the results at 4:28 2. Mr. F was the pivot at that time 3. The method passes IIA 4. The method passes Unanimity You don't need to assume anything else about the specific method. Any method which passes IIA and Unanimity will behave this way. It turns out that the final results of any method that passes these will just be a copy of a pre-defined voter's ballot (the "dictator").

[@Kostas1601]

Fantastic video! Thanks!

[@theeternal6890]

There needs to be balanced system which may work.

[@No0bT4rD]

is this for children?

[@nyjar7915]

for people that had childhoods

[@yan-qu5zm]

NY JAR Amen, sir.

[@lukedavis6711]

NY JAR lol burn

[@anselmschueler]

Why do you think this? Is it beacuse of the usage of names of a franchise somwhat popular with children? In this case, I disagree, because of: A: www.gamefaqs.com/boards/187276-pokemon-sun/74302335 B: medium.com/@sm_app_intel/pok%C3%A9mon-go-demographics-the-evolving-player-mix-of-a-smash-hit-game-b9099d5527b7 Additionally, the usage of "children's ..." doesn't dictate the intended demographic of a certain ... . In fact, considering "children" often indicates particular low age and not necessarily all non-adults, one could argue that this would be the wrong demographic for this video.

[@kennyearthling7965]

This is not how voting works. Everyone votes at the same time, anonymously. Jyst because there's a tipping point, does not mean that any one person has all the power. Poor understanding duspkayed here is disappointing

[@MustSeto]

The method is just a function that can take any possible collection of ballots, so it's not like we're assuming someone sees the other ballots and carefully crafts their own to get what they want. We can always consider a collection of ballots which is exactly the same except that this someone submits any other ballot. The theorem doesn't start out assuming whether or not the method is anonymous. "Anonymity" is another criteria a method can pass or fail, it's basically saying that if you reorder ballots then the result will always be the same. But it turns out that's incompatible with Unanimity + IIA. Because a Dictator is not just a "tipping point", they're in full control of the every election no mater how anyone else votes for as long as that method is used. So dictatorships cannot be anonymous. You're right of course that that's not how any real voting method works. Because every real voting method fails IIA.

[@martinburns7928]

Candidates are problematic as party affiliation drives substantial ideology w/ factors of historical activity may very well exist as actual complete opposite of intentions campaigning dialogue!

[@andriasdickson7129]

I don't play pokemon so the names made it confusing :/

[@DrEhrfurchtgebietend]

Or you could use a range/score system

[@Max-nc4zn]

Better to just use economic means than political means for solving societies problems most efficiently.

[@3snoW_]

But mr F isn't a defined person, it could be mr F or the next guy, all you are proving is that changing the votes one at a time there'll be a guy that "decides" the result because they'll shift the majority from one side to the other. But since that guy could be anyone, that just means that majorities win, and that's what's expected out of a democracy.

[@MustSeto]

It might not be clear from the video, but the dictator is always the same dictator regardless of how anyone else votes.

[@agranero6]

Favorite betrayal is not unfair it is a legitimate choice, you can see it better in elections with multiple candidates. You relinquish your vote for your preferred candidate so other don't win ( In Brazil we call that "useful vote"). Say that is unfair ignores surveys that give you the knowledge (supposing fair pools) that your candidate will not win so relinquish your vote is irrelevant. Besides being fair does not imply in voting for you preferred candidate only means voting against one you fear. Specially if a candidate is seem as dangerous (how many time candidates got to the power by voting and never get out? Hitler was elected before became a dictator). Besides the pivot being the dictator is an edge case, specially on multiple candidates and millions of people voting, and the dictator does not know he is the dictator, meaning that fact that one person will decide the result is irrelevant as long as this person doesn't know it, it is an edge case very improbable to happen specially with the same person over and over (meaning it does not fit your intuitive and even dictionary definition of dictator: meaning the pivot is just legitimately making their choice unaware of his vote will decide all he/she will not change his vote as the example shows to demonstrate he/she will be the pivot, the pivot will be just one of those that voted for the winner their vote is not less legitimate just because it decided the election do this looks a dictator to you?) . *All very abstract but ignores all that: election pools, the right to vote for someone you wouldn't vote for the greater good or at least for the lesser evil. All that makes lots of assumptions that seem reasonable until reality gets in the way: ignores rights of voters, legitimacy of the vote of each one independent of the result it leads, human nature and the semantic of what is a dictator be a twist of language.* Fairness must maximize an objective function lets say happiness about the role of your vote on the result. It also ignores anonymity (or the votes being valued the same or yet commutativity) and others that are also reasonable and continuity: small changes causing big differences in the result and then we will get other problems like Chicilnisky impossibility theorem. Feynman used to say: "I receive letters every day saying this or that in Physics is wrong: space-time should be quantized, etc. But not a single letter says how to do that consistently with math and experiments" (paraphrasing because I am too lazy to search the exact quote. See the relation?

[@MustSeto]

I don't think the main issue with favorite betrayal is whether it's "unfair" or "a legitimate choice" with respect to any particular voting method, the bigger issue is with what higher-order effects it has, such as center-squeeze, and artificially encouraging duopoly. In Arrow's theorem, the dictator must be a consistent voter across elections. If you hold a new election using the same method, then the same voter must be a dictator again and the results will be a copy of their new ballot, no mater how anyone else votes. This does mean that Dictatorships must violate Anonymity.

[@agranero6]

@@MustSeto It is impossible for anyone know the other voters vote so no one can PLAN to be the dictator, besides it is an edge case that only happens in very balanced elections so one vote can make this. It is unreal as an Adam with a bellybutton.

[@MustSeto]

@@agranero6 > It is impossible for anyone know the other voters vote so no one can PLAN to be the dictator Why would it matter if no one knows how any one else will vote? The identity of the dictator does not depend on anyone's votes, so the dictator could easily know that they're the dictator ahead of time.

[@agranero6]

@@MustSeto How if everyone votes at the same time? It is only possible if excluding the dictator there is a tie. Explain how he could know he is the dictator? Dictator is a misnomer: dictators can impose their will no matter what the case. The pivot, the deciding vote is a legitimate as any other.

[@MustSeto]

@@agranero6 Arrow's theorem is agnostic to how such a method could come to be used. But in real life, I suppose they (or their regime) would have to enact and enforce the usage of the method themselves. And the method would have to have fail Anonymity -- it'd need some way of identifying which ballot is the dictator's. Or on a smaller scale, the dictator could just be tricky. Create a method with them as the dictator, but disguise how the method works and convince some electorate/organization to use it. Something like that.