Why it’s mathematically impossible to share fair
ฝัง
- เผยแพร่เมื่อ 19 พ.ค. 2024
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This site was super helpful at explaining the history and math of apportionment. As always though, verify the specifics of the data yourself (but all the general points are definitely correct). www.maa.org/press/periodicals...
You want MY data? Come get my data. www.dropbox.com/s/o8bmzmwcpbl...
George Washington was voiced by my friend Ben Moor. They are in The Queen's Gambit!
The Voice of Alabama is voiced by The Voice of Alabama. They are in Alabama!
Thanks to all of my Patreon supporters who mean I can spend a silly amount of time on a video like this. So much time. So much historical data. You too can help support me: / standupmaths
CORRECTIONS
- I didn't bother showing me locking cell references in the animations. Things like "=B4/B2" should have been "=B4/$B$2" so I could drag the formula down. We took that out in the interest of clarity.
- Yes, at 13:20 I say "the divisor ceases to lose some of its strict meaning" which is the opposite of what I meant! the sentence needs but the one negative. Either of these would work: "the divisor loses some of its strict meaning" OR "the divisor ceases to have some of its strict meaning".
- Sorry, at 16:47 column D is wrong. These are different numbers using 880 but the values over in E use the correct 930. It's just a display issue and does not change the results (despite being a bit confusing!). Spotted by a few people including Thomas Klemm and AverageJon.
- People have mentioned that Theorem 2 (and the conclusion) from this 1974 paper by Balinski & Young has an Alabama-paradox-beating method. doi.org/10.1073/pnas.71.11.4602
- Let me know if you spot anything else.
Filming and editing by Alex Genn-Bash
Animations by William Marler wmad.co.uk/
Voice work by Ben Moor and Destin Sandlin. Yes, it was Destin. Well done on scrolling all the way down here to check.
SUM Music by Howard Carter
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And do check out brilliant.org/standupmaths to learn how to do things like this!
Wyoming wants to know your location.
"Well done on scrolling all the way down here to check." You got me, well played.
Y
@@PopeLando the problem is that this is rounded correctly from the numbers that should have been at the jeff column for DJ=930. but the numbers showed on the jeff column correspond to a DJ of 880, like the last number that have been done.
I have no idea what have happend, but the real problem is not with the rounding, but just the apperent jeff column.
@@hodsinay6969 I noticed this too, I think it’s a problem with the graphics in the edit.
as an Alabamian, I can confidently say our education system never forgave math for this and has actively scorned it ever since
because of re#publicans
another alabamian, i agree
@@grimaffiliations3671 lol. If you say so fam.
@@adrianbundy3249 Same story in west virginia where im from. Poor education, poverty and crime. I guess that poor education is why we keep electing republicans
@@grimaffiliations3671 Children should be taught at better schools. Public Schools should be abolished.
You're really good at explaining complex math concepts. Thanks for letting me play along.
Crazy how my tickets were right next to yours at brain candy live. Felt like I won a lottery all those years ago. Thanks for inspiring me!
We're getting better at math, and I understand that you're feeling pretty good Destin! 😉 Love the cameo
@@MationGaming I still have the poster!
I was pretty sure that was your voice.
You should voice act more!
2:40 Why is Circula so upset? You'd think he of all shapes would appreciate rounding.
ba-dum-tsss
28:06
Fun fact, the folks at Wikipedia have actually gone through the research and found that the original census actually miscounted a single county that had 450 people living in it.
So, it actually isn't a transpose error where somebody flipped a digit.
The original source is off by 450, and it happens by coincidence that 490 + 450 = 940.
How wild.
It's possible that their "miscounting"... was flipping the digits. You don't accidentally count an extra 450 people. You miswrite something.
@@JoCE2305 I agree with you. A transcription at some point is far more likely.
The transcription may well have occurred at a more local level and the difference would be carried forward in the total.
BTW, any transposition (reversal) of two sequential digits in a number will yield a difference that is a multiple of "9".
So the original mistake (if it was a transcription error ) could have occurred by using 160 in place of 610 (which again has a difference of 450) or any of several other number pairs.
[ in book-keeping, back before spreadsheets addition was often checked by doing it twice. If two sums were off by some amount that was a multiple of 9 it was always a good practice to look for a transposition error between the two sets of numbers ]
@@howard5992 See above. They did appear to actually miscount rather than transposing digits. They were unsure about the status of a single county containing 450 people. Or so the story goes.
Still wikipedia has so many errors that people are not realising, because "it is wikipedia, it is on the internet, it must be true"... Well ...
@@radekhavelka3237 so do analog dictionaries, census records, educational material etc... Just as many people fall into the idea of "it is on the internet, it must be worse"
The amount of joy on this man's face when he said "the united shapes" is enough for me.
The United Shapes of Geometry.
I feel patriotic to a mathematical country
2:10 I love the proud smile on his face when he says “United Shapes” and thinks about how funny of a joke it is
Which it is.
It is unusual how he gave a serious meaning to the way "states" sometimes get slurred in normal speech. I did find it funny.
United Shapes of Euclidian.
@@georgelionon9050 the united shapes of archimedes
United Shapes of Mathematica
Speaking of clerical errors, you've got one in the video. At 16:45 the resulting values for =POP/DJ are being displayed using the final divisor, not the initial estimate 930. Which is also how I could tell that the excel/spreadsheet magic was just smoke and mirrors! Clever editors
It also means his presumption for that instance is incorrect; there was no further rounding or adjustment needed. The numbers came out fair after one adjustment.
Yeah, when 24.8614 rounded down to 23 I was like "what?".
Up next: Why it is grammatically erroneous to use the adjective 'fair' in a video title where the adverb 'fairly' belongs.
I thought I was crazy.... but then I saw your comment and was like... Yup... He ducked up....
It is pretty awesome though that they actually go through the effort of showing enough detail to teach the viewers how to use spreadsheets to do this all on their own!
Performance excel has to be perhaps my favourite thing going. I can't quite get my head around what you've done to make it look so slick other than manually updated everything to make it look like excel so I just wanted to say I appreciate the work that went into that!
Are you sure it's not just a customized version of excel screen recorded and slightly edited?
@@mrp0001 some things such as him not using absolute references for formulas he has to drag makes me think it's an animation
I really liked how you showed Excell commands and functions in a more friendly way. It's nicer to be able to follow along to.
Yeah, it's all smoke and mirrors tho, at 16:43 the values for =POP/DJ are shown using the final divisor instead of the initial 930.
If he would have displayed everything correctly, I would agree. But it was just screwed at some points.. :D
The cameo by Destin-I mean Alabama-really made my day. He obviously had a ton of fun recording those bits!
Destin is the perfect choice for the voice of Alabama. 😀
As soon as Alabama was shown, I was like "how cool would a cameo of Destin of Smarter every day and NDQ be?!".
@@joice4042 no
@@kornsuwin robots don’t understand “no”…
Try “else”
@@rossgirven5163 ' OR TRUE; DROP TABLE youtube_spammer; --
Wasn't sure at first if that was him, but had a strong inkling that it was...
Was not disappointed.
“The united shapes” blindsided me like a freaking freight train. I should have seen it coming but I’m still giggling to myself
Motto of the United Shapes of America: "One Shape to rule them All."
@@guardrailbiter and that shape would be the hexagon, of course.
@@guardrailbiter It's United Shapes of Apportionment (USA). He says it in the video (10:45)
I think this explains something I'd been wondering about for a while. In Australia the number of MPs is supposed to be *approximately* twice the number of senators - it's important to maintain a consistent ratio for the case of a joint sitting. I have often wondered why it's approximately rather than exactly two-to-one until now: the quota rule vs Alabama paradox only applies if the number of seats is fixed. If you have a bit of wiggle room to change the number of MPs (as is the case) you can avoid both issues!
Some years ago I was writing code and I came across this exact problem in a different context. I didn't want to spend too long thinking about it, so came up with Hamilton's method and wrote a quick and dirty implementation of it. I remember at the time thinking that there must be a proper mathematical solution somewhere, but that it wasn't important enough here to waste time on. I'm surprised to learn that not only is there no "fair" solution that always works, but also that the US Federal government took the same bodge-it approach I did!
That’s literally how the US Congress does anything. “Oh no the deadline is tomorrow and we’d have to furlough hundreds of thousands of federal employees if we don’t drop our charade, let’s pass a temporary budget that lasts three months so that we can go home for Christmas without people yelling at us”
@@MxSherwood and I wonder what kind of people the US Congress has been full of for the past 50 years....
@@pugdad2555 Do you really need to wonder? One of them became president.
@@ButtKickington Multiple of them have. Unless you want to forget the exact same stuff happening 2016-2020
Are you surprised?
At 16:33, you've claimed that POP(New Triangle) / D_Jeff(930) = 24.8614, rather than the correct ~23.5247. Notably, in your next column, you've fixed the issue with the rounding down correctly recommending 23 seats, but it's still a very baffling error if you don't get out a calculator and do the division yourself.
The error also carries over for Circula, where you've listed the incorrect value of 11.0375 rather than the correct 10.4441, though your floor function still works.
Actually, looking a little closer, that whole column appears to be in error, as though you used a divisor other than 930. It's simply that New Triangle and Circula end up being off by a whole seat, so when you fix the equation in the next column that also takes the floor of each number, it's readily apparent for them.
so glad you said this, thought i was going crazy... why is that 24 rounding down to 23?? oh because it wasn't supposed to be a 24.
I guess the devisor used was 931 instead of 930 but the issue there would have been that the total number of seats after rounding down would add up to the required 43 making the next part of the video obsolete.
Yea I noticed that- was driving me nuts, I paused and rewatched like 10 times
@@FabioNiewelt No, it wasn't 931 - the incorrect numbers were too large, meaning that divisor used to get them was smaller than expected (930). It might've been one of the other lower divisors discussed later in the video, and you can use some algebra to figure out what it was, but I'm not quite that much a stickler for fixing Matt's Parker Squares.
His ENTIRE Excel spreadsheet formulas become convoluted at some point, and it just exponentiates the errors. Later on, around 16:30 ish mark, it is rounding down numbers in an incorrect way. 11.x goes to 10 for example for circle. This throws the amount of seats off by 1. A 2% error.
I loved this video! The concept of "fairness" is absolutely something we should all know more about. I was wondering if you could explore the same type of concept in compating types of voting? Thanks!
ik your comment is a year old but cgp grey has a whole series about different kinds of voting systems
Fascinating video! I'd love to see you build on this by talking about mathematical advantages/disadvantages of implementing something like the Wyoming Rule.
Wow, the production quality of these videos is rocketing up faster than New Triangles fraction under the Jefferson method!
Even the little things, like sharing a cleaned up spreadsheet without all the grid lines and the whole UI up top adds so much to the visual clarity of the math itself. Keep it up Matt!
stop it, scambot,
anyway yep, i agree with you
makes it so much more appealing and motivating to watch the whole thing!
I would very much like this fancy spreadsheet and skills to boot, thanks in advance
@@i_Hally except the values are incorrect around 16:33
@@robertcousins2274 yes. It has been described in detail in the doobly do
A very similar problem comes up in typesetting when splitting lines evenly into a paragraph. A native solution tries to minimise the total divergence of spaces between words from the average, but that can result in one line being very bad (for example, two words on the line with a massive space between them) while all the others are okay. The fix is to minimise the average of the squares of the divergence; I believe that’s what Don Knuth’s TeX system does (I worked with it back in the 90s, so forgive me if my recall is bad). There are also strange Alabama-like paradoxes caused by the fact that although spaces can be stretched and shrunk smoothly, words do not normally stretch or shrink and jump from one line to another unpredictably as wrap width changes or new text is inserted. I spent a happy but confusing 9 years working on typesetting software and have scars to prove it.
It's always interesting when the same mathematical problem has repercussions in such different fields.
This is why justifying a table of contents is my least favorite and most time consuming part of writing any paper. Why don't all the lines end at the right margin? Why are all my numbers misaligned? Why isn't each character "space" the same width, or each line the same number of spaces?
For those who understand it, daily life will turn upside down: The Connections (2021) [short documentary] 💖
@@Lawrence330 Software will do that automatically for you...
Fixed width fonts are a thing, but not really designed for reading well.
There are some pretty decent monospace fonts nowadays, @@someonesomewhere1240, though their usability largely depend on context.
I ran into a similar issue years ago at an engineering company. I wrote a program to generate a monthly report in nice, round dollars. The problem was that the numbers didn't add up do to roundoff and that really ticked off the accountants. They didn't care what it took to make that report balance.
I considered choosing one of the totals and use it for the other side, but I figured if they ever added it up, the accountants would be even madder. The solution was to add up the numbers, and determine a roundoff that would make them balance. So one month, it might be rounding everything up from 40 cents and the next month it might be rounding down from 80 cents.
Why couldn’t you just include the cents?
Just tell them, "ok, assume that every month I will burn a ten dollar bill and that's where any inaccuracies are coming from. So, just accept any number within 10 dollars." lol If only you could do that. I mean, considering the man hours you put into this, that would probably cost the company less.
Gotta love torturing those accountants 😝
Got to love accountants. Guys who will spend $100 to track a penny because every penny has to be tracked. Guys who can tell you to the penny how much money you've lost, but can't tell you how to stop losing it (hint: don't spend too much paying accountants---).
You can tell that my MBA is NOT in accounting.
A lot of times what I've seen is (my parents are/were accountants, is to make all accounts but one be exact numbers, then use the least important one as the "slop" account, that is all rounding errors only affected that account, in terms of making everything balance down to the penny.
😀😀😀 I'm glad I stayed for the unmistakable voice of Alabama. So many of you in this category of Math/Science/Engineering education are doing such good work. It's fun to see the collaboration. It's destin-y!
This reminds me of a similar problem I ran into when writing an RPG engine back in high school (circa 2003 I believe). I was treating character level as a percentage scalar to total attribute values but wanted to distribute attribute increases consistently at every level, so rather than do each attribute individually I decided to increase the total attribute pool itself, then distribute that pool proportional to each attribute's base value. So, for instance, the initial pool was 20 points per character with a 20% increase per level, meaning I wanted 4 new attributes to be automatically distributed with each level up.
The problem I quickly ran into was that using the Hamilton method (without knowing that's what it was; it was just the first solution I stumbled upon), was that the 20% increases led to beat frequencies where occasionally the fourth and fifth highest remainders could be **ties** from attributes with identical bases, meaning no exactly-4 increase was possible with that method. In the end I just scrapped it for a round-down approach that had increment spikes on the beat frequencies, since I felt having every 5th level be the one where **every** attribute increases made for a better RPG style "milestone" feel anyhow.
Or the fact that every data science library divides trust data up using Hamiltons method that every program discovers and is used for non political scientific research
@@gregorymorse8423 why hailton and not hamiltons-hill?
Now, if Hamilton had been designing an RPG, he would've figured this out in the first place.
now did you ever FINISH that game? (yes this is a personal attack on you as a game developer)
@@HesderOleh I'm talking about k-folds cross validation. So k equal partitions of a dataset. This means you can just divide out the remaining ones by fractional % which are all the same anyway. But the fact is you take the rounded down amount and just assign up to 1 more as needed to get the closest to an even partitioning as possible without discarding data
The 3/5's clause was about slaveowners not being able to buy more representation. The free states didn't think the slaveowners should get more representation just for keeping more slaves who couldn't themselves vote or leave. The southern states obviously thought slaves should count as part of the population, not because they believed slaves were entitled to rights or dignity, but because it meant they'd have disproportionate political power.
I always thought that when I was younger. I said "hold on, why would the slave states want their slaves to be worth less when they could have more power"?
@@MOOBBreezy Right, abolitionists were of the position "if they must be slaves, you don't get to steal any of their voting power". Their desire wasn't that slaves are not valued as humans but that the amount of their democracy heisted should be zero.
Naturally a disingenuous person would (still do) try to reframe the situation in the exact opposite of the truth.
I wonder if enslaved people were counted as 3/5 in the last census in the USA.
@@grieske its true. I was enslaved last year
Wouldn't that mean that they just need to buy 5/3 times as many slaves to buy as much representation?
Recently discovered your channel - and love your stuff. Amazing content.
It's a happy coincidence that my brain seems to like the way Matt says the words "number" and "digit" 😄
But how many sides does an “Ore-gon” have?
16:34 - your POP/DJ figures are out here! Although weirdly column E rounds down ‘correctly’ using what should have generated. Did you input Jeff’s divisor as 880 as a test at some point? 🤔
Edit: ah, 880 comes later!
Another Edit: Adams’ POP/DA figures for all states except New Triangle, all jump straight to your 960 conclusion on first entry, same issue.
+
+
Yes, you are right, it seems we have some hold-over numbers in the animation. My fault for not double-checking everything again! But as you thankfully noticed: the final values are all right, we're just displaying 880 instead of 930 in the intermediate column. Annoying and a bit confusing, but at least the results still stand. I've added it to the corrections.
@@standupmaths Not 'a bit confusing'. *Very* confusing for anyone trying to follow along. Undermines the illustration as you're going through it. Worth fixing if you can.
@@standupmaths Thanks for correcting it. You are truly a Stand-Up dude.
Or alternatively, you drop the seat limit entirely and figure out the size of seat based on a divisor of the population of the smallest state, and multiply out. That way as populations change you get automatic switching and remain fairly consistent.
Tbf, we only put in a seat limit to prevent an overpopulation of politicians
Yeah I was thinking of methods where you alter number of seats, but this could quickly end up impractical
@@fresh_dood True, but if you consider that Wyoming has a population of ABOUT 580k, divide by two to have at least two representatives per state and divide the population of the US (331 ish million) by 580 thousand, you get about 1100 representatives. This is a lot more than they currently have, but every single one represents a district of basically equal size, and 1100 is sort of on the high end of what is practical. It's also better as a system because it keeps the representative ratio lower which is generally quite good.
@@Peter-iq9yy 1100 is absolutely not practical. It's impossible to have a functioning legislative body formed out of that many people. Besides the inherent difficulties to getting such a huge crowd to agree on anything specific, the human mind isn't even capable of keeping track of 1000 people so the representatives would be unable to communicate effectively between each other. And at that point power naturally reverts to a small elite among the representatives, smaller committees, and, especially, to political parties. Which is why past a certain point adding representatives is meaningless, as in the end political groups like parties end up having to make the actual decisions so that what actually matters is the fraction of the chamber each group controls rather than the opinions of most individual representatives.
But that requires Americans to be smart and the Republican voter base would be too lost
That moment of pride on your face at 2:11 when you said "The United Shapes" was priceless
I know, right? xD
16:56 24.8614 rounds down to 23. Interesting.
It should have been 23.5247 so rounds down correctly to 23
This was super interesting to see historic real-world consequences of math.
Maths. Both of them.
I was not expecting the history nerd in me and the math nerd in me to both get so excited together.
Am i confused or are the numbers wrong at 16:49? He started talking about the jeff method with dj rounding stuff, but rounded the 24.8 down to 23 and 11.03 down to 10. Looks like he used the wrong box for the dividing
Indeed, there's something wrong there, but he typed D5 and the correct cell got highlighted, so can't be the wrong input value.
And funnily enough those are the 2 missing seats.
26:40 I swear I listen to this every time I have trouble staying focused and it's just too perfect it was used ironically for the reflection bench
A great alternative for the House of Reps is to not have the total number of seats be fixed. Have the "target" be defined, then round normally. The resulting total might be higher or lower than the target, but that's fine.
i agree, remove or add seats to make it a fair number, at the beginning with the 43 seats, justr remove 2 as 41 makes it fair
@@WolfJ so its not done raight, as you need to remove or add seats to make it a fair amound. the point i ment was that when he counted up all the percentages for all the seats he came to 41 seats, thats before he started to explain all the different systems. to make it distribute 43 seats
The problem, of course, is integer representatives. But you don’t have to solve that issue by chopping up representatives, instead you could fractionalise the power of their vote to match the size of their representation.
This then causes problems with voting because the votes are no longer equal, but it could be argued that’s the only way to be truly fair with integer representation.
I like this idea!
Yes that could really work... although that could do interesting things to the power dynamic to the individual representatives in the house. Also worth considering whether you would have 3 full voting reps and one with a 1/3 or everybody gets 2/3 of a vote or something like that.
@@DRicke Or just make the actual house smaller (50 members) and give each member the weight their state brings. This member should be answering to a larger body of actual representatives of their state, who would decide the vote of the main voter.
But with this, you'd lose the ability to split up the state's vote... So nothing is perfect.
Maybe direct representation: every issue pops up on you smartphone screen and you vote with all the population.
(We'd probably have free beer and hookers for about a month, then society would collapse :D )
@@toppantoster
That is basically 'first past the post' and it's the worst election system imaginable.
I'm all for chopping up representatives. **laughs in Robespierre**
Would a “highest averages” method work? Wait that violates quota rule
I usually don't understand everything that these videos show even though it might be a bit simplified. However, I do love watching them, The way you you explain things reminds me of my favourite teacher in school
Oh man must have starred at every single detail in the I spy spooky one! Loved those pictures and just finding anything even if it wasn't one of listed items. Such good memories!
The spreadsheet part it impressively clean! Also love ‘United Shapes’
I always thought the "Alabama paradox" was when you went back in time to meet a distant ancestor, but--in a dramatic twist--it turned out that actually that relative was always you, so you have to stay in the past and fill the role to prevent yourself from disappearing.
That sounds like it should be a plot in a Futurama episode.
My favorite version of this idea was the twelfth Doctor's 'lesson' featuring a hypothetically non-existent Beethoven.
@@JasperJanssen boy do I have good news for you
It really sucks, too.
The Alabama paradox is when you go to meet a recent ancestor but - in an unsurprising twist - it turns out they are also your lover.
This dude is straight up doing master's thesis level work for a TH-cam channel.
I love you Matt
You do deserve all the cookies for the entertaining education that you work so hard to provide!
29:45 fun fact, the "slaves are 3/5 of a person" thing was actually *opposed* by the slave states, who wanted them as full people. The 3/5 would deflate their population and reduce their political power.
True, but it was still messed up.
Yeah, some people are incapable of grasping that the slave states would have implemented a 500/5ths rule if they could have, because that would let them dominate the House. I'm still surprised that Parker did all those apportionment calculations and either failed to notice that the 3/5ths rule *reduced* the representatives apportioned to slave states or forgot that slaves don't get to vote for their representatives.
A -5/5ths rule would have been more appropriate. Actively punish the moral abomination of slavery by making a slaveowner's voting power get canceled out by his own slaves.
And when they abolished slavery and the 3.5ths, they got more representatives but made sure those slaves didn't actually control any by supporessing their ability to vote. So even more power to Mr Jim Crow.
@@vctrsigma So are you in favor, or opposed, to the Trump Administration's rule to stop counting undocumented aliens in apportionment calculations? (Thus reducing the electoral power of states who have a lot of them.) I note for reference that undocumented aliens can't vote, which in and of itself seems uncontroversial.
Yeah, that's why they called it the 3/5 Compromise, because slave states wanted it to be 100% and free states wanted 0%.
I still find it funny that all these methods have different names in different places. In Germany the Hamilton method is called Hare-Niemeyer, Jefferson is called D’Hondt, and Webster is called Sainte-Laguë.
The only channel that can make spreadsheets fun 😂
Considering you have a significantly better understanding of math than I do, I'm shocked that I found TWO blatant rounding errors on your table. It actually through me enough that I had trouble following that section of your explanation. However, overall, I learned a lot and really enjoyed the video. Thank you for all the work you put into it!
That whole problem stems from trying to squeeze between 2 rigid rules: divisor & max seats. If we left the divisor uniform & the max seats flexible, it works fine.
In the beginning, the divisor was 30,000.
The populations round naturally up or down, except when the result is
That's the downside of having to actually fit these people in a physical location.
If we'd actually used the original uniform divisor of 30,000 we'd currently be at about 10983 seats, which puts us into stadium territory at a minimum. Per Wikipedia, all NCAA FBS college football stadiums except for one in Hawaii would hold it, but only 77 out of 360 NCAA Division I basketball stadiums could hold that house of representatives.
Considering the US Capitol building was designed in 1792, they didn't foresee the growth the country would experience. Beyond that, a full stadium complex behind the capitol building might ruin the view for those looking at the Mall.
@@MrSJPowell
Thus as population grows, the divisor has changed: 50,000 then 75,000 then 100,000 to ~750,000 today. That's what I'm saying SHOULD be flexible. If the max seats is 435 (as current), adjust the divisor until the natural rounding gets you as close to 435 as possible without going over.
The paradox comes in w seat 435 when it naturally lands at 434. There is NO extra seat, just leave it at 434.
Actually, I was thinking just that. What if the rules allowed also the addition/subtraction of seats based on the proximity of divisor to the last allocated seats? Pretty much what you said (but I thought allow going over in case going under falls too short): Start with 435 targeted seats and allocate an actual number between 434-436 based on the outcome.
@@IHateUniqueUsernames
I agree it could fall into a range of values, but physical limitations would produce a hard upper limit. I am also wondering how to handle the excessively high divisor issue. 750,000:1 seems a bit excessive to me.
The solution is rather simple: Empires are not meant to be as big as the U.S. is. If the divisor is too big, the nation becomes an elective oligarchy, and if the divisor stays the same, government is no better than a small mob because there are too many people to make effective decisions.
Easy solution: the country needs to split so that the people and representatives can stay as balanced as possible. This won't happen, for obvious reasons, but it is probably the healthiest solution.
Either that or we have tiered representation. Keep 30K. For each 1500ish of the ~11K reps apportion a Mega Rep that represents the representatives at the Federal Level. You would basically end up with "Local Federal" representatives that meet in your State and send your "Federal Federal" representatives to Washington, D.C. The Senate wouldn't need to change, since it already ignores the population mechanic.
The round down at 16:46 seems to be wrong for the first 2 rows (or possibly the pop/dj)
Took too long to find this in the comments.
Wdym?
Wondering the same thing
I noticed the same thing. I checked, and the POP/DJ is what's wrong. Seems like 880 was used for the DJ, though the rounded down values are accurate.
yeah some editing problem maybe - should be 24 and 11
Oh my gosh - I ran into the same problems with the historical data when teaching older adults about election systems. The section on apportionment about drove me crazy as I ran into the same kind of data problems.
When I realized the video was just manipulating a spreedsheet designed to look like a gorgeous background, I went bananas. Dogs were crying, cats were speaking in tongues.
10/10, subscribed.
I love his naked honesty. It’s a breath of pure, fresh air in the atmosphere of bad YT content. I had no real concept of how complicated it is to create and maintain a fair system of governance. (And it’s a shame that recently we don’t seem to care much for fairness.)
Also, I like history.
Well, if its mathematically impossible to actually be fair, then, my side should be where the tilt goes, shouldn't it? ;)
This is just the beginning. Look up arrow’s theorem, and the pros and cons of the shortest split line method and other districting methods, and different ways to count votes such as condorcet, Borda, runoff, range voting etc.
Talking about "naked", the state of Alabama in the first 5 seconds of the video could use some pixels lol.
South Park pfp: of course would be in the bad yt content. It's a natural translation from tv to yt.
Honestly, the issue of representatives is a *TINY* one compared to the problem of the senate, in which 50 senators--potentially representing the 25 smallest states--potentially with the least educated populaces, can bring the entire legislative process to a grinding halt, which is what we see right now.
Democracy has an inherent fatal flaw--in that it gives one vote to one person, regardless of how intelligent that person is. That is, if you had a disease, would you value the opinion of 50 Google MDs, or one actual medical specialist?
The fact that we have absolutely no way of assessing knowledge of civics, ethics, modern-day issues, foreign policy, economics, mathematics--ANYTHING AT ALL--and then let the most easily conned imbecile cast the same vote as a PhD scientist--is the reason we had the orange turd that was Donald Trump.
27:50 had that problem with scientific data (tables of materials properties) recently, when i was looking up a citable source for data used for a publication. Apparently the table from the original publication of those who did the experiment was transcribed with several typos for a publication of a compilation of several materials properties, then that table was adopted by some manufacturers association and widely circulated as service to their customers. Of course those tables are now still widely used although there is newer, and better experimental data.
It happens even with pure math tables. Look for instance at the Savitzky Golay filter.
I feel you!
As far as I know the statistical methods used in CERN counter in mistakes within the code of the instruments.
@@tobiaswilhelmi4819 Would you mind rephrasing that for me? Seems like it's probably a neat and interesting comment but I'm trying to figure out what you mean and I can't figure it out and I'd really like to understand what you're talking about (I know what statistical methods are, I know what CERN is, but I'm definitely still missing something!).
@@idontwantahandlethough I think what he means is that the code in the instruments used in CERN account for the mistakes that arise in the statistical method they use.
I’ve had this come up many times at work. Cheers
22:37 Just realized you used Destin's voice for Alabama, that's great
can confirm that is exactly what Hamilton the musical is about
At least the largest fraction of it, but who knows how it might have been if they had one more actor?
@@Excalibaard hah, indeed. And I guess that means different productions might include all this or not? So if anyone disagrees with Lottie here, they must just be familiar with a different production. Still, I think Mr. Parker should find out directly!
Another interesting thing to look at I think is, what if we don't require an exact number of representatives. What if we just use that number as a target number and if it deviates slightly after rounding (up/down/closest integer) we just have a few empty or extra seats until the next census?
Or, we just remove bits from the fractional representative? I suppose then the problem is, is that you'd need to do a proportional lobotomy but that has the risk of making the representatives more effective.
Or you could just let the representative cast a vote equal to the population of their district.
@@benholroyd5221 "The spleen from New Hampshire has the floor."
The number of seats used to go up as the population did, but it was locked at 435 by the Apportionment Act of 1929. If we had kept increasing it to keep the ratio closer to balanced between the states we would have over 1000 reps now, which would 1)require expanding the physical building where they meet and 2) really change the electoral college.
@@nigh_anxiety #1 doesn't seem like a big deal. #2 seems like a great idea though.
Really great explanation!!!
In Spain we use a method equivalent to the Jeffersen method (D'hont method) for the appointment of representatives to parties in election, I wonder if we should change it to Hill's method. It would certainly help to give representation to small political parties that are never listened to, although it's not like the US has stronger small parties...
I'm embarrassed to admit how much I've enjoyed this video. This is the most nerdtastic thing ever!!!
This video hits curiously close to my interests. Apportionment, and public choice mechanisms more generally, is something I find fascinating - especially all the many, many ways in which our obvious mechanisms and measures are incompatible.
I am also a historian. Historical data - sources - are astonishingly tricky to handle sometimes. It's interesting to get the perspective of someone not used to it. I can't say the level of frustration surprises me. :D
Do you know of a subreddit or something that discusses those things (apportioment and theoretical political structures)? The closest thing I found so far is the worldbuilding community.
@@f_f_f_8142 I mostly remember mailing list and IRC discussions, years and years back. That and books on related subjects ... public choice and decision theory. I've never gotten the hang of reddit.
I also realized this is almost exactly the same thing I had to do when creating a group price calculator. One where I could take any given sale group price and divide it evenly amoungst the diffrent original prices to get the same discount. But I also wanted the sale prices to be rounded to the nearest whole dollar for ease of entering into the system and understanding, which meant a whole lot of rounding and the differences that came with it. Adding a dollar here or taking a dollar off there.
For those who understand it, daily life will turn upside down: The Connections (2021) [short documentary] 💖
As a bit of a history buff, I really enjoyed this.. I never would have guessed that fairly assigning seats in congress could be so difficult to actually do completely across the board, wow.
Should’ve opened with that fact, first veto was maths based, how Rad!
Great vid as always Matt.
One fix is to use non-integer representation. Use one of these methods to determine the number of seats, but each representative doesn’t get exactly 1 vote. They get the precise fraction that, summed with the other representatives of the state, would give the state exactly the decimal percentage they should get.
@@hungrycrab3297 that is a fantastic idea.
Japan has pretended to try and fix its imbalance that favours rural districts, by making some of the small prefectures share one representative.
Using your idea, each prefecture could still have one guy in parliament (it’s always a guy in Japan), but e.g. Tottori’s rep would only have 0.44 of a vote.
That is surely the fairest way to do it.
On the other hand, if a state should have 2.7 votes, then they get 3 reps, each worth 0.9 of a vote
yes, this is what I came here to say. It’s fine to round to an integer number of representatives but then they each contribute the exact fraction of a vote. Or another way of saying it, each state contributes their integer votes times the % of that state’s population to the total. Then rounding essentially has no impact. (well it only affects the granularity of the votes).
@@snivesz32 Yep exactly.
And let’s combine that with multi-member districts, giving each representative the proportion of votes they actually received, and I think I’d actually like the system.
@@hungrycrab3297 And to add insult to an injury, let them earn 0.23423 fraction of the wages also. National laughing stock would be the guy with 0.0072
@@TheHammerchief give the leftover to the youngest rep to balance the power of established career politicians.
Did I miss the part where it's explained *why* apportionment systems have to either break the quota rule or have "Alabama paradoxes"? Where can I go to find the proof that no algorithm can avoid both kinds of unfairness?
I was curious about this too and found something called the "Balinski-Young Theorem". It includes a "population paradox" in addition to the Alabama paradox. In the population paradox a small state with rapid growth can lose a representative to a large state with slower growth. According to the Balinski-Young theorem it IS possible to create a method that follows the quota rule and doesn't have an Alabama paradox but it will have a population paradox. A method can avoid the population paradox and the Alabama paradox but then it will have issues with the quota rule for certain values. I'm no mathematician so I can't explain the proof but it certainly is interesting. en.wikipedia.org/wiki/Apportionment_paradox#Balinski%E2%80%93Young_theorem
@@dking7120 Thanks a lot, I was really wondering about that!
Even just as a once-upon-a-time mathematician (studied maths at uni, haven't touched it in years and lost most of the knowledge as a result) I really wanted to see a proof.
@@masterplusmargarita yes, especially as the title of the video is "why it's mathematically impossible...". It seemed more like just listing problems with some methods with no actual reasoning why it's impossible in general.
You must have missed it, because he definitely showed the proof, and he explained it, very thoroughly. He even wrote a song about it, to make it easier for kids to remember, and he paid Tom Brady and Lebron James to sing it together, along with some old man dressed up like Santa Clause, and Big Bird from Sesame Street. I don’t know how you could’ve missed that. There might be something wrong with your Internet, but I bet you just weren’t paying close enough attention. Try reloading the video, and then watch it again at 0.5 speed. If that doesn’t work, then you might have to upgrade to TH-cam Premium, and eat some wild mushrooms like I did. Good luck.
The basic reasoning is that when you use the quota rule, you’re left with a bunch of fractions of seats that you must round with.
The 1 tenth decimal place value is completely independent and not causally related to the whole number of seats a state is allocated, and it is also the way the remaining seats are allocated.
So the subsequent rounding ends up giving out a bunch of arbitrary or “unearned” seats to states. Since these unearned seats are almost randomly given out, increasing the total number of seats can randomly take them away as well. So you end up with the Alabama paradox.
It isn’t really a paradox though. We only think it’s unfair to lose a seat because we assumed the original allocation was fair in first place. In reality, Alabama just loses the random advantage that gave it a seat it didn’t necessarily ‘deserve’ in the first place.
as a historian it's hard not to chuckle at your section about historical data. YEP, that's part of the hard work that historians do
This video is incredibly helpful, thank you!
Doesn't all these complications come from the fact that we are trying to allocate a fixed number of seats? Why not just round (up/down/nearest), apply the 30.000-rule and change the number of seats to whatever comes out?
That is what Germany does. It leads to physical and logistical problems and not mathematical ones
Quite frankly: too many people. Not only does the house get locked up already with far fewer people but then the issue of finding/creating venues that can seat everyone arise and that need is increased with every passing census. There’s countless more issues but basically it’s good in theory but impractical in execution.
The additional seats in Germany come for a very different reason. We do not add seats to split „correctly“ between states.
Same idea. Came in the comments to see it. Just throw the seats out. Sounds simple. In a math way.
@@wolftamerwolfcorp7465 "Too many people" is easy to solve. Update the divider in every election, to keep the total within set boundaries (e.g. 290-300).
Yes you can have a system that obeys both: you just vary the total number of seats. The problem is having a fixed number of total seats instead of just assigning seats proportionally without an upper or lower limit. Why do you need 299 total seats? Just have seats equal to the number that is most accurately representing the population proportion! If that means one census you have 300, fine, if the next is 298, also fine - as long as the partitions are properly represented in percentage.
As math system it would work. But as a real life situation with constitution changes and other juridical details as well as persons who benefit from how the current system works - it would never happen.
@@thebomber7641 Come on, what could possibly go wrong with a House of Reps with 11,124 seats?
@@thebomber7641 Germany do this (a bit different how it is done, but they do change the number of seats) ^^
The problem with this method is, if done, the states then have to follow a similar system of division... which is often politically gerrymandered to favor a particular political party (both of the American political parties are guilty, just depends on the state in question). So, every 10 years, every state would then have to (get a chance to) redo their house district maps for federal representation.
Ultimately, the likely best solution (since the USA is and likely will remain a two party country for the foreseeable future) is to leave the Senate with 2 seats per state. The House converts to proportional representation based upon the political parties. (Like many European nations.) This... will likely never happen, though.
@@terr281 Would you take me through your reasoning again? I fail to see why a variable number of delegates (according to a fixed "1 delegate per every x-thousand state inhabitants" formula) would increase gerrymandering problems, or how the specific method used to determine the number of delegates/districts should positively or negatively influence gerrymanderbility.
This is chaos and why'd you spend that much time doing this, above and beyond XD
That was greatly instructional! I was thinking how would these methods compare with what's commonly used in Europe (D'Hontd) and I learned it is equivalent to Jefferson's even though the calculations are carried out differently 🤯
I've recently gone from floor level warehousing/logistics to a more administrative role and I've been stunned by what I've found.
Where you can face disciplinary action(depending on the company) for a minor mistake when you are handling the products and maybe cost the company tens of dollars, the number of transposing errors in documentation at the administrative level is absolutely appalling and can regularly cost a company thousands of dollars, or even hundreds of thousands depending on the company and specific issue.
You can loose an entire order because someone between purchasing at your company, sales at the other company, data input, resource allocation, production runs, packaging, warehousing, shipping, to a carrier's receiving, warehousing and shipping transposes or drops 1 number. Or worse, someone gets a little happy with copy and paste and repeats a number
Don't talk about that to me, i work programming software so to automate, validate, give a nice UI,, etc for that kind of stuff, i work programming an ERP basically and let me tell you, sometimes bussiness flows are so illogical, and wild that they are just not programmable, like they have definitions that can be interpreted differently, and each department actually interprets it differently so no matter the implementation, i will always have some departments call to say that is not how they do it. Or even worse, workflows that directly do not have sense, and the CEO did not realize they did not make sense until i pointed out that logically you can't represent a flow to that, and that what they have been doing until now is just do what their own illogical bussness flow wrong, because it is impossible to do it right, at least when this happens i got a call after a week or so with a revised flow that makes sense, and i have to just code that, but when it is one that has multiple interpretations and each part does it differently oh boy, those i know are here to stay, because nobody will admit that the way they do it is the "wrong" way.
And the companies i get to work with are already in the "sane" category, because our boss will not get clients with too wild bussiness flows because we would operate at a loss with them.
My favourite is when their bussness flows where actually widely illegal and not even on porpouse.
The problem i think, is that management is just that hard, if they fired people for that, they would not have employees, very few people is actually competent at managerial roles, and there are way more demand for those roles than adecuate people, so the bar has to be a little bit low, or you would be firing and hiring constantly.
So how do people train people to be good at manegerial roles, or at least specific parts like accounting and bookkeeping?
I had to deal with this when splitting tips between people at a pizza place. Dividing total tips by total hours worked and then multiplying individual hours worked to find how much they earned. Most of the time, the rounding was easy and fair but every so often, we had to give out more tips than we brought in due to rounding. Luckily, I just took money from managers first (I was one, don't worry) and it usually worked out.
It's illegal under federal law for owners, managers, and supervisors to participate in a tipping pool at all, even if they provide direct table service, so I'm still a bit worried...
Matt, a question.
If Jefferson favors bigger states, and Adams favors smaller states, would a middle-ground between the two methods be the fairest? If you calculate the ideal divisor (down for Jeff, up for Adams) and take the average of the two to use as a final divisor. After this of course the question of rounding up or down still remains, but it might result in a per capita fairer system.
Would love to hear your thoughts, appreciate the great content :)
That was a lot of work. Thank you.
You could do a follow-up on seat allocation in the German parliament. It get's really messy due to a mix of proportional and local representation which means not only the allocation of seats between parties is hard, but the number of total seats changes depending on the election results.
yeah... "Überhangmandate" :D
@@m.h.6470 Und Ausgleichsmadate
Well the problem in Germany is, that they stuck to single member districts. If you joined up districts so you had 5 old districts in every new district and then elected 5 members from every district, then the representation error for each new district would be much go down. Therefore you would need to use far fewer members to make parliament proportional.
Then you could set aside a fixed number of extra seats to even it out...
and yes I am Dane in Germany shamelessly advocating the Danish system ;-D
yes I mean it kinda works and gets a somewhat fair allocation but the actual calculations are insane (as is the amount of representatives...)
@@terenzohugel2293 yeah, from all the ways voting systems can be broken ours is relatively harmless
Hi Matt,
I just watched this very entertaining Video and I'm from Germany! Over the last 15 years we had a very strong debate about how to allocate our seats in the Parliament proprtionally to voters. At this point we had 3 different aproaches all passed into law then later being anulled by the constitutional Court for not being fair enough. I thought you could weigh in and explain why its so difficult to figure it out fairly.
Here are some fun facts about the german system that make the maths so difficult:
- Germany is divided into 299 little parts and each of them elects one member 1st past the post (simple) we call that the "Die Erstimme"
- after that we take another 299 Seats and use them to rebalance the Parliament so that every party gets representation roughly equal to their amount of voters. We call that the "Zweitstimme".
- the complication to the second rule is that we do that rebalancing not for whole country at once but for each of the 16 federal states individually. this leads to the problem that the bigger party's get more first past the post seats than they should get even in the full 598 strong house after rebalancing. So tjose bigger partys just get these on top of the 598 which leads to us having more seats overall. we camm the Überhangmandate
- Now that the Parliament has more seats the Divisor changes (YAY!!!) which leads to all parties getting more seats we put the on top again we call "Zusatzmandate"
- After all of that the bigger parties still come out unfairly ahead so we put in even more seats this time only for parties that didn't get any "Überhangmandate" we call these extra seats "Ausgleichsmandate"
In the End we have between 603 (2002) and 736 (2021) but depending on the votes it could even go past 900
Have fun getting all this mess organized maybe you can find a solution. At the Time of me writing this we haven't had a constituionally valid Voting legislation for more then 15 years!
These considerations are mostly independent from the basic method used to turn percentages into number of seats. For that, we currently use Webster's method (i.e. using a divisor and "ordinary" rounding). Webster's method is also used a second time, to further split up each party's seats among the 16 states, according to the number of votes in each state. Also there's no state-level "rebalancing" as you described; the total number of seats for a party only uses the federal result. "Überhangmandate"+"Ausgleichmandate" actually just operate by increasing the size of parliament first, then doing an ordinary round of Webster's method; and the size of parliament was chosen do accommodate for all the direct mandates (in a somewhat complicated manner that does take individual states into account). Well that was how it worked until 2017 election, this time apparently 3 overhang mandates without compensation are allowed again.... weird compromise to reduce the size a little bit.
Anyways, ignoring the new 3 overhang, the procedure is basically
complicated mess of rules -->> determines -->> total number of seats
total number of seats and federal election results -->> use Webster -->> seats per party
seats for one party and votes for that party partitioned by state -->> use Webster (modified, respecting lower bounds for direct mandates) -->> seats of that party for each state
seats for one party in one state -->> hand out direct mandates, subtract to determine remaining number of seats -->> use party list in that state to fill the remaining seats
@@steffahn but aren't our complicated messes of rules still producing an essentially unconstitutional result.
Thats the whole point I was making was that I don't know a solution that would be compatible with our constituition. And all bill so far introduced can't find one either.
It almost sounds like the problem is that lawyers (who draft legislation and constitutions/basic laws) and judges don't usually understand math...
Unfortunately, when constitutional law conflicts with mathematical reality, the latter inevitably wins...
(That said, I'm a bit surprised that the German system - which already adds an indeterminate number of "extra" seats every time doesn't just keep on adding extra seats (by Land and overall) until each Land's share corresponds to the "correct" share within some presumably small allowable margin of error. I understand that there is a limit to the number of representatives that you can accommodate in the Bundestag, but unless a maximum number is specified in the Basic Law, requiring a US-style "least-deviation" approach of some kind, it seems to me that the Constitutional Court needs to brush up on their math skills...)
Grüße von Deutschland nach Deutschland.
Die Frage ist wohl, wie viele Jahre dauert es nach diesem System, bis jeder Bundesbürger einen Sitz bekommt? :D
@@tachzusamm Ich bin mir nicht sicher, ob das System wirklich ausschließt, dass das Parlament nicht auch größer werden könnte als die Gesamtbevölkerung Deutschlands.
Never did I imagine I would of my own free will sit out a 40-minute video about history, American electoral system and spreadsheets - yet here we are.
I mean, I know it’s the whole point, but you’re really funny in this video (more so). Great job!
16:38 The POP/DJ calculation is somehow very wrong here. This is a much larger difference between using D and DJ as the divisor than should be possible here, and it already sums to 43, which it shouldn't.
+
It should have been 23.5247 and 10.444 so rounds down correctly to 23 and 10
The very reason I came to comment:
ROUNDDOWN(24.8614,0) = 24, not the 23 shown. Similarly for 11.0375.
🍌🤔🤨🙄
@@HeinrichDixon Why did you skip over the description then? The correction was there an hour before your comment.
@@Xeridanus Huh, I wasn't aware that people routinely read video descriptions. I usually do if I'm looking for info (for example, if I thought the video contained a mistake), but otherwise never. Interesting!
Something to point out, the House of Representatives has had it's members capped for over 100 years. This means that no more errors can be created from increasing the size of the House, and therefore an error created by increasing the number of available seats is not going to happen at all.
Yes - and I also submit that when the framers of the constitution put in the bit about "no more than 1 rep per 30,000 population" they seem to have also intended to add a House seat for every 30,000 increase in total population. Perhaps this never happened as intended?? Obviously, the cap became necessary as the population grew (for example - based upon todays population 329,000,000 / 30,000 = 10,967 (yikes!) seats in the House). How the "1 rep per 30,000 in population" clause made it thru the drafts into the final Constitution is undoubtedly a story in itself, and I suspect that Washington (possibly Jefferson as well) were personally invested in keeping the clause intact. Thus the Washington veto of the Hamilton plan. If Washington was under the impression that the number of house seats in future were to be added using the 1 per 30k in population, his veto makes slightly more sense as most of these paradoxes and complexity may disappear.
The error still exists though. If your state would have received an extra rep if the house had 434 seats I don't think you would be very happy
Yep. Statehood to Puerto Rico and DC would actually _decrease_ the size of the House, since those two currently have non-voting members and those would be replaced with one or more of the 435 voting seats.
@@cr250rdr I will say there is still an attempt to follow the 1 per 30k Rule... Not only are there 435 Representatives at a Federal level, but there are also 5411 Representatives at a State Level... So 5,846 out of 10,967 is only half way there, but the disparity isn't as severe as it appears. Altho the biggest problematic states that are severely under the 30k requirement are the 4 highest population states, California, Texas, Florida, and New York who have a total of only 500 representatives against 111m pop...
Actually if you remove those states from the figures... 329 - 111 = 218, 5411 - 500 = 4911... 218m/4911 = 44,390 or if we also add in Federal 218m/5203 = 41898... Eh, we are still quite a ways off, but yea...
I can kinda understand why the seat limit, and why these big states want to limit how many state seats they offer too... I guess we just need to divide the big states too
Given our willingness to accept 3 significant figures in the percentages we represent the portion of the population with, we could just lock it at 1000 and solve all of these problems, unless a state happens to fall below .01% pop
Now, that would give California almost 1200 districts and I'm not sure the best way to handle that, but at least on paper, the number of representatives would not be part of the problem that needs to be solved.
OK, that "I'm feeling pretty good." at the very end got me. Made me laugh and almost choke on my toothpick.
Ow.
Solution is to set a fair divisor and rounding method, then let the total number of seats dynamically change.
If the number of seats grows beyond a preset limit, increase the divisor until the seats are below the maximum.
Sounds good except when you need to increase the divisor you end up at square one and someone is advantaged and someone else disadvantaged, and in a country like USA where for much of its history it had a huge population influx you would be forced into this situation after many census rounds.
You also hit issues when the number of seats goes even, because then you can have ties in the House when everyone shows up to vote and that unfairly favors one side or the other, something you generally really don't want. Yes the House doesn't have a tie-breaker at the moment, but the House has an odd number of reps which means theoretically the tie only happens when the yay side fails to get sufficient reps to vote or when the nay side is nominally bigger but doesn't utilize its full vote. Tie's at 100% of reps voting is another and entirely worse thing for the political process.
This can be solved with a tie-breaker position like the Vice President in the senate, but that is a new governmental position which needs to be appointed, and someone is going to get more political power from holding this position.
You will end up having to fight over divisors and points where one state or another loses representatives it had both absolutely and in proportion to others despite itself not changing.
@@NATIK001 "Sounds good except when you need to increase the divisor you end up at square one and someone is advantaged and someone else disadvantaged"
But if you can choose the total number of seats you can choose it such that those differences are as small as possible.
at the timing of 16:48, the rounddown() function seems to have the wrong column as rounddown(24.8614)=23.
I originally thought the same, but then noticed that when DJ value changed the relative POP/DJ changed accordingly and rounddown() correctly displayed the right value.
Point is, the whole POP/DJ column at the very beginning appears to have been divided by a complete different value of DJ (POP/DJ and POP/D values should have been pretty close, as DJ and D were almost identical). To be precise, 880 instead of 930 shown on screen.
Am I missing something here or is this entire problem due to an arbitrary total number of seats to be allocated? Surely the fairest system would be a fixed quota, normal rounding and having however many seats that adds up to?
Well, yes, that's a dependency of the problem. There are practical reasons to limit the number of representatives. Maybe less so in the information age however...
Absolutely, but it'd cause inflation issues. The more representatives there are, the less each one matters, but the less there are the more people that are represented by one vote (which is a negative). The goal is to ensure that each vote represents a correct amount of people without devaluing any singular vote.
@@TrueFlameslinger I don't remember the entire context here given I watched it a month ago, but my suggestion was based on precisely the opposite of what you state. I suggested that rather than work from a fixed number of seats down, it should be a fixed number of people represented by each representative adding up to however many seats that adds up to, there would have to be some amount of rounding given voting districts are per state but the only increase in the number of representatives would be correlated with population size which negates dilution of representation rather than create it as you suggest.
@@badgerfool1980 You could even adjust the number of people per seats for every election to make the total number of seats come to as close as possible to some desired value.
@@kiml42 True enough. Unless I'm missing something (which is far from unlikely) it seems a pretty obvious answer.
Hello Matt,
Thank you for the video.
You said no method can validate both fairness criteria, and that got me thinking of a question for you : here in Belgium, the seats for our house of representatives are apportioned to the different parties according to a method called the D'Hondt key (named from a constitutional lawyer) : the absolute results (number of votes) for each party are divided by increasing integer values (1, then 2, then 3, etc.), and the results of these divisions, called "electoral quotients" are noted. Then, the number of available seats are apportioned by decreasing value of electoral quotient.
Example : 5 seats to apportion ; party A gets 30 votes, party B gets 14 votes, party C gets 10 votes. Electoral quotients are :
A : 30 ; 15 ; 10 ; 7.5 ; 6 ; ...
B : 14 ; 7 ; 4.67 ; 3.625 ; ...
C : 10 , 5 ; 3.33 ; ...
Here, 5 seats are thus apportioned as : 30 (A) ; 15 (A) ; 14 (B) ; 10 (A) ; 10 (C).
I am no mathematician, hence my question : does that method break one of the fairness criteria and, if so, how ?
Best regards,
Antoine
if you look closely, it’s mathematically equivalent to the Jefferson Method. you’re simply decreasing the divisor and always rounding down until you get a whole number, so i would assume that there are circumstances that you would get the Alabama Paradox
you could try running the numbers he uses at 20:29 and you should theoretically get the same result as the JEFF method… might take a while though
Matt you were so proud of that United Shapes joke. And I would be too, it was Brilliant.
I would like to say that this video has used the word Vermont more times than I have ever heard used on TH-cam since I have started watching. I have lived within the state of Vermont all that time and would like to thank you Matt for the internet noise.
Vermont!
Is Vermont the new bamboo?
34:09 "Well the only way to compare the two would be to go back and analyze every single--"
Me, shaking head: You mad lad.
That was surprisingly interesting, kudos and thanks !
You are really killing it with the quality of your videos and humour these days!
Congrats! You just get better and better with each video you make! You should be damn proud of how far you have come! I just love seeing when someone who has been making videos for years, has gone from quick and dirty videos, to an excellent and professionally looking show!
I’m also so glad that with the improvements you have made to your skills, that you still haven’t lost what makes your videos great! Your ridiculous humour is on point! I genuinely laugh more often than maybe I should watching your videos!
Your videos make me excited to learn something weird/interesting/silly/etc… Without triggering my very traumatic memories and experiences with math(s) in high school. It’s such a shame that my experience was unimaginably terrible! I’m just so lucky to have been introduced to you, and many other infectiously passionate teachers on TH-cam! You have encouraged me to see the amazing utility, and ability for math(s) to help explain all the infinitely interesting problems that the world often presents us with! I have learned it isn’t as hard as I thought to have a go at solving questions I may have with a little math(s)!
Thank you!!! You don’t know just how important of a moment it was to me, when I realized I could start to understand and use math(s) to solve or just ponder potential solutions to problems! I’m unbelievably thankful to you for curing the mental and emotional damage I had from some pretty awful math(s) teachers. One teacher physically cornered me in the hallway after I had just gotten out of the hospital, due to some severe health issues I had at the time. He was turning red screaming at me for “always being sick on test day”, and other nonsense…He also would often call me “an idiot who just couldn’t get it, and I was just calling in sick to purposely miss his class”, and he should “just fail me regardless anyways, for missing a few exams…”. Yeah he had doctors notes explaining that I was in the hospital and still said those things to me… So yeah, apparently notes from the hospital were not good enough for him… So you may be able to identify where my “I suck at math(s)”, problem arose…
So thank you so much for changing my view of math(s) and my potential future abilities, if I continue to learn and work at it! I’m just so thankful that you have replaced that false mental perception I had, because of one terrible teacher, and a couple others who were not so great either!
Huntington-Hill can be described the same way you've described Jefferson, Adams, and Webster. Just like those, you vary and divisor and round. Just like those three round down, up, and linearly, Huntington-Hill rounds using geometric rounding.
This means, instead of rounding to the number it is closest to, round to the number which the ratio between the "actual" and "rounded" number is as close to 1 as possible. You can also consider that, for each pair of integers, there is a number between them that separates numbers that round up and down. For linear rounding, that number is always 0.5 above the lower number. For geometric rounding, that number is the square root of the two neighboring integers multiplied (aka the geometric mean). All of these, and the way you described it, give the same results and can be proven to do so.
Thanks, good to know!
Thank you for including those Alabama takes at the end.
Could you explain a little more of the math behind why you can't fulfill the quota rule without potentially causing an Alabama Paradox? Naively, it seems like a hybrid approach of assigning seats according to the minimum quota and then using the Huntington-Hill method to assign the remaining seats would work.
This video seems to misstate the Balinski-Young theorem. There *is* an apportionment method that avoids both the Alabama Paradox and quota violations. (In short: dole out seats according to critical divisors as you would in Jefferson's method, but skip over a state if it would exceed its upper quota. For more details, see theorem 2 of "A New Method for Congressional Apportionment" by Balinski and Young.)
Instead, the Balinski-Young theorem states that if a method follows the quota rule, it exhibits the *population paradox*: it's possible for state A to gain population and lose a seat at the same time that state B loses population and gains a seat.
Balinski-Young theorem seems to state that no method of apportionment can at the same time avoid violations of the quota rule, Alabama paradox and the population paradox. I'm not sure but there might be a method of apportionment that satisfies the quota rule and the population paradox but violates the Alabama paradox.
Huh, I've always heard it as just stating that no method avoids quota violations and the population paradox, and the proof I'm familiar with doesn't require assuming anything about the Alabama paradox.
@@jonahostroff I may be missing something but I don’t think Hamilton’s method suffers from either quota rule violations or the population paradox.
@@aDifferentJT Here's an example where Hamilton exhibits the population paradox, with three states and 10 seats:
First census: populations 1.45M, 3.4M, 5.15M, total 10M, so quotas are 1.45, 3.4, 5.15. Hamilton yields (2, 3, 5).
Second census: populations 1.47M, 3.38M, 4.65M, total 9.5M, so quotas are 1.55, 3.56, 4.89. Hamilton yields (1, 4, 5).
The first state gained population and lost a seat, while the second lost population and gained a seat.
@@jonahostroff nice, you just squashed Johnny boy with that simple counter example
Throughout this video I was waiting for a reference to the D'Hondt method, and it would have been interesting to have seen it mentioned (as it is used in many countries).
What I didn't know, and only discovered because this video prompted to read more about it, is that D'Hondt method is equivalent to the Jefferson method, although the two look different.
This video reminds me a bit of Arrow’s Theorem for voting (though admittedly it’s been many years since I actually looked at it).
On initially hearing it, I like his proposed buddy system for states rounding to half a representative because it could encourage more cooperation and understanding in this time where many people seem bound and determined to villainize the "other side" in their heads.
16:37 somehow the divisor Jeff isn't correct in this particular instance, those numbers are the result of dividing the populations by 880 rather than by 930. The rounding still ended up rounding as if the numbers had been divided by 930, just the numbers for =pop/DJ are using DJ=880.
Not trying to bash or hate, just pointing out in case anyone else was looking at those numbers wondering why 24.86 rounds to 23 and happens to look at the comments to see this.
also at 19:17 the POP/DA of all other States besides New Triangle goes back to the 930 value instead of up to the 960 value
bro something with the math at that point is completely off, the values of POP/DA go up after increasing DA from 930 to 950, except New Triangle. and i thought matt was the expert of Spreadsheets
Right. Getting a pro life state to agree with a pro choice state that wants taxpayer funded abortion up to and including after birth (see Virginia’s governor) is a great idea.
I dont know how to share space with people that can't tell you what a woman is.
I guess that's where overhang mandates actually make sense, just round everything and don't care about how many total seats you have at the end
Is this editing a product of Matt Parker taking on Steve Mould's advice to make things more flashy? Solid effort, man, I dig it!
Loved the Dustin from Smarter Everyday audio. LOL
This entire video is very interesting, but honestly the thing that's stuck in my head the most is "Squaryland"