Why it’s mathematically impossible to share fair 

Yes, I turned every Patreon name into a fake state name. If you cannot find yours: I'll start a post and look them up as requested. patreon.com/standupmaths And do check out brilliant.org/standupmaths to learn how to do things like this!

as an Alabamian, I can confidently say our education system never forgave math for this and has actively scorned it ever since

The amount of joy on this man's face when he said "the united shapes" is enough for me.

2:40 Why is Circula so upset? You'd think he of all shapes would appreciate rounding.

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.

You're really good at explaining complex math concepts. Thanks for letting me play along.

The cameo by Destin-I mean Alabama-really made my day. He obviously had a ton of fun recording those bits!

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

“The united shapes” blindsided me like a freaking freight train. I should have seen it coming but I’m still giggling to myself

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

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!

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!

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!

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.

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.

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!

It's a happy coincidence that my brain seems to like the way Matt says the words "number" and "digit" 😄

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!

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.

😀😀😀 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!

Fascinating video! I'd love to see you build on this by talking about mathematical advantages/disadvantages of implementing something like the Wyoming Rule.

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.

But how many sides does an “Ore-gon” have?

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.

Recently discovered your channel - and love your stuff. Amazing content.

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.

This was super interesting to see historic real-world consequences of math.

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.

"I spent so long going through all this" *CGP Grey*: (coughs)

16:56 24.8614 rounds down to 23. Interesting.

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.

That moment of pride on your face at 2:11 when you said "The United Shapes" was priceless

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

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.

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

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

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.

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 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.

You do deserve all the cookies for the entertaining education that you work so hard to provide!

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?

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 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.

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.

The only channel that can make spreadsheets fun 😂

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 :)

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?

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.

22:37 Just realized you used Destin's voice for Alabama, that's great

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...

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.

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?

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.

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.

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?

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.

I’ve had this come up many times at work. Cheers

I'm embarrassed to admit how much I've enjoyed this video. This is the most nerdtastic thing ever!!!

can confirm that is exactly what Hamilton the musical is about

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.

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.

This video is incredibly helpful, thank you!

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.

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!

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 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 🤯

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.

The round down at 16:46 seems to be wrong for the first 2 rows (or possibly the pop/dj)

This is chaos and why'd you spend that much time doing this, above and beyond XD

OK, that "I'm feeling pretty good." at the very end got me. Made me laugh and almost choke on my toothpick. Ow.

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

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.

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.

That was a lot of work. Thank you.

at the timing of 16:48, the rounddown() function seems to have the wrong column as rounddown(24.8614)=23.

I would like to say that this video has used the word Vermont more times than I have ever heard used on RU-vid 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.

"Imagine you're Alabama in the 1880s" [immediately starts scream crying]

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.

Matt you were so proud of that United Shapes joke. And I would be too, it was Brilliant.

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.

I mean, I know it’s the whole point, but you’re really funny in this video (more so). Great job!

Should’ve opened with that fact, first veto was maths based, how Rad! Great vid as always Matt.

Matt: "Odds of the house getting smaller - very slim" German Bundestag: "Yes" Matt: "Odds of the house getting bigger - also very slim" German Bundestag: "Hold my beer"

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

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

I have a book "Fair Representation" by Balinski and Young. It finds the best method is the Webster method, where you round the exact quota to the nearest integer and adjust the divisor so you get the required number of seats. It is the easiest to understand; harmonic averages confuse people, and so do square roots. It also avoids all of the paradoxes: Alabama, population, and Oklahoma and is "near quota"; that is, it is not possible to transfer a seat from one state to another and make BOTH of them go nearer to quota. I believe that the US should apportion representatives using the Webster method (or method of major fractions).

I was thinking "why not do the Jefferson method, but cap the allocation so it never violates quota" and that is exactly a real thing, called a quota-capped divisor method, and specifically the Balinsky-Young quota method! There can still be issues with these methods too, however

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.

That was surprisingly interesting, kudos and thanks !

Loved the Dustin from Smarter Everyday audio. LOL

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.

At 4:59, I believe he said '93.2' when he meant to say '932', as that is what he puts on screen at the same time, and that matches the calculation done.

Thank you for including those Alabama takes at the end.

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).

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.

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 RU-vid! 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!

I went digging a little bit into the history of the different sizes of the House of Representatives because I was curious what year had the largest House (I didn't actually find an answer) And I think part of the reason that 1920 was significantly different, was that there simply wasn't a reapportionment after 1920 census. There was too much delay from rural politicians fearful of losing power due to urbanization (sounds familiar). They simply kept the apportionment from 1910. So, the population changed but not the representatives. And in 1929 Congress passed an amendment so that it was done automatically, so that it would never again occur that there was no reapportionment.