Voting Paradoxes - Numberphile

How to decide who wins a confusing vote, discussing the likes of Smith Sets and the Condorcet Paradox. Featuring Sophie Maclean. More links & stuff in full description below ↓↓↓
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[@gorpand]

What about KZbin's voting system, where you can choose between "Like" or "Dislike" and nobody can see how many dislikes there are.

[@opensocietyenjoyer]

dictatorial in nature

[@1ucasvb]

Rating comments or products is not the same thing as voting, though. The interface looks the same, but the context is very different. Voting is usually a collective decision-making process, where those voting will be affected by the decision so they have stakes in it. Voting also has the voters giving an opinion on the same set of options at the same time, so we expect the vote to be a comparison between the options. So there's a "shared context" to the information collected. This means some ways to analyze that data are not going to be valid. In comparison, people are not rating every comment or product at the same time, so there's no shared context and no comparative element to it. There are two voting methods called "approval voting" and "evaluative voting" which are closer to the like/dislike idea. In approval voting, you approve/like as many options as you want, the most approved one wins. In evaluative voting, you can approve/disapprove each option, or be neutral about them. It's like giving +1/0 and +1/0/-1 scores.

[@samuelgarrod8327]

​@@1ucasvb🥱

[@NoNowwwell]

They only got rid of dislike when people mass downvoted all of Biden's whitehouse videos. If Trump wins, watch the dislikes come back.

[@ApKieras]

Censorship at work.😂

[@IisChas]

Oh my gosh, Sophie Maclean! She’s my favorite host you’ve invited on. I love her enthusiasm.

[@soundreamerbg]

math crush

[@IisChas]

@@soundreamerbg, real.

[@srwapo]

PBS Infinite did a video called voting systems where they discussed the pros and cons of different methods. In the end, they came up with an election where each system gave a different winner.

[@falutzel7942]

Veritasium did an interesting one too.

[@jarlsparkley]

Bring back PBS infinite series!

[@Kram1032]

There's also a bunch of relatively early CPG Grey videos on various voting systems

[@Anklejbiter]

I also really like Primer's video

[@vlc-cosplayer]

So what you're saying is that we should vote on which election system to use, and turn this into an infinite recursion problem? 👀

[@platypi_otbs]

Sophie is wonderful. I hope you'll have her on the channel more.

[@MrAlanrees]

Love the subtly placed Alice adventures in wonderland quote at the end.

[@Jsjsjjssjs]

Ramon Llull predates the idea of Spain, he was born in the Kingdom of Mallorca, and wrote in catalan, latin and arabic, and is considered one of the first to write literature in a romanic language

[@Frostbiker]

Isn't that splitting hairs? It's like saying that Maimonides wasn't born in Spain, or that Benjamin Franklin wasn't born in the US. Technically true, but nobody thinks of them that way.

[@JorgeHernandez-gm5fk]

@@Frostbiker I'd say its a fine point to make. Ramon Llull didn't speak Spanish, the kingdom he was in didn't unite with Castile for over 100 years after his death. Ben Franklin is not a great comparison as during his lifetime he became a US citizen. A better example would be claiming a random Hawaiian noble who died in the 1800s (and didn't speak English), as an American.

@@Frostbiker Of course, Julius Caesar was Italian, Seneca was Spaniard, Vercingetorix French, Tsar Nicolai Soviet, and Pocahontas a US citizen.

[@brunoparga]

The Phoenicians already called it Spain...

[@londislagerhound]

The day Llull comes up on Wordle will be a bad day.

[@pokerformuppets]

Ha! Having 80% of a word being one letter is kind of impressive.

[@framegrace1]

LL at beginnings and ends of the word is the Catalan way to torture foreign speakers... And this famous guy has both kinds :) . (His names in other languages drop both with no exception: Lulio, Lulius, Lully, Lullo, Lulle ... (And yes we catalans pronounce them) BTW, he is also (maybe) the first to write "serious" literature on any romance language.

[@RFC3514]

@@framegrace1 - Catalans are amateurs compared to the Welsh.

[@yenwastaken]

As if 'corer' wasn't bad enough

[@SupercriticalSnake]

@@yenwastaken I saw that the average number of turns to solve for hard mode players was 5.9 so I had to check online, and sure enough, people were not happy about that day's Wordle. The Wordle Bot didn't even solve it in six.

[@themroc8231]

Spanish is my native language, your "Ramon" was very good and i have no idea how one should pronounce "Llull" either

[@pierreabbat6157]

I think it's pronounced like the start of "Llullaillaco", as pronounced by those who distinguish "haya" and "halla".

[@ButzPunk]

It's [ʎuʎ] (where ʎ ≈ the "li" sound in "million")

[@tristanridley1601]

My ignorant anglophone brain is thinking "Is this pronounced yuy?"

[@vlc-cosplayer]

The intrusive thoughts are suggesting you pronounce it "Juul" 💀

[@RFC3514]

@@tristanridley1601 - A lot of English speakers do seem to assume that "ll" in Spanish is pronounced "y", but that's only the case in some regions. More commonly, it's pronounced like "lh" in Portuguese or "gl" in Italian (basically, a kind of slurred "L").

[@TheNameOfJesus]

@1:44 - all I will remember from this video is that there is a word that contains 80% of the same letter of the alphabet. No word can score higher than that except "I" and "a".

[@stechuskaktus8318]

Eevee ties though. Yes, Pokémon count!!!

[@RFC3514]

Assuming you're talking about Llull, that's not really "a word". It's a name (i.e., a a proper noun). And there are several names (of both people and places) 100% made up of a the same letter. For example, the rivers D, O and Y, or the villages of Aa and Ee, etc..

[@tim40gabby25]

Nooooo!

[@gabrielrockman]

The biggest problem is that we don't even know what's most important in an election. Do we want to maximize how many people get their favorite candidate? Or do we want to minimize how many people get their least favorite candidate?

[@andrewrollason4963]

Yes we do. We want to minimize how many people get their least favorite candidate. The reason for this is government's legitimacy and moral right to use state power is justified and lawful only when consented to by the people or society over which that political power is exercised. As Consent is the most important driver, then you want someone whom the most people will consent to, however begrudgingly; which means you want full ranked choice voting.

[@gabrielrockman]

@@andrewrollason4963 No, it's not clear that we want to minimize how many people get their least favorite candidate. If we have one candidate that is the favorite of 55% of people and least favorite of 45% of people, a second candidate that is the favorite of 40% of people, second favorite of 5% of people, and least favorite of 55% of people, and a third candidate that is the favorite of 5% of people, second favorite of 95% of people, and least favorite of 0% of people, which candidate will win? Which candidate should win?

[@andrewrollason4963]

@@gabrielrockman "If we have one candidate that is the favorite of 55%" Then you have just told me that person is already elected with a majority.

[@gabrielrockman]

@@andrewrollason4963 That depends on what your goal is. If your goal is to get the candidate that is the most people's favorite, yes. If your goal is to get the candidate that is the least favorite of the smallest number of people, no. As I said, we need to decide what our goal should be.

[@felipevasconcelos6736]

@@andrewrollason4963 you literally just said one comment before that we want to minimize how many people get their least favorite candidate. By your own metric, the third candidate, which very few people like, but no one particularly hates, should be the winner! The point is that your choice of voting system isn’t neutral, and neither is your choice of metric to evaluate voting systems! Your metric would make it almost impossible for a non-centrist to win. Conservatives are never going to hate centrists more than progressives, and progressives are never going to hate centrists more than conservatives. As long as both progressives, conservatives, and centrists run, centrists are guaranteed a win, and no one has any incentive to not run.

[@darkpulcinella9690]

About time Numberphile talk about the Mathematics of voting. Other major channels talked about it (PBS infinite series, CPJ Grey and most recently veritasium) and other minors yt channels too. I am looking forward for this video

[@beachboardfan9544]

More Sophie! 😍

[@essentialatom]

People who hold their pens that way scare me

[@antonfelich]

Thank you!

[@GrouchierThanThou]

Sophie is one of my favorite characters on this channel but the way she holds that sharpie while writing kind of hurts my eyes.

[@twwc960]

I don't think the presenters on Numberphile are characters. I think they're actually real people. I could be wrong though.

[@Liwet.]

Her font is nice though.

[@GrouchierThanThou]

@@twwc960 Character, in one of its many meanings, is just a synonym for person or individual.

[@GrouchierThanThou]

@@Liwet. True.

[@goodfortunetoyou]

I would totally like to see more videos on voting systems and their mathematical properties.

[@Liwet.]

CGPGrey did a pretty entertaining series.

[@JimCullen]

​@@Liwet. They're entertaining but extremely shallow and don't delve into any of the mathematics behind voting systems

[@daemoneko]

Veritasium has a recent video(2 weeks ago) about some other contradictory possibilities in some voting systems

[@Simpson17866]

Score voting (where every candidate is rated from let's say 0 = Complete Disapproval to 4 = Complete Approval) doesn't have any weird complexity to work out: The candidate with the highest average score wins :) The only weird thing that can happen - where most people gave Alice a higher rating than Bob, but Bob got a higher average anyway - only happens when many people give A a slightly higher rating (meaning that they don't really care one way or the other) while a few people have B a far higher rating.

[@Liwet.]

@@daemoneko If I'm remembering correctly, he seemed to have an issue where the guy with the most votes won despite only winning with 41% of them.

[@Kayclau]

The Condorcet paradox. AKA Paradox The Condorcet. AKA Condorcet Paradox the.

[@RagingStream]

The Paradox Condorcet

[@5hape5hift3r]

paradox of the Condorcet

[@xocomaox]

I c what u did there

[@stevenogbuehi3846]

I'm curious which major countries have a voting system where you rank all the candidates when you vote. Don't most places simply allow you to pick only 1 candidate when casting your vote?

[@AndorianBlues]

Australia and Ireland are two I know of for sure, there's definitely more

[@xtifr]

Australia and Ireland, for sure, maybe India, and I think there's a handful of others. Although many more countries use ranked voting for some _local_ elections, but not for nationwide. For example, several cities in the San Francisco bay area use ranked voting to choose a mayor.

[@HotCoals]

So the most common voting system on a national level in democratic countries is a system called Mixed-Member Proportional (MMP). Under MMP, voters basically get two ballots: One for their local candidate, and one for their preferred party. The voter ticks a box next to their favourite local candidate, and a tick next to their favourite party. After polls closed and votes are counted, the local candidates are elected as you're probably familiar, under a winner-take-call "First Past the Post" system. Right after, however, the party votes are counted, and the remaining seats are distributed proportionally to each party to match their percent of the party vote. So, let's say the Purple Party got 40% of the seats after the local candidate races, and the Yellow party only got 10% of the seats. But, for the Party vote, the Yellow Party got 60% of the vote, they would gain the remaining 50% of the seats (10+50=60%). (Who gets these seats are determined by what flavour of MMP a country uses. Closed-List MMP has the parties themselves asign seats to their prefered candidates, but Open-List MMP allows voters to choose their preferred Party candidate during the Party vote). Another system, more in-line with what the video is discussing, is Ranked-Choice Voting. This system is typically used more in lower-level elections, like municipal elections. But, STV, a multi-winner variant of the system, is more common, and is also used nationally in Ireland and the Australian Senate. In either system, voters rank their choices from 1 to whatever. In a single-winner RCV system, the votes are counted over and over, which each count eliminating the candidate with the least amount of votes, and transfering their votes to the other candidates based on the voter's preference (if Candidate F lost, many of their votes would be transfered to Candidate G because many voters listed candidate G as their next preference). Candidates are eliminated each round, until all other candidates have been eliminated or until one candidate crosses the 50% margin. In STV, the multi-winner system, the same thing happens, but now any candidate which crosses the vote threshold (simplified as "1 divided by the number of allowable winners" percent. So if there are three winners, the threshold would be 33.3%). This repeats like before until all the winners have been chosen, with multiple rounds of elimination and votes being transferred to voters' next preference.

[@HeroDarkStorn]

In our country, for general state-wide election, we vote for one party but the can add "two favorites" from that party, from our region. So in the end, the parliament is made from voted parties, but each party is represented by their "popular vote" - for as many chairs as they earned. No, there was never any problem with this system, why do you ask? /throws papers, runs into shadows/

[@ArawnOfAnnwn]

@@xtifr India is First Past The Post as well, so no. To its credit tho, unlike the us version, it is actually a multiparty democracy as well as has an independent commission that determines its districts (so less gerrymandering). It's also parliamentary, not presidential.

[@tdm0fr]

In the example finishing at around 7:30 the S set contains 4 players. I get how the set was constructed, but what's the actual purpose of finding a set with the 4 players that are beaten by nobody else ? Especially with the example oriented towards a competition : shouldn't this method help to find a winner ? Again, in the example, since 2 players have 4 points, shouldn't the goal be to find if one of them could win instead of having a tie ? I guess i'm missing something here.

[@phlogchamp]

Could be a compromising method, as it gets rid of the absolute “losers”, keeping everyone that is better than at least those “losers.” From the Smith Set, you could conduct another election to narrow it further I suppose.

[@tdm0fr]

@@phlogchamp Didn't think of it that way. Indeed if your plan is to have a second round of election this could make sense.

[@JDSileo]

I will pass a value judgement: The paradox reveals an issue with single winner elections. If representing interests is a general goal of elections (for office or laws or comapny policies or anything really) A Ranked Choice Vote will pass more voter information than Plurality voting. *Proportional* Ranked Choice Voting (which is multi-winner) ends the paradox by sending a series of representatives roughly matching voter interests. Proportional multiwinner elections are more representative of the interests of the whole by mapping voter interests of the whole. All who get locked out of power by being the minority voter are not represented. Since we now know that systems can exist that can ensure that everyone gets represented, then maintaining a single winner system, and for sure maintaining a single winner plurality system, is objectively and demonstrably immoral. And the legitimacy of any decisions made from such systems become suspect. And the motivations of those in power through such systems to suppress reform attempts for proportional representation, delegitimize themself and are also suspect. I call on all who lose out consistently, especially those in heavily gerrymandered single party districts, to make note of this and make a note of those who claim to love democracy yet love to keep you locked out.

[@matejlieskovsky9625]

As someone who is hoping to one day get a paper on social choice theory published, I completely agree on the need for proportional representation, but: Sometimes you do need exactly one winner (like a president). Multi-winner systems get complicated quickly, making them hard to explain. Even working on this stuff gets complicated - to have a chance of getting some results, I am focusing on single-winner elections with three candidates and hoping the results will then be applicable to larger elections.

[@HotCoals]

​@@matejlieskovsky9625 Just have 3 presidents, boom done. You could also have a system where an elected body chosen through the Single Transferable Vote (STV)(proportional RCV) is the body who elects the president, maybe even from the body itself. Like, say, a parliamentary system! As for the system being difficult to explain... yeah. The actual process of how votes are counted in STV elections is a bit wonky and requires a few asterisks compared to "which number is biggest?", but on the voter side all thye really need to do is ranking, which is so easy that there used to be an entire genre of youtube video about it.

[@JDSileo]

@@matejlieskovsky9625 I find the need for a single winner dubious. You don't need a president. And even if you needed a snap decider you can have the snap decider be the chair of the small group with the nominal executive control in a system. (And you would want this especially if you want to distribute executive power between multiple interests) The complications of proportional ranked choice voting for the voter is: Put a 1 next to the one you like best. Assume they are going to lose and put a 2 next to your next favorite. Keep going until you don't want to anymore. " This concept (ranked choice voting) was explained to me in third grade at summer camp when they asked what kind of snacks we wanted and we had to pick a first second and third choice. If you can explain ranked Choice voting to 3rd graders then you should be able to explain ranked Choice voting to adults.

[@JDSileo]

@@HotCoals re: counting STV votes. I agree with you it is the most complicated part of the process, but even that is explainable. (Attention STAR nerds: I'm sorry but PR-STAR is madness [to count] and comes out about where PR-RCV does anyway in terms of proportionality. Sticking with PR-RCV/STV)

[@xtifr]

Multi-seat elections have their own lengthy list of problems, which are probably worth an entire video on their own. I'm a big fan of proportional representation, but it's far from being any sort of perfect solution! For example, it assumes that everyone fits into a small number of boxes (usually called "parties"). But a person's choice of party is often as much a compromise as their choice of a single candidate might have been!

[@germalganis]

“I’m pretty sure someone is going to correct the pronunciation” after perfectly pronouncing Ramón Llull.

[@McMxxCiV]

Aggressive gate on the audio there.

[@p07a]

Editor must have not enjoyed the reverb spending too many hours on it haha. I like the reverb of that hall.

[@andrewmirror4611]

Wait wait wait This is the kind of analysis we do in fighting games! Determining the strongest characters in a game based on their match up chart which this table clearly is. Important to note from our observations, the simple sum (which is very ironically apparently called Copeland) is very much like the most basic and thus worst way to analyze a match up chart, it doesn't give out expected results aside from maybe top spot in the most egregious of cases. And in general we deal with a lot of weighed RPSs and thus the common mathematical method of match up chart analysis, much the same as with weighted RPS analysis, is through matrix manipulation, finding the Nash Equilibrium. I have a friend that knows way more than me on that, I personally went with the simulation approach. The simulation is not built for "voting" match up charts, but I reckon it will produce much comparable to the expected results still.

[@LeonardTavast]

So you're saying we should use ELO ranking to select politicians with 🤓

[@andrewmirror4611]

@@LeonardTavast that's a different kind of analysis, match up charts are about characters regardless of players, while Elo and other derivatives are about ranking the players themselves. I had another simulation program to check how tournament formats line up with hypothetical "true Elo" rankings, but that project got shafted as that's where I had my only true contribution to the world of math, my own OEIS sequence A355729.

[@matejlieskovsky9625]

I saw the talk about using linear programming to find stable metagames! Brilliant solution, but does not work too well for elections. Would love to invest more time into social choice theory.

[@andrewmirror4611]

@@matejlieskovsky9625 I know the one you're talking about, in my testing it was a bit too vague, it would rather underexaggerate than overexaggerate. On the other hand my program tended to overexaggerate the results. (and their program was a bit more than just finding one equilibrium, it was finding every possible equilibrium for every position or smth, to say that not like matrix calculations don't work, more that their specific program did a very specific kind of analysis)

[@JordanBiserkov]

A, E, C, F can form the CAFE coalition. Or the FACE coalition.

[@TheNameOfJesus]

Is one of these methods how they determine the winner in chess tournament round robins, when two players end up in a tie? (Ties are common in chess tournaments.)

[@FlesHBoX]

As an American I am confused by having more than two possible winners...

[@danielsmerdel8214]

Sad, innit? I'm an American who is tired of being presented with "only two possible outcomes" and choosing "the lesser of two evils". But maybe not the best place to talk US politics...

[@goodfortunetoyou]

Why? Think of how many representatives there are per state. We already have multiple winners.

[@Giantcrabz]

​@@goodfortunetoyou where all the winners are careerist neocons

[@shugaroony]

Seems a lot of you were confused that Biden wasn't running...

[@killymxi]

Combined Approval Voting (CAV) needs a video. And it needs to replace our current terrible voting systems. It seems to strike the best balance between simplicity and fair results.

[@brunoparga]

It would still likely be vastly disproportional, as all single-winner methods are.

[@rajasekarm]

There was a vertiasium video on the same topic. Do watch that as well. It was quite succinctly illustrated.

[@janesda]

What does the second 'i' in MINIMAX represent?

[@MustSeto]

It's part of "MINI", as in minimum or minimize. If you do the min and max in the opposite order, it's usually called MAXIMIN. I don't know why they aren't called minmax/maxmin or maximini/minimaxi, except that I guess they don't sound as nice?

[@srwapo]

The letters look so sad every time they lose. 😢

[@aceman0000099]

Because of the rise in fascism they will see as a consequence.

[@ArawnOfAnnwn]

@@tbird-z1r Honestly I don't care about that guy, as I'm not American. But I do hope you feel that way soon, cos I am sick of your ilk, with your endless attempts to spread your politics everywhere you go.

[@AndersHaalandverby]

Hey, I have a question for numberphile/Brady and the math folks. In trackmania, a racing game, the player wirtual(famous streamer) got a score on a map that included all of the digits from 0-9 once with no repeats, so something like 01 hours, 23 mins 487 sec, 956 milliseconds *mind you the actual score was NOT this exact thing, but something like that. But the question is the same, what are the chances of hitting a score in a racing game that includes all numbers?

[@AndersHaalandverby]

granted, in this partiular map, for instance, he is almost 100% likely to finish under 2 hrs, so the first 2 digits will always be 0 and 1... so the question is really about the last 8 digits

[@MustSeto]

ts = sequence of all timestamps that have one of each digit of length k p(t) = probability of finishing the rate at some time t. For example p(x) where x is anything > 3 hr would probably be closer to 0 since it generally shouldn't take that long. If trackmania allows it you could probably check the track's history and create a probability distribution based on it. I think your answer would be "Sum from i = 1 to k of p(ts[i])"

[@AndersHaalandverby]

So , what you are saying you need essentially all possible combinations of digits , and a subset of those will include all different digits.. and the p will be the that relation

[@AndersHaalandverby]

So one possible combination could be 01.333.333 etc which is one of the larger subsets that needs to be compared to the number of subsets that included all digits

[@MustSeto]

@@AndersHaalandverby Well, I was saying the elements in Ts already have exactly one of each digit. So it'd contain 01:23:48.7956 but not 01:33:33.3333 p would be a probability distribution for _any_ timestamp. You could use it to find the probability of 01:33:33.3333, but _we_ wouldn't end up checking that one in the sum since it's not in Ts. But yes, you'd have sum the probabilities of every possible _valid_ timestamp (where "valid" means each digit appears exactly once, etc) One caveat here is that you presumably wouldn't have a timestamp like 01:98:76.5432 since you probably can't have 98 minutes, nor 76 seconds (well, your example has 487 seconds, so I'm not sure how trackmania's timestamps work, but there'd probably be some kind of restriction here) Creating the Ts sequence could be interesting. You should be able to build up every possibility digit by digit. Each string keeps track of which digits it hasn't used yet, then creates a batch of new strings each using a different, as-of-yet unused digit, etc, checking other conditions for relevant place (like tens of minutes), until you runs out of digits

[@ensamplescreens]

I've not watched a Numberphille video in a while, but i miss Brady's questions

[@Manisphesto]

I wanna see G to Z in this style of letters

[@ZandarKoad]

The only good part about election season: cool math videos.

[@RFC3514]

Is there something special about mathematicians that prevents them from holding a pen like a normal human?

[@BlameTaw]

That's a very common pencil grip. It's not "correct" but certainly quite common nonetheless.

[@RFC3514]

@@BlameTaw - It certainly seems common among mathematicians, hence why I asked. It's like they want to _carve_ the numbers into that brown paper. P.S. - Maybe image-generation AI was trained on extensive footage of mathematicians holding pens, that would explain why it's so confused about how fingers work. 😜

[@xocomaox]

​@@BlameTawI would argue this is an uncommon grip

[@MrMctastics]

Years of writing has caught up to them and now they have arthritis

[@tom.parryjones]

Probably autism

[@Lord_Skeptic]

1:22 that is what I like to call an ouroboros.

[@koenth2359]

4:10 Does she mean here 'there is always a non-empty Smith set' ?

[@gregoryfenn1462]

Yes but this is obviously because the whole set of candidates/options would be smith set, just maybe not the smallest, so just keep removing options while still being Smithh until you get to a true smirh set

[@brunoparga]

The biggest voting paradox of all is why people spend so much time and energy discussing bad single-winner methods when you can just go multi-winner with proportional representation and be done with it.

[@soofgolan]

The audio noise gating on Sophie's microphone channel feels so wierd!

[@EneldoSancocho]

Yesterday was the presidential elections in my country, I must have mentioned STV to at least 50 people. Nobody knew what it was nor was interested at all😢

[@brunoparga]

Was it Uruguay? If so you already do better than STV, there's no reason to change.

[@EneldoSancocho]

@brunoparga Yes it was. Well we are doing better than lots of other countries, but still I don't think bipartidism (or however you say it correctly) is a form of democracy, it means the same few people will be in or around "power" for many years. The whole thing would be very long to explain, but I'll say, as a mathematician myself (that's why I'm here) that such a stable system is a bad thing for many reasons. Pd: with some "Economic axioms" one can prove that inequality grows forever with probability 1. I'd like to encourage mathematicians ands students to try to formalize some aspects of economics and sociology, since math people hopefully care the most about truth, and if anyone else says this kind of things they're often called communists or fascists, etc.

[@brunoparga]

@@EneldoSancocho I think the solution is parliamentarism. The problem with Uruguay (and Brazil, my country; and all other Latin American countries) is presidential government. Having separate, potentially conflicting sources of democratic legitimacy between the legislature and the executive is a recipe for bad governance. Uruguay kinda addresses that by linking the two elections, but the single-winner nature of the presidential election distorts what could otherwise happen - a more multiparty system. Make the president purely ceremonial and have a Prime Minister. Your two-party system will instantly go away and be replaced by coalition government.

[@Sofronas]

Would love if you captured some ambient "silence" in the room and had it play in the gaps between the audio clips

[@vkjaencjhewbacjenajcean]

seconded

[@marc.lepage]

Thirded.

[@Sophet]

I don't know what I'm missing but how is the first scenario an actual paradox when you deliberately choose a scenario where all candidates had the exact same configuration of voted position results (meaning every candidate had exactly 1 vote for 1st, 2nd and 3rd preference)? Shouldn't it be obvious that in this case you'd have a draw? Like, having two candidates with just exactly 10 votes of "yes" you also would not think about a paradox, it's just a draw?

[@rosiefay7283]

Yes it does look rather trivial when there are only 3 voters and they vote like that. But in a real election there'd be many voters. You'd get the same paradox with n1 voters voting ABC; n2, BCA and n3, CAB, just so long as the largest of n1 n2 and n3 is smaller than the sum of the others.

[@michaelbodell7740]

The fact that it breaks transitivity is what makes it a little more unusual than a normal draw. Normally if I told you A is preferred to B and B is preferred to C you'd assume that A is preferred to C as well, but here C is preferred to A. And that can lead to lots of weird things like adding extra candidates changing the winner or the order of pairwise comparisons changing the winner. And obviously this is a simplified example but you can add differing numbers of votes or candidates so that it isn't an exact duplication but still has this non-transitive effect. Like if you had 10 A,B,C and 9 B,C,A and 8 C,A,B you might say there is no obvious draw and maybe A should win (first past the post the most votes) but while 10 people prefer A to B or C you have 17 who prefer C to A so it seems like C should beat A. But of course there are 19 who prefer B to C and only 8 who prefer C to B so while C > A, B > C, but lastly 18 prefer A to B so A > B. All without having the "exact duplicate tie" situation. So if A is the winner due to first past the post it is weird that a majority prefer a specific other candidate than the winner, and that is true no matter of which A, B, or C you pick.

[@Leyrann]

It's a somewhat poor representation of the paradox, but it can occur with a great many votes as well, where the number of first place votes may not be equal between all candidates. For example, if someone gets ranked first and last often, someone else might be ranked above them more than below them, and yet have fewer first place preferences. So if you were to elect the candidate with the most first place votes, there might be a candidate out there that is ranked _above_ the winning candidate by a majority of people. Which arguably means they should win instead. But they might hardly have any first place finishes, so they're not actually anyone's preferred candidate, so should they really win? Etc.

[@sirati9770]

the video still is very limited. in terms for most elections you do not elect a single candidate and for that the question of voting because a lot more complex but some of the paradoxes do not need hold. also this only looks as rankings which do not include what voters think about the distance between their choices which is very important information. for tasks where it is to elect for example a president it is much more important that the winner is a unifying candidate rather than controversial. so for that election systems based on median may be preferable

[@robertolson7304]

Probability is separate from and or after you establish the ratio.

[@Liwet.]

There are more methodologies named in this video than there were candidates to vote for.

[@tomholroyd7519]

I like the Rocking Horse Winner

[@AkiSan0]

if llama is yama, i think llull would be yuy?

[@Faroshkas]

It doesn't work like that

[@opensocietyenjoyer]

and mallorca would be mayorca and tortilla is tortiya 🤯🤯

[@jmarvins]

@@Faroshkas actually, that's almost exactly how it works - it is somewhere between an "L" sound and a "Y" sound (as considered by English), closest to the English sound from "biLLIon" for the linguistic technicalities, the sound is a voiced palatal lateral approximant - IPA symbol ʎ note that this is the case only for european spanish (as Llull spoke) and the dialects of the Andes and Paraguay - most of American Spanish underwent a sound changed called "yeismo" where the sound distinction between "ll" and the more direct English "y" sound dissolved, and it is pronounced exactly as a "y" would be in English

[@Faroshkas]

@@jmarvins at the end of words, ll doesn't make that sound in spanish

[@jlljlj6991]

​@@Faroshkas lluvia is usually transcribed [ˈʎu.bja], though (for Castellano at least).

[@HL-iw1du]

Take it off

[@nemesisurvivorleon]

I wish this is how election news coverage was

[@tuppyglossop222]

But what about systems where there are more voters than candidates?

[@brunoparga]

You can also have the Condorcet paradox there. You can have A beat B 60:40, B beat C 55:45, and C beat A 57:43, for example.

[@robins7357]

The moment she started the matrix my head was like: oh, hey: pokemon xD

[@CC-gg4oj]

I would suggest that Luis Carroll was not mad, but in all likelihood a monster.

[@Pfhorrest]

Now factor in ranged instead of merely ranked choices.

[@toddhunter123]

Alas for the Ramon Llull played by Sean Connery movie we will never get.

[@Krekkertje]

Here’s another one. Imagine an election where the voters are very divided. Everyone is far left or far right. There are three candidates: A (left), B (center) and C (right). If every voter would get two votes the clear winner would be B because that candidate would get the second vote from both left and the right wing voters. But in reality either A or C is going to win.

[@HotCoals]

@Krekkertje That's an example of vote splitting which typically only happens in First-Past-The-Post, which is different from the ranked-voting shown in the video. Ranked-Choice Voting actually trends towards centrism, because of the very reason you specified. This trend towards centrism is one of the issues observed in single-winner RCV, as it gives a particular political leaning more preference.

[@MustSeto]

​@@HotCoalsRCV (or at least IRV, which is the most common version nowadays) usually is thought to still have voting-splitting and is often argued to still encourage two-faction domination Giving everyone just two votes could also have this issue, if there are more candidates. But allowing everyone to indicate whether or not they approve of each candidate arguably doesn't have that issue

[@Brownyman]

As an anarchist I find this video absolutely fascinating!

[@Giantcrabz]

🏴

[@brunoparga]

anarchy is definitely a way to solve all voting paradoxes!

[@alejo_324]

My country just had elections yesterday!!

[@DevilishScience]

And there's the Russian Method where it doesn't matter who you vote for Putin wins

[@Giantcrabz]

"The United States is also a one-party state but, with typical American extravagance, they have two of them." - Julius Nyerere

[@gregoryfenn1462]

Speaking of democracy? Am I banned? None of my comments ever ever get replies or even likes/dislikes...

[@MustSeto]

I think KZbin just has some kind of problem. A lot of my comments seem to disappear hours after I post them, almost at random. It's not always political ones. My guess is that KZbin has some particularly aggressive automatic flagging or something, maybe to combat a flood of bots.

[@awaredeshmukh3202]

You exist, don't worry

[@yuvalne]

sad to not have had Schulze get a mention

[@deleted-something]

Math is so lovely

[@schnapsdrossel78]

Amazing project and looking forward to seeing more of it.🎉

[@AtomicAndi]

The real paradox occurred Nov. 2016. A repeat might constitute the most pathological paradox we witness in our lifetimes.

[@JackTheAwesomeKnot]

Nah, nobody wanted crooked Hillary 😂😂

[@AtomicAndi]

@@JackTheAwesomeKnot well, yet another paradox that he's about the only one facing women

[@AtomicAndi]

@@JackTheAwesomeKnot then there was the whole Clinton paradox of course. US politics seems to be the breeding ground for paradoxes

[@lucianchauvin8587]

good quote at the end

[@jimallen8186]

Alaska 2022. Burlington 2009.

[@AliJardz]

Having CS50 tideman flashbacks

[@psbpsbp]

1:46 it's pronounced like "yui"

[@cz19856]

nope, that's not really the pronunciation

[@ubertoaster99]

@@cz19856 The Catalan 'll' is a little bit like a 'y', but subtly different.

[@cz19856]

@@ubertoaster99 you pronounce it like an l and a y simultaneously but dragging the y. It's difficult to explain in the comment section. The double ll in Llull is a killer even for me xd.

[@RFC3514]

No, it isn't. English speakers keep getting "ll" wrong and then "teaching" it to other people...

[@RFC3514]

Just look for KZbin videos explaining how to pronounce "ll" in Spanish, "lh" in Portuguese or "gl" in Italian (those are all basically the same sound). There is no equivalent in English, so writing "it's pronounced like [anything meant to be read as English]" will never be accurate.

[@andyman127]

B is up to something... just cannot put a finger on it! LOL

[@AlexDudeOriginal]

I really just love her... Please feature her more!

[@whatarewedoing0]

Score each one to three, lowest wins

[@theograice8080]

I think voting should take two rounds: first round is nominations (each person may nominate at most one other person), and the second round is the ranked choice vote after all nominees have been recognized. Everyone has equivalent power to introduce a candidate - that right shouldn't be limited to the leaders of parties.

[@Mastersword56]

It should definitely use Condorcet Ranked Choice Voting instead of Hare Instant Runoff Ranked Choice Voting which is what American elections use.

[@IlPoetaDeiGiovani]

Ramon LLULL???? LOOOOOOOOOOOOL

[@InfiniteWithout]

Complex Ballots Animated

[@PureZOOKS]

SAFETY WARNING! DO NOT PLAY THE SMITH SET DRINKING GAME (Taking a drink every time she says "Smith Set" in this video). VERY DANGEROUS I CANNOT SEE

[@CaptainHandsome]

My man's freaking out about the subject of the video being referenced in the video

[@mathphysicsnerd]

Lay off the wood alcohol

[@5hape5hift3r]

Despite these problems i still think condorcet methods are the best systems.

[@elettramelodia8990]

We're all mad here, and it's okay!

[@spindoctor6385]

I missed the paradox.

[@funkydiscogod]

The paradox is that it is possible for Biden to get elected, even though nobody voted for him.

[@modernkennnern]

​@@funkydiscogod🤔

[@idontwantahandlethough]

@@funkydiscogod what?

[@spindoctor6385]

@@funkydiscogod You just have no respect for votes from deceased people.

[@thomasr2472]

The paradox is that A > B and B > C, but also C > A. In other words, the relation ">" ("majority prefers … over …") is non-transitive, which is counter-intuitive if you are used to working with (real) numbers. A similar paradox occurs with certain dice, see the video "The Most Powerful Dice" on this channel.

[@iamrepairmanman]

Louis carrol was a mathematician, him doing maths is obvious

[@RFC3514]

And a political satirist (a lot of AIW characters are inspired by politicians of the time), so it makes sense he'd look into the mathematics of voting systems.

[@seanwhat6]

It's actually pronounced "Ramon Llull".

[@megaladonas]

She holds the pen in her hands very strangely

[@SuviTuuliAllan]

Yet is able to write. How strange.

[@thadoc5186]

You’re assuming people are educated on who they want as a second or third choice.

[@pedrocasella1327]

There is also one more curious fact: minimax theorem was first proposed by no one other than John Von Neumann.

[@syrmo]

Tom Scott!!!

[@abe_ismain]

Minmax lifestyle 8:45

[@MrGeekGamer]

The paradox is really that, whoever you vote for, you lose.

[@tristanridley1601]

Only if all your options and corrupt or ideologues for the rich and powerful. ... Let me know if you find a country where that's not true, eh?

[@aceman0000099]

​@@tristanridley1601Iceland, Switzerland, Faroe isles, San Marino may be places you're thinking of.

[@Giantcrabz]

​@@aceman0000099 lol

[@azaria_phd]

You never notice the job of filling pauses with ambient noise until someone forgets to do it.

[@peterk822]

Just remember, whenever anybody tries to say "Voting System A > Voting System Z", Arrow's Impossibility Theorem shows there is no universally best methodology - it's all about tradeoffs.

[@trimeta]

@@peterk822 True, but there is a quasi-"Smith Set" of voting systems where every system in the set is objectively better than all the systems outside the set.

[@JDSileo]

That only applies to single winner elections. Arrows theorem does not apply to proportional ranked-choice voting.

[@aksela6912]

But there is a universally worst one.

[@goodfortunetoyou]

And yet somehow US elections still use a method which is worse in all respects than alternatives.

[@JDSileo]

@@aksela6912 agreed. And if the metric of mandate (or legitimacy) is measurable (percentage over threshold of victory.) then that means voting systems are comparable and some are, in objective fact, better than others at achieving that maximum representation.

[@merlindrake5557]

I like how all the theories assume that you know the full list of preferences of every voter regarding every candidate.

[@mytube001]

You do in the voting systems where voters rank their candidates/parties on the ballot!

[@JNET_Reloaded]

when 2 have = score we should have a coalition government like we had once before when it was a close result! problem solved!

[@skyscraperfan]

Those candidates look much more friendly than actual candidates like the black H and the orange T.

[@TailOfThePup]

No comment about Veritasium already covering this?

[@FLooper]

I was surprised as well.

[@Giantcrabz]

is it really surprising to see multiple big channels covering this topic two weeks from the election?

[@AstroTibs]

Condorcet paradox is just a more complicated version of "the three voters each voted for a unique candidate; which one should win?" It's a tie.

[@culwin]

Yeah, this doesn't seem like a paradox or very interesting either, really.

[@JDSileo]

And doesn't apply to multi winter elections which this video is so far ignoring

[@milobem4458]

@@JDSileo Brits and Americans are stuck with the idea of "constituency", where one person represents their village in a far-away parliament (or congress). It made sense in the middle ages, when it was established, but most of the world moved on. Multi-seat districts, as used in most other countries have some issues too, but they are much better than single-seat

[@JDSileo]

@@milobem4458 America wasn't stuck when they decided to leave the monarchy...just took a lot of effort. But it's doable

[@JDSileo]

@@milobem4458 I acknowledge the lack of perfection. But as you said, multiseat is better.

[@Cacuofa]

Ok

[@danieldare2640]

I am going to but would love you to do an examination of why Japan is so undecided in its voting and it seems to always be in the knife edge of negotiation between the parties.

[@justforplaylists]

I wonder if political parties in Japan are more willing to change their policies if they lose, so the parties end up with policies that get them close to the same number of votes.

[@brunoparga]

Most elections in Japan result in clear majorities for a party, usually the LDP. The recent one is a strong exception.

[@scotttroyer]

ROOM REVERB

[@markshiman5690]

Ruined a perfect chance to endorse your candidate. What the hell man?

[@E1craZ4life]

That was the objective.

[@Giantcrabz]

Nobody for President 2025

[@Speed001]

Thank you for not making me listen to the writing sound, it makes me cringe.

[@johnchessant3012]

sophie

[@gregbard]

The one thing you said we shouldn't worry about, is something we actually should be very mindful of! All elective systems handle the landslide elections the same. You judge an elective system by how it handles the very close elections.

[@heronstreker]

Only people in the USA and the UK think voting is complicated because of your election systems. For the rest of the world it is simple and not worth discussing. Your math in interesting though, but is has not much to do with elections.

[@26IME]

.... What?