Congressional Apportionment I: Observations

2020 June 26
  1. Part I: Observations
  2. Part II: Theory (unfinished)

With the US Senate and electoral college heavily biased against states with larger populations, there sometimes arises the misconception that the apportionment of seats in the US House of Representatives is similarly biased. A brief glance at the number of people in each district for every state shows that there is no such significant bias for or against larger states. (Later, we will consider the matter in more detail and explore the tiny biases that do exist.)

We leave aside questions such as US citizens who have no voting representation in Congress, how districts are drawn within each state, and how the population of the states is determined, and only examine how many seats in the House each state is allocated.

US Congressional apportionment algorithm

A brief overview of US Congressional apportionment and its history can be found in this CRS report. Since 1941, the fixed number of seats in the House has been apportioned amongst the states after each decadal census according to the Hill method of apportionment. Let p_i be the population of the ith state, and for positive integers j define

 \alpha_{i, j} = \frac {p_i^2}{j(j - 1)}.

Then the Hill method apportions n seats by taking the n largest of the \alpha_{i, j}, with state i gaining one seat for each \alpha_{i, j} so taken. As law requires that each state is allocated at least one seat, we require that the \alpha_{i, 1} are all taken before any \alpha_{i, j} with j > 1.

Equivalently, if n_i is the number of seats allocated to state i, then the Hill method minimizes

 \sum_i \frac {p_i^2}{n_i}.

Let p be the total population. Then this is equivalent to minimizing

 \sum_i n_i \left( \frac {p_i}{n_i} - \frac pn \right)^2.

The quantity \frac pn is sometimes called the ideal district size; it equals the population-weighted harmonic mean of the district sizes \frac {p_i}{n_i}, which in general is not equal to the arithmetic mean of the district sizes.

Is the 2010 apportionment biased?

Define the voting power of each person to be the number of House seats their state has, divided by the population of the state; each person in state i has the same voting power v_i = \frac {n_i}{p_i}, which is the reciprocal of the district size within that state. The average person’s voting power is v = \frac np, and so does not depend of the choice of apportionment. In the US after 2010, v = 1.407 \cdot 10^{-6}, corresponding to an ideal district size of \frac 1v = 710767 people.

We consider the 2010 House apportionment, and are interested in whether a person’s voting power v_i depends in some way on the size p_i of their state. A simple test is to do a linear regression of v_i against the independent variable p_i; while there are p = 309183463 data points, each person in the same state has the same data, so we can simply do a weighted linear regression on 50 data points.

However, almost all of the variation in voting power occurs in the smallest states, which are most subject to the restriction that each state has an integer number of seats. The state with the lowest voting power is Montana, having 1 seat for 994416 people and thus 1.006 \cdot 10^{-6} voting power. The next larger state is Rhode Island, having 2 seats for 1055247 people and thus 1.895 \cdot 10^{-6} voting power, the largest of any state and 88% more than Montana.

A linear regression might pick up on these significant variations amongst the small states, but when we speak of bias in House apportionment we are often most interested in the small states collectively compared to the large states collectively. To measure this, we sort the people by the size of the state they are in, and group together those in the first half as the “small state” sample and those in the second half as the “large state” sample. (The median person is in Georgia, which will thus lie partially in the small states and partially in the large states.) Then we find the difference in the average voting power of the people in these two samples: this equals the difference in number of seats those states received, divided by \frac p2.

The results can be seen in the figure:

2010 US House apportionment. For each state, we plot its population p_i and voting power \frac {n_i}{p_i}. The black line shows the average voting power \frac np. In green is the average voting power within the small states and within the large states. In red is a linear regression of voting power against population.

The large states have an average voting power 1.255 \cdot 10^{-8} larger than the small states, an improvement of about 0.9%; we call this difference the voting power gap. The slope 4.885 \cdot 10^{-16} of the regression line is in units of voting power per person (that is, per person squared), so we must multiply by a population to make it the same units as the voting power gap. We will find that multiplying by the population of New York, 19421055, makes a good comparison with the voting power gap, so the scaled voting power slope is 9.486 \cdot 10^{-9}.

Thus we see that, by either measure of bias that we investigated, the 2010 apportionment is biased a very small amount in favor of large states.

Different House sizes

Of course, if the House had only 50 seats, then each state would have one seat, and it would be massively biased in favor of small states. Clearly for small number of seats a bias exists, and for 435 seats it appears to be insignificant. We investigate how the voting power bias changes as the number of seats in the House changes.

Voting power gap (green) and scaled voting power slope (red, scaled by the population of New York) with the Hill method of apportionment for each number of seats from 50 to 600.

We see that the two measurements of voting power bias substantially agree with each other, up to a constant scaling factor. While there is quite significant bias for small states when the number of seats in the House is small, this rapidly decreases with increasing numbers of seats. While it happens that the 2010 census data apportioned with 435 seats favors large states very slightly over small states, this is a bit of an anomaly, as the Hill method tends to favor small states slightly. Looking at seat numbers from 400 to 500, we see that at these sizes with the 2010 census data there is typically a voting power bias of about 10^{-8} in favor of small states, which is a bit less than 1% of average voting power.

This is an advantage of one seat for every 100 million people, which put into absolute terms is about 1.5 seats. As the number of seats is increased further into the thousands and beyond, there continues to be a slight but diminishing bias in favor of small states. As the number of seats increases, the total voting power of all people increases, while the voting power gap decreases, and thus the proportional bias relative to total voting power decreases quite rapidly. That is, even as the number of seats increases, the difference in how many of those seats go to small states or large states decreases.

In part 2, we will consider other apportionment methods and their theoretical properties.

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