The proud nation of Examplestan has 20 members in its Congress. Each member of Congress represents one of its three states: Bigland, Middletown, and Smallton. Bigland has a population of 405 people, Middletown has 145, and Smallton just 81. How many representatives should each state get? Numbers are hard so let’s put that in a graph:

In total, there are 631 people in Examplestan, so each state should earn a representative for every 31.5 people on average. This is called the quota, and I’ll use the x-axis to show the number of quotas each state has:

Unfortunately, the states do not have an exact number of quotas. In fact, Bigland has 12.83 quotas of people, Middletown 4.60, and Smallton 2.57. We cannot divide representatives in two, so we’ll need to do some rounding. If we round these all to the nearest whole number, they all round up and we’d have 13 for Bigland, 5 for Middletown, and 3 for Smallton – 21 seats total. Since we only have 20 seats, we need a different solution.

It seems like the most fair way to resolve this is to round down the state with the smallest remainder and round the other two up. In this case, Smallton has the smallest remainder at 0.57 quotas and so will be rounded down to 2 seats. On the graph, I’m going to use a dotted line to show the rounding threshold we are using – past the line and we round up, below and we round down. In this case the line will be just past 57% of the way between the quota lines (since the remainder we round down is 0.57). We can see that Middletown is just barely above it while Smallton is just below:

Seems fair right? In isolation, this seems like a great solution. But cracks start to show as the population of this nation starts to change. At the next census, Bigland has grown by 30 people. Now with 661 people divvied between 20 seats, that amounts to a quota of about 33 people per Congressperson. I’ve made the next graph interactive to let you see visually how this effects the allocation of seats. To see what happens, drag and drop the top bar on the graph below to increase the population of Bigland. Compare the seat allocations when Bigland is around 405 and 435 people. As you drag, you increase the number of people living in Bigland. Since there are more people but Congress is the same size, the quota will increase, meaning the black lines for the x-axis will also increase.

Did you notice how the rounding threshold decreases? This is easiest to see by looking at the dotted line farthest to the left as you are dragging the top bar. That means the cost of the final seat is decreasing even when the total population increases. Weird.

There’s a reason the dotted line has to move though. Our old rounding threshold of 57% of a quota would only give away 19 seats with Bigland’s new population, so we must lower it to still give away 20 seats. In fact, this rounding threshold drops to 42%, but Smallton rounds up instead of Middletown. Consequently, Middletown loses a seat and Smallton gains one. Bigland, despite the extra work, gains nothing.

“Wait, that’s not fair!” protests the representative from Middletown who will soon be unemployed. “Why should I lose my seat to Smallton solely because Bigland grew?” Indeed, it does seem like strange behavior.

In the figure below, all of the state’s bars can be dragged around to change the times. Play around with it and see if you can spot any other interesting or unusual behavior. Can you find another case where increasing a state’s population causes other states to lose seats? Hint: the dotted line will move left as the state’s population increases, just like with this example.

Examplestan’s Congress sees this problem, but doesn’t know how to fix it. To make sure all the states are happy, they decide to just increase the size of Congress by 1 seat for Middletown to replace the one they lost. But Bigland insists the calculation be done again using 21 seats, as the law demands. Begrudgingly, Congress agrees, and allocates the 21 seats following the same procedure as before, finding the ideal number of seats and rounding up or down so exactly 21 seats are given away. Increase the seats in the figure below and watch what happens.

Congress hoped the extra seat would give back a seat to Middletown, which it does. But strangely, Bigland also earns a seat, and Smallton loses one!

“Wait, that’s not fair!” the Smallton representatives argue. “Why should our state lose representation when there are more to give away?” Indeed, it does seem like strange behavior.

Can you explain what’s happening here? Feel free to play with other seat counts as well and see if that helps your intuition.

The motion of the rounding threshold is once again an indicator of something amiss. Normally, increasing the number of seats reduces the rounding threshold. But going from 20 to 21 seats causes the rounding threshold to increase instead, making seats more expensive despite there being more of them. This is most obvious if you look at the far left dotted line when you bring the seat count up and down. Hopefully, you are starting to feel that in a fair system, the rounding threshold has to move in the same direction as the black lines.

In the midst of the bickering about the seat count, Congress receives a letter from the Examplestan colony of Tinyville demanding representation, lest they revolt. They have 45 people and should be entitled to a seat. Congress is persuaded by the argument and adds a seat for them. Then they recalculate the apportionment of seats once more to arrive at this:

Somehow, despite Tinyville taking the one new seat, Smallton has gained a seat at the expense of Middletown. At this point, it isn’t even surprising. The Benevolent Dictator of Examplestan realizes that the states will never agree on the allocation, so they override the claims of injustice and finalize the result.

Unfortunately, Examplestan got very unlucky in how the numbers turned out. These strange behaviors aren’t that common, but they hint that there may be a subtle unfairness inherent to our methods that we overlooked. So what did the nation do wrong?

At any rate, it’s just an example so it isn’t that big of a deal. It’s not like we use this to determine the fate of real nations.

The Fate of Real Nations

Every ten years after the US Census is conducted, Congress must determine how many representatives in the House each state gets. The Constitution dictates that the number of House seats is proportional to the population of the state, but leaves Congress to figure out the exact method of apportionment. You can see that this is the exact same problem that Examplestan faced. Many other nations have similar methods for divvying up legislative seats among states and provinces.

Control of Congress has always been one of the most contested issues in American history. The contest between big states and small states led to Congress bifurcating into the Senate and House. The conflict between slave and free states, which we eventually fought a war over, was assuaged when the union was formed with the infamous 3/5ths compromise. Apportionment was also the target of the first veto in US history and a Supreme Court case. Needless to say, the pathologies we examined in the above examples might be a bit of a problem in a highly charged political arena.

Across the Atlantic, Western European democracies elect representatives using proportional representation. That means a party with 20% of the votes gets 20% of the seats. Once again, trying to allocate seats proportionally requires solving this apportionment problem. Since the party or parties with the majority gets to pick the Prime Minister in the parliamentary governments, the choice of apportionment can and does determine who controls the nation.

Generally, we have to deal with apportionment whenever we want to allocate some resources (in this case, House seats) proportionally to some value (population or party votes) and that our resource is indivisible (you can cut a cake in half, but not a House seat).^[1] The solution must require rounding, which is what caused all our problems in the last section. While not limited to politics, the apportionment problem most commonly occurs in the political arena, and so we want a solution that is both fair and politically feasible.

So, how can we assign House seats fairly?

Alexander Hamilton

When not busy writing, running the national bank, or rapping in a Broadway musical, Alexander Hamilton was trying to figure out how to distribute House seats for the nation. Hamilton’s solution, also known as the Method of Largest Remainders, is exactly what Examplestan did when allocating the seats amongst its states.Here are the steps of the method:

1. Take the national population and divide it by the number of representatives to get the quota.

2. For each state, divide its population by the quota to get how many seats each is owed.

3. Round up the number of seats owed for the states closest to rounding up and round down everyone else. The number of states rounded up should be set so that the correct number of seats are awarded.

This is generally considered easy to understand and seems like it should be fair. But as we saw in the intro, this method can easily lead to lots of unfair outcomes. In fact, each of the examples of unfairness from above corresponds to a well-known paradox of the system.

First is the Population Paradox, where increasing the population of one state can cause another state to gain representatives. We saw this paradox when Bigland growing by 30 people caused Smallton to gain a seat, despite them not growing.

Second, increasing the size of the house can also cause states to lose representatives, like what happened when Examplestan increased the number of seats from 20 to 21 and Smallton lost a seat as a result. This is also sometimes known as the Alabama Paradox because it was first discovered when investigating an apportionment that took away seats from Alabama when the House expanded.

Third, when a new state joins the country, it is possible that another state gains or loses representatives as a result. This is known as the New State Paradox. We saw this in the intro when Tinyville decided to demand representation.

Betrayal on the Shining City on the Hill

The 1791 apportionment Congress made using this method was vetoed by Washington, the first veto ever in the United States. You may think it was because of this paradoxical behavior, but in fact these paradoxes were not known at the time. So why prevent it? His veto message is short and to the point:

Gentlemen of the House of Representatives:

I have maturely considered the act passed by the two Houses entitled "An act for an apportionment of Representatives among the several States according to the first enumeration, " and I return it to your House, wherein it originated, with the following objections:

First. The Constitution has prescribed that Representatives shall be apportioned among the several States according to their respective numbers, and there is no one proportion or divisor which, applied to the respective numbers of the States, will yield the number and allotment of Representatives proposed by the bill.

Second. The Constitution has also provided that the number of Representatives shall not exceed 1 for every 30,000, which restriction is by the context and by fair and obvious construction to be applied to the separate and respective numbers of the States; and the bill has allotted to eight of the States more than 1 for every 30,000.

George Washington.

Let’s break this down. Washington didn’t look at the calculations that led to Congress’ apportionment, just the final result. He looked specifically at how many people per representative there were for each state. To show this in the graph, I break down the red bar representing seats into as many boxes as seats the state was allocated. So Bigland has 13 boxes, Middletown 4, and Smallton 3. The width of the boxes tell us how many people each state has for a single seat. Take a look:

This does look a bit unfair now! Middletown has over 9 more people per seat than Smallton, for an impressive 26% error. Something similar happened with Hamilton’s apportionment – some states had way fewer people per seat than others, and Washington found this unacceptable. Some states even dipped below 30,000 people per seat, which is a minimum written in the Constitution, so Washington thought that this bill might also be unconstitutional.

But there was another force at play here. Washington decided to veto only after a contentious debate within his cabinet. While Hamilton was one member of the cabinet, there was another who had his own preferred method, one that could benefit the large Southern states he and Washington came from.

Thomas Jefferson Doing Math

Washington didn’t like that the proportions were different for each state and could dip below the quota. Thomas Jefferson had his own method that attempted to solve these problems. The key idea is that rather than changing the rounding threshold to allocate the right number of seats like in Hamilton’s method, Jefferson’s method adjusts the quota up and down until the right number are allocated.^[2] Jefferson also thought rounding up unfairly overrepresented states and so always rounded down. This was the proposal Jefferson made, and what was ultimately used for the 1790s. Let me break down Jefferson’s method step by step:

1. Pick some quota that is needed per seat.

2. Divide each state’s population by the quota and round down to the nearest whole number to get the number of seats owed.

3. Total the seats awarded to each state. If too few, try again with a smaller quota, and if too many try again with a larger quota.

As you can see from step 3, there is some guess and check involved in this method. There is a more complicated way to define this method that removes the guesswork, but I’ll save the details for later. In this figure, adjust the quota up or down using the slider to try to get 20 seats allocated. The total number of seats currently allocated is displayed above the slider to help you.

After playing with that slider for a bit, do you think this method will suffer the same paradoxes as Hamilton’s method?

Washington ultimately accepted this solution, but it looks nearly just as bad as before. You should find that setting the quota between 29 and 31 will correctly allocate exactly 20 seats. There is still a 9 person gap between the number of people per seat of the worst and best off state (though the relative error is somewhat better). What changed are the winners and losers. In Hamilton’s method Smallton had an advantage, though generally the advantage is random. In Jefferson’s method, Smallton got the worst deal, and this is because Jefferson’s method is consistently biased in favor of large states.

However, we found that Hamilton’s method was unfair because of all its paradoxical behavior. So even though Jefferson’s method is biased, can it avoid the paradoxical behavior?

Testing for Paradoxes

We identified three unfair cases with Hamilton’s method in our earlier examples. Let’s check them one at a time and see if Jefferson’s method suffers from them. First, can increasing the population from one state cause a different state to gain seats? Try manipulating the times here and see if you can either find an example where it happens or convince yourself it’s impossible. The quota will automatically adjust so you are always allocating 20 seats.

Notice how when you are dragging the times around, sometimes the quota is adjusting and sometimes it isn’t? Try to see why this is occurring, since building an intuition will help you understand the method.

The answer is that no, a state cannot gain seats when another grows. Increasing a state’s population can only cause the quota to increase, since we have to pick a bigger quota to prevent giving away too many seats when a state tries to cross the quota line. And increasing the quota will never give seats to other states. Don’t worry if that doesn’t make sense – we are going to look at these methods with a different perspective soon that provides another justification for why this cannot happen.

Next up, can increasing the number of seats cause a state to lose seats? Play around with the seat count here:

Hopefully you can convince yourself that this method is immune to this one as well for similar reasons. I’m not going to provide an example for the New States Paradox, but I hope you are seeing the pattern here: Jefferson’s Method solves all the problems we had with Hamilton’s.

Computing the Results

In the above explanation of Jefferson’s method, there is a guess-and-check process for finding the right apportionment. There is a more systematic method for computing the results that doesn’t require any guessing. The basic idea relies on the fact that we can compute where we need to move the quota slider for someone to gain or lose a seat. We can repeatedly adjust the quota to these thresholds to increase or decrease the seats until we have allocated the right amount.

Let’s use an example to demonstrate. We are going to start with this allocation. We are going to increase the seat count from 20 to 21 and see who gets the new seat.

When we increase the seat count, either Bigland will get their 15th seat, Middletown their 5th, or Smallton their 3rd. Let’s start with Bigland. What does the quota need to be for them to gain a 15th seat? Well, we would need to be able to fit 15 boxes inside their population bar that are quota sized or smaller, which means it happens when the quota is no more than $1/15$ of their population, which is $\frac{435}{15} = 29$. If the quota is any higher than that, Bigland will only have 14 seats. Likewise, we can find the quotas for Middletown and Smallton to be $\frac{146}{5} = 29.2$ and $\frac{81}{3} = 27$.

We are only giving away one seat, so we want to keep the quota as high as possible. The highest quota that gives away a seat is 29.2. Adjusting the quota to 29.2, we get the following distribution, with Middletown’s populations right on the quota line.

Notice how the the size of the boxes for Midland is $29.2$, the quota we just calculated? That’s not a coincidence! We are basically calculating what the box sizes would be if we gave a new seat to each state, then picking the state whose new box size would be the largest. As a consequence of this, Jefferson’s method finds the allocation that maximizes the minimum box size. ^[3] This value, the new quota, is called the priority of the state, with the general formula of

$$\text{priority} = \frac{\text{population}}{\text{seats} + 1}.$$

If you always give the state with the highest priority the next seat, you get the allocation Jefferson’s method would have given. You can start with every state having no seats and giving them away one at a time to get the right allocation, and without any of the guesswork that was involved before. Here is the new process explicitly:

1. Start with no seats assigned.
2. Find the priority for each state to earn its first seat.
3. Give the state with the highest priority a seat, then recalculate its priority.
4. Repeat the previous step until all seats are given away.

This technique also lets us see why the method is immune to Hamilton’s paradoxes. Increasing the population of one state affects their priority but not any of the others, so the other states can’t exchange seats between them. For similar reasons, new states won’t cause any exchanges either. And finally, increasing the number of seats doesn’t affect the priority calculations (we just have to compute them one extra time), so states still receive the original seats in the same order before the new seats are handed out.

A New Kind of Paradox

Jefferson’s method might have avoided all the paradoxes of Hamilton’s method, but it has its own strange behavior that needs accounting for. Take a look at this new allocation, this time with 4 states and 22 seats. Bigland steals the final seat from Middletown by a single person.

Imagine for a second that the seats were divisible. Then we could simply give away the seats in a way that was exactly proportional, so a state with 20% of the population would get 20% of the seats, which in this case means that a state earns a seat for every 32.14 people (1/22nd of the national population). Calculating for the states, we find that Bigland has earned 13.57 seats, Middletown 4.51, Smallton 2.52, and Tinyville 1.40. These are, in fact, the numbers we round with when using Hamilton’s method. We can see a visual of this here:

Wait one second though. Bigland would ideally earn 13.57 seats. Since we can’t split up seats, we can’t give them exactly this amount, but surely we should give them either 13 or 14 seats as a closest approximation. Yet Jefferson’s method gives them 15! This rule, that the number of seats a state can earn is no more or less than one away from the amount they would get if seats were divisible, is called the Quota Rule, and Jefferson’s method breaks it flagrantly.

Why did this happen? Well, the Quota Rule is based on the claim that the quota should be 32.14 people, which it would be if seats were allocated independent of state lines. But if we set the quota to that value, we don’t give away enough seats, which we saw when exploring Hamilton’s method. This means the only fair way to give away the remaining seats is to conclude that the states actually don’t need that many people to earn a seat. Instead, if we lower the quota to just over 29 people, we give away the right number of seats.

This makes clear the philosophical difference between Hamilton’s and Jefferson’s methods. Hamilton thinks its fair when everyone is as close to the ideal quota as possible. Jefferson, on the other hand, doesn’t think there is an ideal quota, and that it’s fair as long as everyone is using the same quota. Both views have their merits.

Even though Jefferson doesn’t fully keep with the Quota Rule, it satisfied part of it. It only breaks the Quota Rule by sometimes giving states more representatives then they deserve, but never fewer then the minimum the Quota Rule demands. ^[4] Giving a state less representation then they deserve seems far more egregious then giving them too many, so this at least seems reasonable. It will also only break the Quota Rule by at most one seat, so no state could earn two more seats then it deserved.

But maybe we can find a method that is the best of both worlds? Could we find a way to keep in the Quota Rule without succumbing to the paradoxes that plagued Hamilton’s method?

A Web of Solutions

I’m going to talk a bit more about the history of apportionment in the United States. All of this information comes from the book Fair Representation: Meeting the Ideal of One Man, One Vote by Balinski and Young. Page numbers for specific facts are in parentheses.

Washington’s veto was a death blow for Hamilton’s method, at least for a time. He had basically said the method was unconstitutional, and nobody wanted to say Washington was wrong.^[5] So Jefferson’s method became a regular method in the United States. However, unlike today there was no fixed House size: Congress chose the quota for Jefferson’s method, and however many representatives it calculated became the size of the House. This state of affairs became a festering ground for political scheming. Setting the quota to different values would make some states slightly over- or under-represented, so congresspeople would painstakingly tweak the numbers by hand until they found a solution that benefited their coalition. As a most egregious example, Congress debated 10 different quotas for apportionment in 1832… on the same day (p. 25)!

The large state bias of Jefferson’s method had become apparent, and some experimentation was done with Hamilton-like methods. But Congress rejected Hamilton’s method for the final time when an example of the Alabama Paradox appeared from the results of the 1900 Census, this time involving Maine. Maine would earn 3 seats in the House if the number of representatives was between 350 and 382, 4 seats if between 383 and 385, 3 seats again with exactly 386 representatives, 4 seats again for 387 or 388, 3 seats for 389 or 390, and finally 4 seats for over 391 representatives (p. 40). If this sounds ridiculous and confusing, Congress agreed. The injustice of the method had become too blatant to ignore.

Fortunately, during this time Americans had found new methods that were very similar to Jefferson’s but without its strong favoritism toward the big states. One method that sprung up is Adam’s method, from founder John Adams. It is the exactly like Jefferson’s method except, after dividing the populations by the quotas, we round up instead of down. It consequently has the opposite properties of Jefferson’s. It tends to bias in favor of small states at the expense of large ones. It will never break the quota rule by giving states too many seats, but it can break it by giving big states fewer seats than deserved. And it will always produce the apportionment that minimizes the maximum district size (while Jefferson’s maximizes the minimum district size). Replacing the large-state bias for a small-state bias doesn’t seem like the correct solution though, and so this method isn’t much more favorable than Jefferson’s.

Enter Daniel Webster, a well-renown orator and polarizing political figure that was part of a trio that dominated US politics in the 1800s. Unlike Jefferson, Webster didn’t think rounding up seats based on the quotients was inherently a bad thing, so he proposed a simple tweak: simply round the quotients to the nearest whole number, rather than always down like Jefferson or always up like Adams. This, he argued, was a much fairer method and might bring an end to the dominance of the larger states.

The best way to understand this method is by playing around, so below is an interactive figure for Webster’s method. Unlike Jefferson’s method, we will round up once a red bar passes half a quota, which is indicated by the dotted lines. Try switching between the different apportionment methods to see how they differ. You can also adjust the populations to see how that affects things.

From playing around with Webster, do you think it solves the problems that Jefferson’s method has? Can it still violate the Quota Rule?

The Impossible Solution

These three methods are all a kind of apportionment method known as divisor methods. This is any method that involves dividing each state population by a quota (the divisor) then rounding using a consistent rounding rule. These methods have some useful properties:

• No divisor method suffers from the Population Paradox
• No divisor method suffers from the Alabama Paradox
• The divisor methods are the only kind of method that can avoid these paradoxes (p. 70)^[6]

This last point is very interesting, a fact proved by Balinski and Young in the aforementioned book “Fair Representation.” It means we can limit our search for a fair method to simply choosing the fairest rounding rule. Anything else (such as methods more like Hamilton’s) will inevitably suffer all the paradoxes that we believe make methods inherently unfair.

That wasn’t the only fact about divisor methods they proved though. They in fact made another important discovery:

All divisor methods must violate the Quota Rule in some cases

This seems to make the opposite point, claiming that the divisor methods will have some unavoidable unfairness in them. Combining these facts yields an impossibility theorem, the Balinski-Young Theorem:

There is no apportionment method which satisfies the Quota rule and avoids the population and Alabama paradoxes.

This doesn’t mean we can’t minimize how often the Quota Rule is violated. In fact, seeing how often this rule is violated will be a good way for us to measure how fair the various divisor rules are. But we will never be able to get rid of all violations unless you are willing to reintroduce some of the paradoxes we had before.

As a consequence of this, politicians debating apportionment are going to have to agree on an method, where any method they come up with must be unfair by some metric and the result may tip the balance of power. I’m sure they’ll come to a reasonable conclusion.

1920: The Decade Without an Apportionment

Despite having a constitutionally-imposed obligation to reapportion after every Census, Congress was unable to do so in 1920 when they deadlocked over which apportionment method to use and what the house size should be. Until then, it was common to increase the size of the house whenever a reapportionment would otherwise take seats away from a state. But reportedly, the capitol building was getting full and there was opposition toward further increases, while population movement made several rural states set to lose some seats. Unwilling to lose their power (or their jobs), these representatives got their party to kill all reapportionment bills that were brought forth (ironically, the bills passed in the House and were killed in the Senate instead, even though senators were unaffected by reapportionments). They also stoked the debate about the merits of the different methods proposed (p. 51).

Mathematicians now found themselves with a math issue making national headlines, which led to a lot of investigation into the nature of these methods. The goal was to see if the most fair one could be determined once and for all. One of those people was Joseph Hill, the chief of the US Census Bureau, who came up with a new divisor method he could show mathematically to be fair.

Huntington and Hill: The Relative Difference

We need to figure out what the best apportionment method is. But it is difficult to define what the “best apportionment” even means. How can we tell if one apportionment is better than another? Hill’s idea was to ask a simpler question: given some apportionment, is it possible to transfer a seat from one state to another to make the states more equal? If so, he reasoned, doing the swap results in a better apportionment. Continue swapping until we run out of options and we have the best solution.^[7]

However, there are multiple ways to measure how equal two states are. I find the most intuitive measure to be the difference in their district sizes, or the number of people per representative. In this case, Hill’s criterion would be that if giving a seat from state 1 to state 2 reduced the difference between the size of their districts, then we should do so.

Arguably, a better method for measuring a state’s representation is to look at how much a single voter can shift the vote percentage in a single district, which is the reciprocal of district size. We can think of this as a kind of measure of “voting power,” with a higher value meaning voters have more personal say over who their representative is.

These seem like they are measuring the same thing. After all, if you know one you can compute the other by swapping the numerator and denominator. Unfortunately, they produce different solutions for the best apportionment. The former measure yields an apportionment method known as Dean’s method, which I won’t be covering here (though Montana thought it was so good they went to the Supreme Court to argue that we must use this method), while the latter measure yields Webster’s method.

Hill wasn’t satisfied with either of these solutions though, since they all involve absolute differences. This means that the difference between a district size of 100 and 110 is the same as the difference between 10,000 and 10,010. Hill thought the former case is much worse, since there is a 10% difference between the values, where in the second case the change is 0.1%. In other words, Hill cared more about the relative difference between values instead of the absolute difference.

Unlike before, relative differences in district size and voting power do produce the same solution, but it is neither Dean’s nor Webster’s method. In fact, Hill looked at a bunch of different measures of inequality, and when using the relative differences they all converged upon the same method. The math professor Edward Huntington was shown the method, corrected a small error, and took half the credit, which is why we now call this the Huntington-Hill method.

This method is a divisor method, so it works exactly the same as Jefferson and Webster but with a different rounding threshold. It’s a bit weirder too, since the rounding threshold is different for each seat. We round from 1 to 2 seats if the state has at least 1.41 quotas, yet we earn a 3rd seat after the number of quotas reaches around 2.45, and our 4th seat at 3.46. The general rule is that if a state is between $n$ and $n+1$ quotas, we round up if the number of quotas is at least $\sqrt{n(n+1)}$ and round down otherwise. Since this is $0$ when $n=0$, this means any state with at least 1 person is entitled to at least 1 seat (so we never round down to 0).

If you’re confused why there’s a square root all of a sudden, that’s totally normal. My investigation into this is how this entire article came to be in the first place. Unfortunately, I don’t have an intuitive explanation for it. I can show it more formally, but this involves going into the mathematical weeds so I’ve left a proof in the appendix. I encourage you to read it if you are interested, even if you can’t follow every step.

Using our priority formulation of the divisor methods, we compute a state’s priority in Huntington-Hill as

$$\text{priority} = \frac{\text{population}}{\sqrt{\text{seats}\times(\text{seats}+1)}}.$$

In practice, however, we generally compute the squares of the priority instead, which removes the square root while keeping the ordering the same. When a state has zero seats we have a division by zero, which we treat as infinite priority.

Aftermath

The fact that Huntington-Hill minimizes relative differences is a persuasive argument, though the other method that has a claim to fairness was Webster’s. Balinski and Young back this up with several arguments and some statistical analysis showing that Webster’s method is least likely to break the Quota Rule, and would only do so in the US apportionment once every 16,000 years, on average (p. 81). Huntington-Hill, by contrast, is five times worse and will break it “only” every 3,200 years. Balinski and Young also can prove through various arguments that no other method can beat Webster’s. But alas, their book was not written in the 1930s and 40s when this debate was raging, and at the time it was Huntington who was making the persuasive arguments. Committees and hearings were held, reports were written, and many came out in favor of the Huntington-Hill method. Nevertheless, prominent figures were also making strong arguments saying that Webster’s was best after all. This being politics, the debate devolved into petty name calling and heated op-eds in the newspapers.

Below you can switch between the Webster and Huntington-Hill methods. Which one seems more fair to you? You’ll find it very hard to see examples where they even disagree.

The Webster and Huntington-Hill methods agreed in their apportionment based on the 1930 Census, and Congress agreed to set that result as the new apportionment. They also passed a bill saying that if they didn’t decide an apportionment in time again, the President should use the same method they did the previous decade to compute it, which would prevent the deadlock of 1920 from occurring again. Now all they had to do was cut through the mathematical debate and pick between Webster and Huntington-Hill once and for all.

Congress pondered all the mathematical arguments and was able to come to a bipartisan decision on the merits. Just kidding! In 1940 the Huntington-Hill method gave one more seat to a blue state at the expense of a red one, so a bill setting that method as the way to apportion the House passed on a party-line vote. Congress has never changed the method since, and apportionment has turned from a heated issue of importance every decade into a routine part of census procedures. The House was also set at it’s current size of 435 and has never grown since, despite a large growth in the national population.

Playground

I hope you had fun with the interactive diagrams! If you want to play around with the diagrams some more, here is the final version with all of the interactive elements turned on:

Conclusion and Further Reading

This article focused on apportionment methods for allocating seats to different states. But it is also used in Europe to allocate seats to parties. There are different trade-offs in that case which I won’t be covering in depth here. When used to allocate seats to parties, Jefferson’s method is known as the D’Hondt method and it has some desirable properties for apportioning seats to political parties based on votes, most prominently that it is the only divisor method to guarantee a party with a majority of votes will get the majority of seats. But this article is long enough, so I will leave it there for now. The main purpose of this article was to use the interactive diagrams to build an intuitive understanding of the divisor methods and why we use them over the largest remainder methods. Further discussion of which divisor method is best will have to wait for another time.

Most of these interesting facts, both mathematical and historical, are in Balinski and Young’s book “Fair Representation” and I recommend reading it if you were interested in what you read today. The Wikipedia articles on the various apportionment methods and paradoxes are also fairly good, and I recommend specifically this Wikipedia article on the mathematics of apportionment and this writeup from the American Mathematical Society.

While the diagrams here were meant for showing how an apportionment changes as the populations change, it isn’t as effective for understanding how the methods of apportionment differ. For that, I recommend this excellent interactive visual which shows you the apportionment of every different method at the same time. I almost didn’t finish this project after I discovered this; it is a truly wonderful visualization and I highly recommend playing around with it a bit if you found this article interesting.

The interactive diagrams were also new to me and quite a bit of work to get right. I hope you found them useful. I’d love to hear your thoughts on how they worked for you. I haven’t gotten around to making a comment section for this blog, so you’ll have to email me at ryan@absentmindedandroid.com. If you found this article through my newsletter, you can reply to the newsletter email and that should get routed to my inbox as well. If you want to subscribe to my newsletter so you get an email when I make a new post, the sign-up form is below the appendix and the footnotes. Thank you for reading!

Appendix: Math for Huntington-Hill Method

Here I’m going to prove why the Huntington-Hill method is the one you get when you cannot swap a seat to another state without increase the relative difference between their district sizes. The strange formula with square roots that we use to allocate House seats was so strange to me that it prompted my initial investigation into apportionment, and hence this entire article is only written because of it. I hope if you feel the same way that this article will help make it clear how we ended up with this apportionment formula. This argument is based off Huntington’s original proof (page 89).

Before we start though, I want to offer a piece of intuition on why the square roots show up. The square root used as the rounding threshold is the geometric mean of $n$ and $n+1$, which is a kind of average. Webster’s method, by contrast, rounds at $n + 1/2$, which is the arithmetic mean of $n$ and $n+1$. While Webster’s threshold is the middle of a linear scale, Hill’s threshold is in the middle of a multiplicative scale (i.e. a log scale), which is exactly what you want when considering relative differences.

Now, let’s prove this a bit more formally. Suppose we have two states with populations $p_1$ and $p_2$ with $n_1$ and $n_2$ seats apportioned to them respectively. Let us further suppose that the first state’s district size is larger than that of the second state, so $p_1/n_1 > p_2/n_2$.

Suppose also that this apportionment satisfies Hill’s criterion: no transfer of seats between these two states can improve the relative difference of their district sizes. For the purposes of finding the relative difference, we will always divide by the maximum value, though similar arguments work for other options like the minimum or average. If we transfer a seat from the second state to the first, Hill’s criterion implies the following inequality:

$$\cfrac{\cfrac{p_1}{n_1} - \cfrac{p_2}{n_2}}{\cfrac{p_1}{n_1}} < \cfrac{\left|\cfrac{p_1}{n_1+1} - \cfrac{p_2}{n_2-1}\right|}{\text{max}\left(\cfrac{p_1}{n_1+1}, \cfrac{p_2}{n_2-1}\right)}.$$

Note that we can drop the absolute value and maximum on the left side since we know that the first state’s districts are larger. We are taking away a seat from the state with small districts and giving it to the one with big districts, so their district sizes should approach each other. But by Hill’s criterion, their sizes have to be further apart than before, which implies that the district sizes “pass” each other on the number line and the first state now has smaller districts than the second. Let’s prove that formally. We want to show that $p_1/(n_1+1) < p_2/(n_2-1)$. Suppose otherwise, which lets us simplify the inequality as follows:

$$\begin{align*} \cfrac{\cfrac{p_1}{n_1} - \cfrac{p_2}{n_2}}{\cfrac{p_1}{n_1}} &< \cfrac{\cfrac{p_1}{n_1+1} - \cfrac{p_2}{n_2-1}}{\cfrac{p_1}{n_1+1}} \\ \\ 1 - \cfrac{\cfrac{p_2}{n_2}}{\cfrac{p_1}{n_1}} &< 1 - \cfrac{\cfrac{p_2}{n_2-1}}{\cfrac{p_1}{n_1+1}} \\ \\ \frac{n_1}{n_2} &> \frac{n_1+1}{n_2-1} \end{align*}$$

Since in the right fraction the numerator is larger and the denominator is smaller than the fraction on the left, its value must be larger. Hence, the final inequality must be false, giving us a contradiction. Now that we know the second state has the larger districts, we may instead simplify like so:

$$\begin{align*} \cfrac{\cfrac{p_1}{n_1} - \cfrac{p_2}{n_2}}{\cfrac{p_1}{n_1}} &< \cfrac{\cfrac{p_2}{n_2-1} - \cfrac{p_1}{n_1+1}}{\cfrac{p_2}{n_2-1}} \\ \\ 1 - \cfrac{\cfrac{p_2}{n_2}}{\cfrac{p_1}{n_1}} &< 1 - \cfrac{\cfrac{p_1}{n_1+1}}{\cfrac{p_2}{n_2-1}} \\ \\ \frac{p_2^2}{n_2(n_2-1)} &> \frac{p_1^2}{n_1(n_1+1)} \\ \\ \frac{p_2}{\sqrt{n_2(n_2-1)}} &> \frac{p_1}{\sqrt{n_1(n_1+1)}} \end{align*}$$

That final inequality is equivalent to saying that the second state’s priority for its $n_2^{th}$ seat must be higher than the first state’s priority for its $(n_1+1)^{th}$ seat in the Huntington-Hill method. In other words, Huntington-Hill will never give us the apportionment with swapped seats where the districts were further apart, because the second state has higher priority to get their last seat than the first state has to get an extra seat. The Huntington-Hill method is the only method that can make this guarantee. Therefore, it is the only method that satisfies Hill’s criterion.

Running the same argument in reverse proves that Huntington-Hill actually satisfies Hill’s criterion. That proof is provided in the linked paper and is effectively the same logic in reverse. I find this proof more insightful though because it shows where the squares come from without presupposing them to be needed, and I hope you found it interesting too.

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Footnotes

1. There is a subtle third requirement here: we cannot compensate for an unequal distribution, such as by paying money for the extra seats. We don’t want to let people pay for House seats for obvious reasons. ↩︎

2. Congress would instead increase the number of seats in the House if the quota dropped below 30,000 people since that was required by the Constitution. ↩︎

3. In practice, this means minimizing how overrepresented a state can be. It pays no heed to underrepresentation though, so Jefferson’s method waits until small states are very underrepresented before giving them a new seat. This is why it is biased toward large states. ↩︎

4. To briefly explain why: the sum of the seats each state earns proportionally (including the decimals) equals the total number of seats. Since Jefferson’s method rounds down, using the ideal quota cannot yield too many seats. Hence, we must use a lower quota and thus cannot take away seats the state has earned according to the Quota Rule. ↩︎

5. This was before the Supreme Court gave itself the power to strike down unconstitutional laws. Washington saw part of the duty of the President as making that constitutionality judgment, and no one could claim Washington was wrong. ↩︎

6. There are some technical conditions that go along with this, like assuming that the only valid methods are not biased toward particular states and give the exact solution in cases where an exactly proportional solution is possible (i.e. no rounding). So while it is technically possible to get around this, any method that does will be have to break even more important rules for fairness. ↩︎

7. Technically speaking this only guarantees a local maximum, but I believe there is some proof that there is only one maximum, which would guarantee this is also the global maximum. ↩︎