It’s election day, and tomorrow there’s going to be a lot of complaints about the results, with good reason: there are many problems with American style democracy, ranging from the electoral college, to gerrymandering, to the presence of big money in politics, to the effective non-existence of third-party candidates. I’ll talk about those later.

This time, I’ll be more positive and point to one thing we get right in this country, and that’s apportionment. (And in a later post, I’ll talk about how we can use this to remedy many of the problems above)

Every 10 years, we have a census, and on the basis of those numbers, each state gets a certain number of Congresscritters. The problem of apportionment is deciding how many.

One problem is that we have, for rather stupid reasons, decided there will be exactly 435 Congresscritters. That causes problems I’ll talk about later, but for now we’ll take it as a starting point.

Any such limit necessarily requires us to proceed as follows. First, make an initial assignment. Then fiddle with the numbers until you’ve apportioned the correct numbers. For illustrative purposes, suppose you have two states, one with a population of 4000 and the other with a population of 16000. Since there are 4000 + 16000 = 20,000 persons all together, then state 1 should receive of the delegates, and state 2 should receive of the delegates.

That works…if the number of representatives is divisible by 5. But what if it isn’t? Then we might try to get as close as possible to the ideal ratio of for state 1 and for state 2. For example, suppose we have 8 representatives. We could begin give the first state 2 representatives and the second state 6.

That’s just common sense. The problem is that common sense doesn’t solve problems: it merely repeats a solution to a different problem. It’s great if the two problems are the same…not so great if the two problems are fundamentally different. In this case, the residents of the second state might object: Even though they have *four times* the population, they only have three times as many representatives. Surely this is unfair.

Common sense can’t resolve this issue. But mathematics can. We’ll adopt the following strategy:

- Begin with some initial apportionment, leaving some number of representative to be assigned to the states.
- Assign priority values to each of the states, which measure which state is “most deserving” of a representative.
- Give that state an additional representative.
- Recompute the priority values, and give out the next representative.
- Repeat until all representatives have been given out.

Again, suppose we only have two states with populations A and B. We’ll need to make an initial assignment. Rather than try and figure out what that initial assignment is (which will, in effect, require us to solve the apportionment problem!), we’ll start each state with its mandated one representative.

Now, we need to choose on a method for deciding which state deserves the next representative. Since, in theory, our Congresscritters are supposed to represent the people in their districts, then it seems we’ll want to base our decision on how representative they are.

To that end, consider three comparisons:

- and , the persons per representative in each of the two states.
- and , the representatives per person in each of the two states.
- and , the ratio of the size of the congressional delegations and the ratio of the populations.

In general, these won’t be equal. But this means we can frame the apportionment problem mathematically: assign the next representative in such a way as to reduce the inequality.

Here’s the general approach: First, choose one of the comparisons. For example, we might take the first (which corresponds to the size of the congressional districts in each of the two states). We’ll give each state one representative, then assign the remaining six by determining which state is “more deserving,” based on the goal of trying to make the congressional district sizes equal.

For example, consider our states above, with 4000 and 16,000 people, respectively. If we start each state with 1 representative, we can assign the next (third) representative to either of the two states:

- Currently, the representative of state 1 represents 4000 people, while the representative for state 2 represents 16,000 people. That’s a difference of 12,000 people per representative.
- If we give it to the first state, that state will have 2 representatives for 4000 people, so each representative will represent 2000 people. This
*increases* the disparity, to 14,000 people per representative.
- If we give it to the second state, that state will have 2 representatives for 16,000 people, so each representative will represent 8000 people. This
*decreases* the disparity, to 4000.

If our goal is to make the representation more nearly equal, we should give the third representative to the second state. This is a “no brainer case”, because one assignment causes the disaprity to decrease and the other causes it to increase.

What about the next representative?

- Currently, the representative of state 1 represents 4000 people, while the representatives (two of them) for state 2 represents 8000 people, a difference of 4000 people.
- If we give the next representative to state 1, then the two representatives will represent 2000 people,
*increasing* the difference to 6000.
- If we give the next representative to state 2, then the two representative will represent 5333 people (rounded),
*decreasing* the difference.

Again, this is a “no brainer case”: the second state should get the next representative.

The “no brainer” cases are easy to decide. But what if neither of these are true? If we follow the above, eventually we’ll get to a situation where the first state has 1 representative and the second state has 4. However, if we’re apportioning 8 representatives, this means we still have representatives to assign. So which state should get the next representative? At this point, we have a problem:

- In both states, the representatives represent 4000 persons.
- Giving the extra representative to the first state means there will be 2000 persons per representative,
*increasing* the disparity to 2000 persons.
- Giving the extra representative to the second state means there will be 3200 person per representative, which
*also* increases the disparity (to 800 persons)

Here *both* assignments cause the disparity to *increase*. Since we *must* assign a representative, we have to decide which state is more deserving *even though this will increase the disparity*. Again, since our goal is to minimize the disparity, it should be clear that we want to choose the assignment that causes a smaller increase. Thus, we’ll give the second state the next representative.

Continuing in this fashion, we can assign all eight representatives. We’ll leave, as an exercise for the reader, the actual assignments.

With a little algebra, we can compute a formula that gives us the priority values for each state. We can then solve the apportionment problem as above.

Unfortunately, we run into a problem. If we choose to minimize the difference between persons per representative, we get one rule for computing priority values. But if we choose to minimize the difference between representatives per person, we get a different rule. And if we choose to minimize the difference between the population and representative ratios, we get a third way to compute priority values.

One surprising way to get around this is to use the relative differences, expressed as a fraction of the smaller value. I’ll leave off the math again (it’s not particularly difficult, but it is a little messy, and a blog isn’t the best medium…)

The punchline is this: *all three relative differences are reduced by assigning the additional representative to the state with the higher priority value, when the priority values are computed by comparing * and .

And this is *exactly* what we do, in the United States, to compute the number of representatives each state receives.

Of all the features of American-style democracy, this method of apportionment is a shining example of how democracies *should* be run.