Voting Systems and the Condorcet Paradox | Infinite Series
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What is the best voting system? Voting seems relatively straightforward, yet four of the most widely used voting systems can produce four completely different winners. Get 10% off a custom domain and email address by going to www.hover.com/...
*Correction: The ballots at 1:20 were labeled incorrectly. At 1:20 the top ballot should read 1 Green, 2 Blue and 3 Purple and the bottom ballot should read 2 Green, 3 Blue and 1 Purple. Thank you to Hoarder who first noted this.
*Correction: What's stated is the converse of the Condorcet Criterion. Oops - Stating conditionals can be tricky! For more details, see: / diyft53
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Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
With access to a complete set of ranked ballots - which means we know every person’s opinions - it seems like a clear winner should emerge. But it doesn’t. The outcome of the election depends critically on what process you use to convert all those individual’s preferences into a group preference.
Further Resources:
Voting and Election Decision Methods
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The Mathematics of Voting
www.math.ku.ed...
The Mathematics of Voting, Power and Sharing
web.math.prince...
CGP Grey Voting Playlist
• CGP Grey Voting Series
Comments answered by Kelsey:
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CGPGrey just got excited.
+Austin Chesnut I was just about to mention him in the comments, but you beat me to it :)
(Edit): I just noticed that he was referenced at the end
i don't know... may not be enough jungle animals to hold his interest
I wonder if het would take the time to make more videos in his election series, it has been some time...
Maurice Downes Maybe we could coax him out with a few asterisks ✳ ✳ ✳
You mean because of the hover ad? :P
Note that in real cases of a two-round run-off voters can usually change their vote in the second round. (Even if the candidate they placed first in round 1 went through to the second round, they can vote for his competitor.) For strategic voters, this can make a difference.
Maybe this is why two-round runoffs are more often multiparty while IRV still enforces two-party domination?
Yes, simplifying assumptions were used, I would argue the best voting system
a) does not induce strategic voting
b) no true minority (pP/p) within an area where this is true.
I would love to add a third, but I fear a paradox
c every true minority in an area gets some representation in that area
From the popular systems of ccp greys series, it seems is IRV with with [100/(n +1) +1]% as the threshold is the only one that satisfies A and B, but it does not satisfy C and I fear no finite representative algorithm will.
Having two separate elections bring in strategic voting. You only want you candidate to barely make 2nd (makes ng the other part assume a sure win, deincentivising voting. Then in the second election everyone comes in thinking every vote counts.
I consider a the most important criterion
It's already a mess trying to figure out where a representative stands on an issue. It's all tribalism and rhetoric, if you add in wondering how everyone else might vote. Well, this is why I don't vote on America. Too complicated, I admit I haven't spent the time needed to know if I'm helping a faud. Besides, Thier practically the same party. (Let's fight about unborn babies and if people can possess certain weapons, and how a few people get in; but we all agree the military is worth more than education, that inflation is necessary, that politically incorrect thoughts should be silenced, the territories can't succeed ........ WHEN WE DONT.)
Correction: What's stated is the converse of the Condorcet Criterion. Oops - Stating conditionals can be tricky! For more details, see: www.reddit.com/r/math/comments/6hh9sb/voting_systems_and_the_condorcet_paradox_infinite/diyft53/
@ 6:44 - It is INCORRECT to claim in such an example: "We have all the possible information about each individual's preferences". No information was given about the strength or polarity of preferences. Three people who chose the exact same ranking might for example consist of one voter who was almost equally AGAINST ALL OPTIONS BUT ONE and not actually for any of the options at all, plus another voter who was almost equally IN FAVOR OF ALL OPTIONS, plus another voter who was STRONGLY AGAINST TWO OPTIONS, and WEAKLY AGAINST ANOTHER TWO OPTIONS, but STRONGLY IN FAVOR OF THE REMAINING OPTION. There are ways of collecting such additional information. For example, the voter could be allowed to vote using a point system where they could give as many points as they choose in opposition to or in support of any candidate, with their total absolute number of points given either way being considered 100% of their vote, and the points cast for or against each candidate (or color in this case) converted to a percentage of their total vote for the purposes of counting. This is one version of what's called a weighted net approval system, and counting the total votes in such a system is very straight-forward. For example, you could count the percentages in favor of each candidate for a total number of in-favor votes or "provotes" and then count the percentages in opposition to each candidate for a total number of in-opposition votes or "antivotes" and then simply subtract to the total antivotes from the total provotes for a total net vote count for each option on the ballot. The highest net vote count would be the winner. A tie would only happen if the group was actually tied in their references and the results would never include a cyclic ranking.
You bring up some good points. There are also practical considerations - for instance, the more complicated the ballot, the more likely the ballot will not reflect the voter's preference due to voter error. Plus, a voter's preferences fluctuate, particularly in relative strength of candidates, less so in absolute ranking. So we do have to accept some limitations on the amount of information we can really get out of voters. Let's give them the benefit of the doubt and say the simplifying assumption was actively made here to call this a good approximation of "all the possible information about each individual's preferences" - emphasis on "possible" - but for the sake of getting to the point, that was left to others, like yourself, to highlight :)
You should pin this comment at the top so everyone has a better chance of seeing the correction.
@@DonaldKronos Good point. A 5 star ballot for example would show all the same preference info, but also level of support. It allows a very nuanced level of information without being overwhelming or confusing.
We had STAR Voting tested for representative accurate results and included variations of STAR with a 0-3 star ballot, 0-5 stars, and 0-10 stars. 0-5 was expressive enough to get best in class results, and got basically the same results as 0-10. Plus it's in the sweet spot for cognitive load.
At 6:31 you made a mistake.
G: (18*4)= 72
B: (12*4)+(14*3)+(11*1)= 101
P: (10*4)+(11*3)+(34*1)= 107
R: (9*4)+(18*3)+(18*1)= 108
O: (6*4)+(12*3)+(37*1)= 97
Red still wins but not by that large of a margin.
The votes changed after the 1:20 mark. At the 1:30 mark it's differentent.
Thank you Hoarder. The ballots at 1:30 are correct and the ballots at 1:20 were labeled incorrectly. At 1:20 the top ballot should read 1 Green, 2 Blue and 3 Purple and the bottom ballot should read 2 Green, 3 Blue and 1 Purple. We'll also add this note to the description.
Awesome. Love the videos on this channel!
The Russians!
It's not only the bottom one that changes, so does the top one , it goes from 1 3 2 to 1 2 3
Thanks Fauxma, I've amended the explanation.
I was literally, LITERALLY thinking about this problem the moment that I turned on the video. I've been thinking about it for a while, particularly in relation to sorting algorithms.
In relation to sorting algorithms? Care to explain?
7:52 Orange was the condorcet winner but didn't win the popular vote
maybe the russians helped orange!
That's fake news!
@Jannes Fransen the Russians all voted for red.
Yeap ... it claims you won a game that wasn't being played, next time, read the rules and win the game being played. But joke was good nonethless
The popular vote is only as good as the voting method being used.
Yay! I can finally comment on one of these videos with something topical! :D
The Condorcet paradox also really highlights the importance of agenda setting in the study of social choice.
Imagine having three people (A B C) with the following ranking on 3 choices (X Y Z) :
A B C
---------
X Z Y
Y X Z
Z Y X
Now, as described in the video, look at what options are better than the others:
X>Y
Y>Z
Z>X
It's obvious that this order creates a cycle (as seen in the video). The group preferences are not transitive despite individuals having transitive preferences. But what's not as obvious is that you can make the outcome *whatever you want* based on how you pair the options in a single elimination set-up.
Want X to win? Have Y go against Z then have X go against Y.
Want Y to win? Have Z go against X and then have Y go against Z.
Want Z to win? Have X go against Y then have Z go against X.
In short, when you have cycling and institutional structures demand pairwise consideration, the order with which you address the options will determine the winner. The person who has the ability to determine the order has an immense amount of power, then. This is super important in studying legislators and other institutions as, often, they force issues to be considered one on one. So this creates a number of counter-intuitive strategies like the introduction of "killer amendments."
Definitely looking forward to Arrow!
Condorcet "paradoxes" aren't a big deal, though. They're just a tie. All kinds of elections can have ties, it doesn't mean much.
Yey! I'm very excited for next video. Since CGP Grey started my interest in voting systems I've been wanting to know more about the properties of different voting systems.
I hope that video will have a fun challange to work on
+
Yay! Voting systems! I wanted to know more about them since Grey's videos, (And I now I know more, because of my research, but now, this video and the next too). This was an introductory video, let's see if the next is more in depth!
I love you guys! The mathematics of voting systems are what got me into politics
STAR Voting Rocks!
People Ready to Post Results: So, who won
First Past Post: Red
Double Round: Blue
Run-off: Purple
Borda: Red
Condorcet: Orange
Thank you for taking time to talk about this super important and widely unknown topic, but I really miss Approval voting in this list. It's way simpler than all the systems you presented, except maybe for majority and plurality, and it has very nice properties to provide contrast with the other systems. People are also used to it -- it's the "likes" system in Facebook or Twitter, or alternatively, voting by raising hands.
Or any other rating method. Condorcet published his results in 1785, and Arrow got the Nobel Prize in 1972. The video should mention something from this century.
Approval probably wasn't included because it is arbitrary and different voters will have different ideas on what approve means.
6:47 - "We have all the possible information about each individual's preferences."
No we don't!!
"We know exactly which colors they like more than other colors."
Yes, we know that, but we don't know *by how much* they prefer each color over others!!
Suppose the ballot were on a rating-system basis - each person's vote for each candidate could be anything from, say, 0 to 100.
Would that make the result, if not completely independent of the scoring method, at least somewhat less dependent on it?
That's called range voting (or score voting). The underlying assumption behind this method is that preference or utility is a cardinal measure and not ordinal like in the methods shown in the video, which is debatable. Another assumption is the interpersonal comparison of utility, which again is a big assumption.
What makes FPTP (or any other system, it's just broadly regarded as the worst) seem equivalent to the others by this metric is that there's no relation between the colors. Someone who votes blue can vote red just as well as they vote purple (if we assume purple is a mix of red and blue policies). When we draw the analogy to actual voting that's obviously not an accurate model anymore. The extreme left doesn't vote extreme right second choice as much as they vote moderate left and viceversa. So something like FPTP is arguably not inferior for populations with excessive ideological inconsistency.
But I love the idea of scoring systems because it would allow for inter-policy expressions in voting. Which I feel the need for in my proportional government. But it's all moot in the places with the least representative systems already and the topic isn't even brought up elsewhere.
@@noneofyourbusiness6269 How does range voting compare utility interpersonally? It just compares the utility of a choice relative to that same person's other choices.
Say a 100 rating for me is life or death, and a 100 for you is mild preference. That fact is irrelevant. All the system does is find out how much more I want one choice over another. It doesn't try to compare one voter's level of investment to another's.
However, one of the obvious problems with this system is that it incentivizes lying about one's actual preferences. For example, if I really want to vote A: 100, B: 20, C: 0, but I'm afraid A can't win and I really don't want C to win, then the way I might end up voting is A: 100, B: 100, C: 0, or similar.
So yeah it might help but it doesn't solve all the problems.
As Devin said, it incentivizes lying super clearly
@@DevinDTV but with the ordinal system you don’t need to lie right? So it feels like the superior of all.
I love voting systems! Great to see you do videos on this topic!
I gotta say though, that plurality vote and 2-round runoff are just shortcuts, so it's not surprising that they can misrepresent the group preference.
Schulze and ranked pairs are the two best voting methods that we know of right now. All others are mathematically more vulnerable to failures due to strategic voting.
I don't know what the best method is but the worst is clearly plurality. Of course that's the one we use here in the UK for elections.
Why are cardinal voting systems NEVER mentioned in popular presentations of the issue? The idea that we should be ranking candidates is arguably THE issue with the voting systems we use so far. It's exactly why all of them show poor representation, polarization and oppressive use of political force: because they are all based on the idea of "the one candidate I want", not "which candidates I can support".
Cardinal voting systems, where candidates are RATED, suffer none of these issues, and the criticisms are all based on silly epistemological arguments, superficial strategic voting concerns that are even worse in other systems, or completely superficial projections of what people will do under such a system which go against all current data on the matter.
For example, score-runoff (a variant of plain score, where the most favored of the top two rated candidates wins) has shown to be the best among all of these proposed systems under election simulations, and under some pretty reasonable criteria of what these systems should do. It neatly addresses the three main criticisms against score voting: strategic voting, no majority rule and unknown candidate risks.
The point is: we live in a society. Elections shouldn't be about "having it your way", but about "having a say".
Any voting system based on an order of preferences will break that principle, because no matter how you set it up, at any step your full support will be completely concentrated on a single candidate. The rest of your opinion is never taken into account fully and simultaneously in conjunction with everyone else's full opinion. That is the major flaw here.
Score voting has strategic voting incentives that reduce it to simple approval voting. That said, there are good arguments to be made for approval voting (including, to jump the gun on the next video, that being a series of boolean choices, it circumvents Arrow's Impossibility Theorem).
I'm guessing the examples were picked so they can show off 4 different results that don't match the condorcet criteria. I'm pretty sure there's issues with cardinal voting too, though I can't remember ATM. I think it encourages strategic voting (which is a big problem for a lot of systems, but is worse in this case).
I think you'd have a hard time getting people to correctly and fairly rate each of their preferences. I'm fairly certain most people I know would rank their first choice 10 and everyone else 0 (considering the lethargic attitude towards politics with the only real motivation for many people being fear of the "terrorists on the other side of the aisle". I guess it would at least stop the spoiler effect, which is great. But good luck convincing anyone to do it. I grew up in rural PA and I know many people that consider the electoral collage god's gift of wisdom passed on by the founders (as if they were profits of God), to protect America from the evil liberal city-dwelling terrorists. These people are also kings of double standard though, so if it ever manages to elect a democrat when a conservative won the popular vote you can be bet they'll be ready to change it. So basically that's what we'd have to wait for, and that would probably never happen the way politics are split atm.
You are talking about the majority judgment, right ? I totally agree with you, but i guess many peoples don't know about it. It seems to be quite recent.
josh white, it doesn't matter, there will be people doing that and there will be people scoring what they think is resonable, if we want a democratic election, why not make the election more democratic itself.
Josh White: How much of those fears you are talking about are a byproduct of the current voting system? You seem to be thinking just because politics is highly polarized (particularly in the US), that people are ideologically highly polarized in nature. Actually, your statement is even stronger than that: that ideological leaning is ALWAYS highly and irrationally polarized.
Every single study about political ideology has shown there's a very huge ideological spread in populations. The choice system is what encourages the polarization, not the other way around. I mean, let's use the US as an example: what does gun rights, climate change, economic regulation, tax rates on the wealthy, and gay marriage have intrinsically in common with one another? Pretty much nothing. But strong polarizing instances in all those things are currently bundled together under the highly polarized Democrat/Republican divide. Does any of that make any sense to you?
A proper voting system would allow room for these political positions to be decoupled, as they should be, which naturally changes the entire relationship people have with the voting process and politics. That's the importance of a truly representative voting system: it must be able to reflect this multi-dimensional nature of political ideology to the highest degree available, and how the general population deal with it. (Ranked systems are not really appropriate for this task, and it's not difficult to understand why when you consider what exactly the ballots represent in such systems.)
Put a better representative voting system in place and in a decade or two you'll have highly polarizing candidates and platforms struggling with winning, as the system cannot favor that strategy anymore. This means candidates who better represent the interests of the population, in general, will be more likely to win, which is what we all implicitly seem to assume that representative governments are all about.
Right now, we have systems that are basically forcing voters into saying "it's my turn to rule over everyone I disagree with", instead of "this is how much I care about these issues". Can you understand how problematic that is and how it has led to the issues we face today? The heart of this difference is at the ballot system, not even the voting system level.
Also, this whole issue is WAY beyond electoral college and US politics. Don't mix the challenges faced by voting systems and representative governments with the distorted and highly irrational political and cultural landscape of the United States. Keep the eye on the ball.
The solution is simple. Use any Condorcet method. If there is a cycle of top preferences, use sortition to choose between them, because any of them is as good as any of the others (according to the electorate's ranked preferences) so it doesn't matter.
I like Condorcet methods, but I think there are still problems with tactical voting.
I'm going to have to watch this video one more time, but I gotta say it's very relevant, as some countries are thinking of adopting a ranked ballot system.
Relevant to some, yes. However, a little bit of information can be dangerous. In the USA, it is not wise to put the following idea at the top of people's minds: "ranked choice voting systems have flaws". Of course they do, but they have fewer flaws compared to first past the post systems.
The kinds of examples used to illustrate differences in tallying ranked choice ballots happen very rarely. Yes, these issues should be discussed and balanced, but these factors not be cause for concern among most members of the public.
Cant't wait for Arrow's Theorem. I once gave a speech to fellow students about it, and I myself was amazed!
Please make more videos on this, I can't get enough of it.
This is one of my favorite topics. It is fascinating!
Peter de Blanc and Marcello Hershoff invented what they called Hay Voting, which has the unusual property that any voter's best strategy is always to vote sincerely. I had some small involvement with it after they invented it, theorizing about Multi-stage Hay Voting.
Hay Voting gets around Arrow's Impossibility Theorem by using randomness to select a voter, basically voiding the "no randomness" and "no dictator" clauses. It also had the problem that there was a small but real chance that a candidate that nobody liked would be elected.
I was able to eliminate the second problem and ameliorate the first with Multi-stage Hay Voting, but I was never entirely satisfied with it.
What about the Schulze method? I've been trying to find an explanation on that but I'm pretty sure it also ensures a Condorcet winner.
It does, it's a Condorcet method.
Didn't tipping point math make a video on this. Btw you are my favorite channel and you have inspired me to really get into math. You make math so simple that I can understand all of your videos and I am only 9 years old. Thank you.
I don't think I've been this excited about a video since your first higher dimensional sphere video. Voting systems is something I have a small passion for as well.
1:52 it's already better than 2-3 other videos i watched on voting systems.
showed the cyclicity, showed the proper term of art.
How about adding NONE OF THE ABOVE to the ranking? If NOTA wins, then NONE of the candidates can be elected, and the process starts over with a new field of candidates. Note that in ranked choice, NOTA might not win on the first round, but could be the "winner" on second or third or lower rounds.
You guys should totally do a show about the mathematics of gerrymandering and the recent court decisions in North Carolina, Wisconsin, and elsewhere.
In Czechia we've got a project called Demokracie 21, which is based on the rule that every voter has 3 votes for and 1 against. Right now there's an ongoing online experiment simulating the election of the the next president using this system :)
The adventage of this project is that you eliminate extreme choices that arise in Plurality and Two-Round Runoff (choices that divide the nation, like Trump vs Hillary) while not requiring a full list of candidates.
6:48 - You do *not* have all possible information about each individual's preferences when you only use order of ranking. You are missing any absolute measure of a person's like/dislike of the candidate. If there is only one candidate I feel strongly in favor of and don't care about the rest, I cannot represent that here. Range/score voting is a much more straightforward way of avoiding these paradoxes.
@Tab hear*
That's what I was thinking. Perhaps use borda count only the numeric values aren't determined by rank order but by deliberate placement on a spectrum of support-oppose.
You would probably like range voting. In range voting, voters give a score to each candidate and the candidate with the highest average score wins.
Unfortunately, range voting is likely vulnerable to tactical voting. It has not been tried on a large scale before so we can't know for sure how voters would behave but we do know that the incentives to underscore or overscore competitive candidates, lying about your true preferences, are high.
A good solution to this is something called Range Runoff Voting, also known as STAR voting. In STAR, voters give a score to each candidate like in range. The two candidates with the highest average scores go onto another round. Out of those two, the candidate which has a higher score on more ballots wins. This disincentivizes tactical voting. This system has also not been tried on a large scale either so we don't know how it would perform.
Also, although I can't speak to this, I remember reading somewhere that STAR and range voting are really good at better at producing condorcet winners than even condorcet systems themselves because of how condorcet systems effect voter behavior.
@@ben8557 Indeed I would. Which is why I mentioned it in my last sentence ;) Will have a look at STAR
@@cogwheel42 I didn't notice you mentioned it. I was a bit tired while replying, lol
It's worth noting that one can't simply take sincere preferences and generate Plurality ballots naively. Voter strategy results in differences between sincere preferences, discussed preferences, and how one actually votes in various systems (I prefer X, but I vote Y). Take a look at Duvuger's Law for an example of how Plurality distorts voter expression in successive elections.
you just shattered my world
So are you going to spit out the same solution CGP Grey said, or is some other system objectively mathematically superior.
It's weird having to qualify the 'mathematically' qualifier with an 'objectively' qualifier
so excited you are doing one on social choice theory!!
Very interesting. I watched this about a year ago.
18 April 2022
1:42pm NZST
The Condorcet criterion made me think: What if everyone stated Orange as their second option? Should Orange win then?
Also, you could make an Inverse Instant Runoff system, where the person that is ranked least the most, gets eliminated first and so on. Thinking about it: The possibilities are endless. I always was a fan of the Instant Runoff vote and the Two-Round Runoff (which is essentially a slimmed down version of Instant Runoff, requiring less effort) but probably, there is no ultimate voting system.
1. Yes Orange should win then. 2. Inverse Instant Runoff is called Coombs method. 3. Correct that there is no ultimate voting system, but some are better than others. You can measure their goodness using Social Utility Efficiency, Condorcet Efficiency, etc.
The Condorcet Paradox reminds me of Rock Paper Scissors. Every choice is better than one choice but worse than another choice.
That's the analogy that is used, for sure. I'm surprised she didn't say that here.
The left side of equations at 6:34 is wrong; if it were right the red sum is 108 and the orange sum is 97. Looking back at the ballots shown earlier before all the scoring methods, I see Orange is 6*4 + 12*3 + 37*2 = 134. Red is 9*4 + 18*3 + 18*2 + 10*1 = 136, so while the totals came out right.
YES! VOTING! IM ODDLY EXCITED ABOUT THIS!
Just calculate the average Rank:
Add up (Number of Votes)*Preference Rank for each Color and divide it by the Number of Ballots. For Example:
Green:
18x1 + 12x5 + 10x5 + 9x5 + 4x5 + 2x5 =203 => Average Rank is: 203/55 = 3,69
Calculate the same for all other Colors and you get this (hopefully without calculation errors ;-):
Green: 3,69
Blue : 3,16
Purple: 3,05
Red : 2,53
Orange: 2,02
So the Winner is: Orange just like the Condorcet Winner. QED
The fundamental problem is that the information you're collecting is rankings. Rankings destroy information about *degree* of preference, and aren't comparable between voters (if I hate B slightly more than A, and you love A and hate B, our preferences will both be A>B, but they don't mean the same thing).
We should be focusing on voting systems that choose the Utilitarian Winner, not the Condorcet Winner (though they will often be the same). The UW is the candidate with the highest overall approval rating, or whose platform is closest to the average voter's desired platform (in multi-dimensional opinion space).
UW means?
@@siebren005 UW = "Utilitarian Winner" = the candidate with the highest approval rating = the best representative of the will of the voters
I feel required to say this: There is no perfect voting system. Which is exactly why you should limit the winners (gov't) power.
No. The wisdom of limiting governmental power is not necessarily coupled to the method by which members of the government are selected.
The idea of a voter paradox is incoherent: it is an attempt to figure out how to "agree to disagree." If we reject this contradiction, and simply accept that people disagree, all voting systems are fair.
Condorcet "paradoxes" are just ties. Ties aren't a big deal.
8:06 Did anyone else notice "Borda count" turn from red to orange?
I did not notice... impressive detective work my friend.
"Voting is complicated."
Me: "I've got it, let's vote on the best method! --Oh, wait..."
A group of 31 people are trying to figure out what ranked voting system they should use. Six different voting systems are suggested: Plurality, anti-plurality (whoever has the least number of last place votes wins), Top 2 Runoff, Instant Runoff, Borda Count, and Copeland (the winner is whoever has the most paired wins minus paired losses; this is perhaps the simplest Condorcet method out there). They decide to hold an election with ranked ballots, and here are the ballots cast:
4 ballots: Plurality > Borda > Anti-plurality > Top 2 Runoff > Instant Runoff > Copeland
3 ballots: Plurality > Borda > Anti-plurality > Top 2 Runoff > Copeland > Instant Runoff
1 ballot: Plurality > Borda > Copeland > Top 2 Runoff > Instant Runoff > Anti-plurality
5 ballots: Plurality > Copeland > Borda > Anti-plurality > Instant Runoff > Top 2 Runoff
2 ballots: Plurality > Top 2 Runoff > Anti-plurality > Copeland > Instant Runoff > Borda
6 ballots: Top 2 Runoff > Copeland > Instant Runoff > Borda > Anti-plurality > Plurality
4 ballots: Copeland > Instant Runoff > Borda > Anti-plurality > Top 2 Runoff > Plurality
3 ballots: Instant Runoff > Borda > Anti-plurality > Copeland > Top 2 Runoff > Plurality
2 ballots: Borda > Instant Runoff > Anti-plurality > Copeland > Top 2 Runoff > Plurality
1 ballot: Anti-plurality > Instant Runoff > Copeland > Borda > Top 2 Runoff > Plurality
In order to get a fair result, the people decide to figure out the election winner using each of the 6 systems they voted on. But when they do this, they notice something interesting. Plurality wins under the plurality method. Anti-plurality wins under the anti-plurality method. Top 2 Runoff wins under the Top 2 Runoff method. In fact, every single voting system wins the election under itself!
Frustrated, the 31 people decide to abandon ranked ballots altogether and switch to range voting. Then they live happily ever after. The End.
as a country, we use majority since a candidate has to win a majority of state electors, but the electors are chosen by the plurality in each state.
This is why it is so annoying that instant runoff voting advocates call their system "rank choice voting." There are many ranking systems and IRV is the worst.
Also, approval voting deserves a mention.
What are you talking about? Plurality voting is by far the worst.
@@oppenheimerfaaaaaan5547 I agree, but plurality voting isn't a rank choice voting system. I said it is the worst of the ranking voting systems.
@@oppenheimerfaaaaaan5547 Plurality voting is the worst voting system. Plurality voting is not the worst rank choice voting system because plurality voting is not a ranking choice voting system. You don't rank candidates under plurality voting.
@@Mutex50 Contingent vote and Supplementary vote are worse ranked systems than IRV (and are also used in the real world)
PBS Space time has not uploaded their video yet :(
Hi Fourzin, Space Time is taking the week off to prep for the big Quantum Field Theory series. Thanks for your patience. -Rusty Ward, Producer of Infinite Series and Space Time
Oh my god!!! you replied! this is literally the best notification I have received from CZcams 😊... your show is awesome! both you and spacetime are my two favorite channels...Oh I see it is ok. I bet it is gonna be awesome...keep the epic work!!!
PBS Infinite Series i was just thinking of QFT done by spacetime,ty.
But they certainly would, very soon if not at the time of this reply.
Quantum Field Theory explained by Space Time, OMG, hype just went through the roof
everyone enter a rational number from -1 to 1 for each option and tally up sums
with integers, some call that "3-2-1-voting"
It would be great if you could do an episode addressing the mathematics of selecting voting districts and gerrymandering.
Was going to mention CGP Grey after seeing the title, then you linked to his series! Love his channel and podcast as well as this channel.
One of the flaws of your analysis is assuming Avery voter votes for every candidate. In many ranked voting systems, you vote for a limited number of candidates, rather than rank every candidate. For instance, you rank your top 3 candidates, even if there are 6 candidates. This alters the evaluation for each type, and makes it much harder to manipulate a vote using strategic voting.
Borda Count is Supreme, change my mind.
An outcome that most people are ok with is better than an outcome that some people love but most people hate.
Borda is one of the most strategically vulnerable & sensitive methods out there, not just to voter strategy but to nomination strategy (teaming)
But rating-based methods like Approval, Score, and STAR (not covered in this video) also accomplish the goals you say you like, and are less sensitive to strategy
@@MustSeto Hmm interesting. I will do some research on those other methods. Thanks!
I would just love to have a voting system where people don't just vote against who they don't like.
I stand by the idea that the condorcet criterion is useless. If one candidate is neither hated nor loved by anyone they would beat anyone 1 on 1 yet could theoretically stand for literally nothing.
It's really interesting when simple problems don't have obvious answers.
In my lab we use a number of decision making methods such as Analytical Hierarchy Process and Technique for Order of Preference by Similarity to Ideal Solution to make system design decisions in multi dimensional spaces (e.g., is it better to have a faster but more expensive airplane or a cheaper but slower airplane?) and things get even weirder. Imagine you're electing someone to be your cook and driver. How are you going to balance tasty food against road safety? Ho are you even ranking and measuring them in the first place? It gets very messy.
When I was explaining the instant runoff to someone, i thought of borda count system all by myself when he said that the most acceptable candidate should win what if the person eliminated has all 2nd preference votes. And now app
I really miss this series.
How many people are revisiting this video after the new Veritasium dropped?
I've been tossing around an idea for a similar voting system to the Borda count, except instead of assigning a point value directly relative to the ranking of the vote (c-n, where c is the number of candidates and n is the ranking given), you use an exponential system that avoids consecutive integers altogether but still remains close enough so that the numbers don't get way too big to conceptualize or explode off into runaways -- powers of 3: 3^(c-n); or even powers of thirds: (1/3)^(c-n)
What I found after plugging in the same test ballots from the video above is that green wins that election but with an order unique to the other listed methods in the video:
Green (1495, or ~18.46)
Red (1407, or ~17.37)
Blue (1401, or ~17.30)
Purple (1209, or ~14.93)
Orange (1143, or ~14.11)
How does this method fair for the Condorcet Criterion?
Similar to Dowdall system ?
i basically invented the borda count in my head before watching this lol
an easier method is to not have a winner. every person voted for takes a seat in power with as much influence as they got votes
@May Day the phrase "20 presidents" is a bit stupid because obviously a group of 20 doesn't function the same as a single person, they would be a Presidential Council that have to debate, and *vote* their decisions with each hand carrying the percentage of power that they got through elections.
@@blop-a-blop9419 Australian parliament
I'm surprised you did not mention my favorite, and one of the most mathematical, voting system - Primal Voting. Each voter casts a ballot selecting only their top choice. The votes are tallied and each candidates vote count is factored. The candidate whose vote count results in the largest number of prime factors is the winner. If there is a tie amongst two or more candidates, they fight to the death and the survivor is the winner.
Here's my method:
1. Do a borda count to determine the median candidate. In this case, Purple wins the center.
2. All voters that chose purple as #1 or #2 (half of voting length rounded down) is collected, and using only those cards, determine the worst candidate through another borda count.....which is Green.
3. Repopulate the entries as if the loser (Green) and the center candidate (Purple) didn't exist, and repeat the steps until you have 3 left.....in this case we do (blue, red, orange), so we'll just move on....
4. Borda count to determine the center candidate (orange), and of those voters (in same manner), choose the winner (blue)
The reasoning? Cheating and overt bias. If you have a rigged election with candidates pumping too many on a single position, those positions are faded to the outside of the counts....away from the center, and the centers determine the losers and finally the winner. This means that the "independent" voter is more valuable. If any candidate attempt to center himself, then he automatically loses because centers are disqualified in the following rounds for selection.
This is how any election should happen, just saying. And it took me all of 5 minutes to come up with.
LOL - I worked out somewhere that if you also include voter intention (e.g. suicide vote) that with just 2 candidates you have something like over 20 combinations of meanings for all the possible outcomes.
This video was incredibly illuminating. I had no idea!
Can someone please tell me how well the APPROVAL VOTING system measures up with Condorcet?
I think it's fairly favorable. For one, maximally strategic Approval is effectively a Condorcet method, and since Approval isn't limited just to preference order, but also incorporates strength of preference, with more honesty it elect a better-than-Condorcet winner, if they exist
For a proof sketch of Strategic Approval being a Condorcet method, imagine a case where an election has a would-be Condorcet winner C, but the election is held with Approval and the winner is A instead
Because C was the Condorcet winner, this means they were preferred to A by a majority, which means for A to win either, some of those who preferred C must have either approved of both A and C, or neither, which is not perfect strategy. If they had used perfect strategy, they would have approved of C but not A, and A could not have won
Not only does each system have bias, but each system can be gamed with strategic voting. Therefore you will never have a "fair" system because you can never force someone to vote "honest". Probably the "fairest" system is one in which the majority of voters end up with a candidate who is at least marginally acceptable to them. Allow voters to vote either 1 or 0 on each candidate on the ballot, based upon if they feel a candidate is acceptable or not. Whoever gets the most "1"s wins.The upsides are that your "vote" is never wasted (since you have more than one vote to give), and it tends to favor moderate candidates who appeal to a broad spectrum of the electorate. It also greatly limits the ability to game the system by voting insincerely, since it is hard to find a scenario where voting against your true preference who help you.
>-- Non sequitur. It's not necessary to have honesty to have fairness. All you need is that the voters have equal power over the outcome. This is the case in systems where for each possible vote, there is another vote that will exactly balance the first vote.
Your definition of "fair" is not shared by others in this field -- that may be what is confusing you. It is well established via Arrow's Paradox that in elections with 3 or more candidates, there are no "fair" systems due to quirks of each system and the ability to engage in strategic voting. You can watch the next episode of the current series or visit the following website to learn more: skeptoid.com/episodes/4281
Basically fair (as defined by everyone but you) is defined as:
1. If every individual prefers X to Y, then the group prefers X to Y.
2. If every voter's preference of X over Y stays the same, then the group's preference of X to Y stays the same, even if other preferences change: such as Y to Z, or Z to X.
3. There can be no dictator, as Arrow called him; a single voter with the power to dictate the group's preference.
Hope this helps. Agreement of terms is a must before engaging in discussion, so that everyone is on the same page.
@ 6:44 - It is INCORRECT to claim in such an example: "We have all the possible information about each individual's preferences". No information was given about the strength or polarity of preferences. Three people who chose the exact same ranking might for example consist of one voter who was almost equally AGAINST ALL OPTIONS BUT ONE and not actually for any of the options at all, plus another voter who was almost equally IN FAVOR OF ALL OPTIONS, plus another voter who was STRONGLY AGAINST TWO OPTIONS, and WEAKLY AGAINST ANOTHER TWO OPTIONS, but STRONGLY IN FAVOR OF THE REMAINING OPTION. There are ways of collecting such additional information. For example, the voter could be allowed to vote using a point system where they could give as many points as they choose in opposition to or in support of any candidate, with their total absolute number of points given either way being considered 100% of their vote, and the points cast for or against each candidate (or color in this case) converted to a percentage of their total vote for the purposes of counting. This is one version of what's called a weighted net approval system, and counting the total votes in such a system is very straight-forward. For example, you could count the percentages in favor of each candidate for a total number of in-favor votes or "provotes" and then count the percentages in opposition to each candidate for a total number of in-opposition votes or "antivotes" and then simply subtract to the total antivotes from the total provotes for a total net vote count for each option on the ballot. The highest net vote count would be the winner. A tie would only happen if the group was actually tied in their references and the results would never include a cyclic ranking.
What about Score Voting?
I think that's what Kronos described.
In score voting, there is no limit on the total points you give. In Kronos' system, ballots would be reweighted so the total points is effectively always the same (100%).
If you feel that in order to best beat your least favorite candidate, you're best off to support your second favorite as strongly as possible, then it is to your advantage to vote that way. This will be the case when your favorite is perceived to be in 3rd place (or further back).
I made a ONE MINUTE video (today) that shows 2 examples of flaws in the instant-runoff method.
The video also has a list of objectives for a voting system.
"Ranked Voting Explained in 1 minute (Ranked-Choice/ Preferential Voting)"
czcams.com/video/uVOBk8p9N08/video.html
You have a rank not a weighted preference which is additional information. I happen to be a fan of score voting.
Yes, this is my favourite episode so far. But if I will see the next one, my favourite will be the previous episode ;)
I'm still confused. Is it like a first come first serve where if the higher vote is first than you just compare it to the next? My explanation doesn't make sense either
Anyone else notice "Condercet" at 8:03
What about cardinal voting systems? Does the Condorcet paradox or the Arrow's impossibility theorem apply too?
Nope. Arrow himself admitted it.
Still, *rock* is better than _paper_ and _scissors_
Billy Willy, I agree. You can make planes with it, too.
PlayTheMind "Good old rock, nothing beats that!"
@PlayTheMind: Probably true when the majority who prefer Rock over Scissors and the majority who prefer Scissors over Paper are larger than the majority who prefer Paper over Rock. The larger a majority, the more likely they're right, all else being equal. That's the "majority rule" heuristic.
PlayTheMind, this is why I play rock scissors _dynamite_.
Please make a video about range and approval voting. There's a lot to say about voting system's mathematical properties (especially concerning how vulnerable each of them is to interference) and these systems are the best I've ever seen.
That 'Orange' winner though. Very nice.
Scienceforall has an entire serie about this subject.
It would be interesting to have a electoral vote based on issues instead. If one could in some way tell the system your opinions and the system could find your candidate. Obviously too tedious, same flaws mentioned in the video, reliant on technology that could be exploited, and people don't like to be confronted with reality. However, it would rid us of polarisation, by removing the bias to a candidate's image.
One could also vote by asking who you think could solve issue x on a range of issues. Has the same problems as the previous one though.
With two colors (choices) its still wrong to pick based at plurality voting system.
3 brothers, the choices for pet is cat and spider. Kid 1 is anarchnophobic give an score of 1 to spider and 8 to cat. Kid 2 and 3 give 10 to spider and 9 to cat. Under plurality voting (also called first past the post) spider would win and thats a bad thing. Under approval system (people give yes to candidates they like) and assuming kid 2/3 give yes to cat too, cat wins, under scoring voting spider get 7 and cat 8.66666... and cat wins.
At 6:50 the video said "you have all the information about each individual preferences", at this specific example of the video you don't, it would require each score between 1 to 10 or 1 to 100, each person gave to each candidate, to have the full information.
As said before, even between just two candidates plurality system (called also first past the post is bad), so this means wining all 2 candidate battles using plurality system may not exactly a desirable thing. And if plurality was desirable they would be using it and not the system they are using now. Maybe someone could invent some name for something like Condorcet winner but where the guy win each pairwise battle, using the voting system you are testing, instead of plurality.
Borda Count with weighted scores. The weights would have to be figured out by smarter people than I, but it feels to be to be the best. From a purely American Voting perspective, being able to vote non-primary party and not 'waste your vote'... That seems tremendously beneficial.
"being able to vote non-primary party and not 'waste your vote'... That seems tremendously beneficial."
What about Score Voting? It satisfies favorite betrayal, so there is NEVER a reason not to give top support to your honest favorite.
In fact I don't think Borda accomplishes this even if you used a different pattern for point distribution (it definitely fails if you use the normal versions). In Borda, if you put a third party first, that means a front-runner gets less points than they could have, which could cause them to lose to a third candidate.
I've always wanted CGP Grey to go more in depth into the mathematics of voting, so this is like a dream come true. My only complaint is the lack of jungle animals.
CPCGrey, Infinity Series and xkcd having the same topic? Awesome!
Britain uses a "First past the post" system, where the post is 60%. the problem with that is the latest election resulted in the #1 being less than 60% (For the second time recently) which results in a "Hung parliament". After this happened the first time, and we ended up with the ConDem party(2 separate parties formed a temporary alliance to bring the total votes up to the goal), we were supposed to shift to ranked voting (Like your Method 3 here), but that didn't happen. Now we're in the same Jam again.
thx for the video! if i ever found a state I will use this knowledge to manipulate the elections for sure!
How did u transfer 6 to blue and purple but u didn't give 3 for each of blue and purple in second round?
This video would be much more visually comprehensible if each ballot were re-arranged into voter rank order.
I disagree. Would find very disconcerting and distracting to have distinct listings of colors. I gues there is no perfect visualisation system either
The six ballots shown at 2:35 should be normalized to "1 2 3 4 5" rather than normalized to "green blue pink red orange". She even verbally describes the ballots in ranking sequence, "eighteen people ranked green first, red second, orange third, purple fourth, and blue last." Yet visually, we must painstakingly parse each numeral to decode the meaning of the ballots.
@I agree with Park Tamaroon's initial point: Each ballot should list the candidates in order of preference. I prefer listing them top-to-bottom rather than left-to-right, because top connotes best. Either top-to-bottom or left-to-tight, it's easy to see when a vote ranks one candidate over another. And there's no need for any numbers. Instead of asking each voter to write a number next to each candidate, the voter could drag the candidates up or down until they're in the desired order.
"We can simulate the second round". If only it was so simple. In reality, other candidates give support to the remaining top two candidates, depending who won the first round. Furthermore, voters tend to not turn up in round two if their preference did not win.
Australian system is - count totals for all options, discard last place. Count all totals for remaining options using discarded votes 2nd preference, discard last place ... rinse repeat.
Australia uses IRV for its lower house, and STV for its upper house. STV is proportional and the upper house is multi-party, but despite this the lower house is still two-party dominated.
Does expanding from a single winner to a proportional system eliminate some of the issues with selecting a winner that satisfies the Condorcet criterion? Does it complicate the problem if you introduce a proportional system instead of a single winner? What did Condorcet have to say about proportional systems for deciding voting results?
PR was just starting to be invented at the end of Condorcet's life. "In February 1793, the Marquis de Condorcet led the drafting of the Girondist constitution which proposed a limited voting scheme with proportional aspects." He died in prison a year later.
Boda count where instead of count×4+count×3+count×1 you have (100×count/total)^1.5+ (100×count/total)^1+ (100×count/total)^.5 has interesting properties.
How about this one: Use a process similar to Instant Runoff, but in reverse. Instead of eliminating the one with the least first-choice votes, eliminate the one with the most least-choice votes. In your 5-color example it works like this:
Round 1: Blue ranked worst by 18, Green ranked worst by 37, Green eliminated.
Round 2: Blue ranked worst by 29, Purple ranked worst by 16, Red ranked worst by 10, Blue eliminated.
Round 3: Purple ranked worst by 34, Red ranked worst by 12, Orange ranked worst by 9, Purple eliminated.
Round 4: Red ranked worst by 28, Orange ranked worst by 27, Red eliminated, Orange wins.
Orange was both the winner of this system and the Condorcet winner in this case. How often will this happen? Does it always find a Condorcet winner if it exists, or does this fail in other cases?
This is known as Coombs' method sometimes. It can fail to find the Condorcet winner.
Okay, thanks!
6:45 "We have all the possible information about each individual's preferences"
Not quite, you have their ordered preferences but you don't have their strength of preferences. Score Voting and to a lesser extent Approval Voting can take these into account. They also pass Arrow's Impossibility Theorem, as later admitted by Arrow himself. At that time he also said he thought Score with "three of four classes" was probably best. (electology.org podcast)
@Skyval Ream: No, they fail the spirit of Arrow's theorem, which is what really matters. The reason is that you can't expect every voter to be non-strategic when their best strategy is extremely obvious. For example, suppose Trump would narrowly win a 3-candidate Approval election, with Clinton a close second and Sanders third. Suppose some voters prefer 'Sanders over Clinton over Trump' and would vote "approval" of Sanders and "disapproval" of Trump and Clinton. But now suppose Sanders doesn't compete (because he doesn't want to be a spoiler). The obvious strategy when only two candidates (Clinton & Trump) compete is to approve the candidate you prefer more and disapprove the candidate you prefer less. Thus some strategically minded voters who prefer Sanders will approve Clinton and disapprove Trump when Sanders doesn't run. This lets Clinton win when Sanders doesn't run. This shows a failure of Arrow's consistency criterion: "If X finishes ahead of Y given some set of candidates that includes both, then Y must not finish ahead of X given another set of candidates that includes both." All voting methods that are even slightly democratic fail this criterion. With Range Voting (sometimes called Score voting) the obviously best strategy when only Clinton and Trump compete is to give one of them the maximum score and the other the minimum score, and it too fails the criterion. The reason Arrow's theorem has an ordinality criterion is because if the voting method uses info about voters' strengths of preferences to affect the outcome, it gives voters a strong incentive to exaggerate the strengths of their preferences, which can leads to eliciting less information rather than more information about voters' preferences. Also, many voters aren't strategically sophisticated and have a tendency to rate compromise choices worse than they really are, which would tend to defeat the best compromises and lead to (or maintain) polarization. Dropping the ordinality criterion is unwise.
Strategy in ranked systems tends to involve explicitly misordering candidates, while strategy in Score tends to involve compression. So in this sense highly strategic Score ballot is still more honest then most minimally strategic ranked ballot.
I disagree that exaggeration of strengths of preferences leads to less information than ranked ballots. Even an Approval ballot expresses some information a ranked ballot cannot. Because a strategic ballot is a function both of strengths of preference and candidate chances, the reported strength of preference are distorted by the candidate's chances of victory, but some preference strength information remains even in Approval. Ranked ballots have none.
SteuTube222
I think that has its own theorem, no need to call it the "spirit" of Arrow's Theorem:
en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem
YES! you cited the voting god of youtube.
(please tell me if you know of any other voting gods on youtube, there need not be a monotheism on voting gods)
If we find other voting gods we should vote on whos the best
There is also problem with representation of the preferences. With rank preferences we assuming that difference between option are equal that not always is the case. E.g someone may prefer blue slightly more than green, but both prefer much more than red.
In these case, I think, is better to store for each option rank value from 0 to 1 (or from -1 to 1 ) . In that case the easiest option would be to choose option with highest sum of ranks.
Exactly! While I don’t quite understand the second part of your comment, I completely agree that the speaker in this video is wrong when saying they have “all possible information”. I advocate for the fractional voting system, even though it is quite obscure. Essentially, the way it works is that each individual is given one vote, but they can split that vote between candidates in amounts they choose. For example, they can give sixty percent of their vote to one candidate, thirty percent to another, and ten percent to another. This system allows for people to quantitatively represent the amount they support each candidate, while ensuring that each vote still has equality, the lack of which is a major problem in the similar score voting system. In that system, if a candidate scores five out of ten points in total, the ballot of someone who tallied them a five will not change the final result at all, making it essentially worthless, whereas someone who gave that candidate a zero or a ten will have more influence on the outcome. Therefore, I advocate for a fractional voting system, which I think is the best.
Brazil has a Two-Round Runoff
but it's ironic since voters can simply not vote, or vote blank, meaning there's actually 3 options in the second round and therefore the election winner may not have a majority
How can you say that we use the plurality method for US president, when the Constitution requires a majority of the 538 Electoral College votes (270) to be elected?