The Mathematics of Gerrymandering and the Supreme Court

By Dr. Rachel Levy, Deputy Executive Director of the Mathematical Association of America with an excerpt from Dr. Jeanne Clelland, University of Colorado, Boulder

One of the interesting things about working at the MAA in Washington, DC is that people talk politics and policy. Honestly, I find it hard to keep up or even to connect the names of people with their jobs. But I like being more politically aware and thinking about connections between policy and our MAA work to advance the understanding of mathematics and its impact on the world.

I don’t often think of social media as the best source of political information. So I was delighted when I saw a post from mathematician Dr. Jeanne Clelland discussing the recent Supreme Court decision on gerrymandering. I had enjoyed a plenary talk this spring by Jeanne at this year’s MAA Rocky Mountain Section meeting.  After seeing the post, I asked Jeanne if we could reprint it here and she said yes!

The original gerrymander. In 1812, Elbridge Gerry, the governor of Massachusetts, signed a bill that redistricted Massachusetts to benefit the Democratic-Republican party. The Boston Gazette coined the term "Gerry-mander" for a salamander-shaped district in the Boston area. Image Source:

The original gerrymander. In 1812, Elbridge Gerry, the governor of Massachusetts, signed a bill that redistricted Massachusetts to benefit the Democratic-Republican party. The Boston Gazette coined the term "Gerry-mander" for a salamander-shaped district in the Boston area. Image Source:

Dr. Jeanne Clelland’s post

After reading the entire text of the recent Supreme Court ruling on gerrymandering and commenting briefly on a friend’s Facebook post, my friend asked me to write a longer, public post with more details. What follows here is (with minor editing) the resulting post, written quickly and late at night after a long flight. I was very pleasantly surprised when Ray asked me if she could post it here on Math Values; I never imagined it might reach such a wide audience! 

First, a bit of historical perspective: Over the last several decades, the Supreme Court has ruled many times on racial gerrymandering cases under the auspices of the Voting Rights Act. As is often the case, it’s not so easy to articulate a precise standard for what does and does not constitute illegal racial gerrymandering, but the courts have gradually worked it out through a series of cases, and this process is, of course, ongoing. Meanwhile, in cases involving partisan gerrymandering, the Supreme Court has repeatedly indicated that it considered such claims justiciable, but due to the lack of a clear and manageable standard for measuring it, the Court has never actually declared any particular map to be unconstitutional on the grounds of excessive partisan gerrymandering. Justice Kennedy, in particular, clearly indicated that he would very much like for someone to come up with such a standard, but then he retired last year before identifying a standard that he found satisfactory.

Enter the mathematicians! Just in the last few years, several groups have pioneered a strategy for quantifying gerrymandering based on statistical sampling and outlier analysis. The basic idea is to take all the rules that a particular state requires a districting plan to satisfy - e.g., districts must be contiguous, relatively compact (whatever that means), they should try not to divide communities of interest, etc. - and have a computer draw a large random sample (an “ensemble”) of districting plans that satisfy all the necessary criteria. Then take precinct-level voting data from recent elections, and for each plan the computer drew, compute the number of seats that each party would have won with that plan and the actual voting data. This typically results in a bell curve describing how many seats one might expect each party to win under a politically neutral plan. If a particular plan - say, one drawn by a highly partisan state legislature - yields a result that’s way out on the tail of the curve, that’s pretty strong evidence of gerrymandering. For North Carolina in particular, much of this analysis was carried out by Jonathan Mattingly’s research group at Duke; their work is described in detail here, with lots of graphs to illustrate their analysis:

What’s really powerful about this approach is that it takes the political geography of a state into account in a way that simpler measures (like, say, the much-touted efficiency gap) cannot, and it provides a more nuanced picture of what’s actually reasonable to expect. The mean of the ensemble may or may not reflect proportional representation; indeed, it often does not. A particularly striking example is Massachusetts, where Republicans consistently get 30-40% of the statewide vote in Congressional elections, but no Republican has been elected to Congress since 1994. A recent paper by Moon Duchin’s group, available at, shows that this is not due to gerrymandering, but rather to the fact that Republicans are simply too spread out throughout the state, to such an extent that it is, in fact, mathematically IMPOSSIBLE to draw a majority-Republican Congressional district no matter how you draw the lines. A simple measure like the efficiency gap would flag Massachusetts as an egregious gerrymander, but ensemble analysis shows that this outcome is simply a consequence of how Republicans are geographically distributed throughout the state.

In this week’s decision, Justice Roberts basically threw up his hands and declared that the search for a manageable judicial standard for measuring gerrymandering is hopeless, and therefore such claims will no longer be considered justiciable in federal courts. He seemed not to clearly understand the mathematical argument; he repeatedly referred to the proposed outlier analysis as attempting to measure deviation from proportional representation, which it absolutely does NOT do. More significantly, he opined that it was not the Court’s business to decide how much deviation was permissible, and that therefore the entire question should be left up to the states and to Congress.

In Justice Kagan’s scathing dissent, on the other hand, she made it clear that she understands the math and believes that it could and should form the basis for a judicial standard. She did not attempt to set a clear threshold for how much deviation from the mean should be permissible, but she thinks that in the North Carolina and Maryland cases at hand - both of which are way out on the tails of their respective bell curves - the Court should say, “This much is DEFINITELY too much.” Then it would be up to future litigation, legislation, etc. to work out the question of where to set limits on how much deviation from the mean is permissible, similar to the process that has played out for racial gerrymandering.

The upshot is basically that the status quo will remain, at least for now. The decision explicitly cedes the power to regulate gerrymandering to the states and to Congress - so the Court will not, for instance, strike down the initiatives that have passed in some states to create independent redistricting commissions. I would like to think that Justice Kagan’s dissent might provide some states, and maybe even (some future) Congress, with a template for legislation that might put some sensible limits on partisan gerrymandering - and meanwhile, we mathematicians will continue to work on developing and improving these methods so that we can contribute to the conversation at any and all levels!

Want to learn more? Check out the Metric Geometry and Gerrymandering Group: MGGG is led by Moon Duchin of Tufts University and Justin Solomon of MIT.

Jeanne Clelland.jpg

Jeanne Clelland is a Professor of Mathematics at the University of Colorado, Boulder.  She is a signatory to the Amicus Brief of Mathematicians, et. al., that Justice Kagan cited in her dissent.