Supreme Court: Not Possible! Math: Possible!


This week Occupy Math takes a trip to the land of Freedonia which is beset by a vile dragon that menaces its democracy. A small African nation with a diverse population and the magnificent port city of Great Haven, Freedonia is a constitutional democracy modeled on the American experiment, an active, participatory, free democracy and, until recently, with a vibrant and open economy. Founded in the early 20th century by a largely peaceful revolution organized by tribal leaders — advised by the famous explorer and polymath, Captain Spaulding — Freedonia has been synonymous with hope for generations. Recently, however, the economy has been experiencing problems with corruption. Nepotistic awards of government contracts to incompetent nephews and corrupt back-room deals have taken the economy away from the hard-working farmers, shopkeepers, and craftsmen who have been the backbone of Freedonia society. UN monitors certify each of the biannual elections as free and fair but, somehow, in spite of public outrage, the Lucarian party ekes out a bare majority and restores the corrupt Prime Minister, Joseph Cagliostro to power. What dark force is subverting democracy in Freedonia? Let’s ask no lesser authority that the Governator himself!

Gerrymandering is a subtle way of subverting democracy — and the vorpal sword that can slay it is edged with mathematics.

The idea of gerrymandering is simple. In the United States we choose our representatives in geographic districts. This means that the choices about where the district boundaries are have a lot of influence on how the election might go. If the authority that does the redistricting plays its cards right, it can break up its opponents across multiple districts to dilute their electoral power or gather them all into big districts to limit the number of representatives they can elect. Occupy Math has already covered the basics of gerrymandering in an earlier post. The topic of that post was the Elections Minister of Canada making fun of math used to support fair elections. There have been a number of developments that merit another post.

To jump to the punchline for the Freedonia story, the assignment of electoral districts is one of the jobs held by the prime minister’s office. Until that power is taken away and given to some sort of independent commission, Freedonia will continue its downward spiral with Prime Minister Cagliostro ensuring the election of his minority party. Let’s establish that we have some of the same problems as Freedonia.

  • Donald Trump lost the popular vote, but won the 2017 Presidential
    election. Implacable Republican gerrymandering of congressional
    districts — used to select members of the electoral college, itself a hold over from the days of slavery — were at least part of what made that victory possible.
  • The State of Texas was recently whomped in a federal district court for trying to fix the next election with a racial gerrymander. In this case Latino citizens were the targets of this anti-democratic move.
  • The State of North Carolina — which was just stopped by the courts from stripping power from the Governor’s office after a Democrat won the office — was
    just whomped by the Supreme Court for a racial gerrymander — one hundred and fifty two years after the end of the Civil War. As one might expect, the target in this case was black citizens.
  • The Republican districting plan for the state assembly in Wisconsin was whomped for being a partisan gerrymander. The theme that unifies these example is: “if you cannot win based on your merits, manipulate the rules”. Sad!

The problem of gerrymandering is not new — the term gerrymander dates from 1812 and was invented by the Boston Gazette to describe Governor Elbridge Gerry’s redistricting plan that transparently opposed representative government. The word gerrymander is a portmanteau of the governor’s last name and the word salamander (the mythological fire elemental, not the amphibian). The picture at the top of the post is a political cartoon depicting that first gerrymander.

The mythological salamander

Math to the rescue?


There have been a number of developments in the mathematical world that address Gerrymandering. Occupy Math is most heartened by the work of Moon Duchin, a math professor and director of the Science, Technology and Society program at Tufts. Recognizing that her work in metric geometry might apply, Professor Duchin put together a summer school and working group which caught on in a big way. The aspect of gerrymandering that metric geometry attacks is compactness. The most compact shape going is a sphere — it has very little surface are given its volume. Look at the picture of the very first gerrymander at the top of the article. Among other things, it is an excellent example of a shape that is not compact.

An article in Wired magazine notes:

“The problem is that there is no such thing as a perfect map–every map will have some partisan effect. So how much is too much? In 2004, in a ruling that rejected nearly every available test for partisan gerrymandering, the Supreme Court called this an “unanswerable question.” Meanwhile, as the court wrestles with this issue, maps are growing increasingly biased, many experts say.”

and this is a key problem.

Another example of this type of problem is that the Bush administration mothballed satellites to prevent enforcement of environmental regulations by making sure it was impossible to tell if they had been violated. The Wired article goes on to document that math is coming to the rescue — both Professor Duchin’s work on compactness and also the packing and cracking discussed in the last Occupy Math on gerrymandering. There is another nice Wired article on packing and cracking. The title of this article warms Occupy Math’s heart: “Gerrymandering Has a Solution After All. It’s Called Math”.

Who else is working on gerrymandering?

The excellent blog mathbabe discusses algorithms for Gerrymandering. This article makes the point that — in mathematics — designing and detecting a gerrymander are pretty much equivalent activities. The New York Times weighs in with a statistical approach — look at the difference between the distribution of voters and the distribution of outcomes. The Princeton Electoral Consortium has an app for that. You can try to see if you’re being gerrymandered with your smartphone. The Stanford Law Review weighs in with a very dense document about what a gerrymander is in law and how to document it. This link is useful only if you’re starting a lawsuit and need to brief your lawyer.

What’s this sampling thing?

The sampling approach is worth more exploration. Suppose that, in order to establish voting districts, we assign counties in a state to congressional districts. The only clear rule is that they have to form connected groups. It would not be hard to get a computer to sample 1,000,000 possible districting schemes. If we know the partisan makeup of these districts, then it’s not hard to score the probable election of Democrats and Republicans from each districting. As correctly noted by the Supreme Court, a completely fair districting is a miracle — it just won’t happen. What, then, can you do with 1,000,000 districting plans scored by their partisan outcome?

  1. You can pick one that favors your party. Not the best one — that would be too obvious. Rather a districting plan about 80% of the way up the distribution. One that gives you a solid advantage without being so obvious that you will get whomped by the courts like North Carolina, Texas, and Wisconsin.
  2. You can compare any proposed plan to the distribution to see just how biased it is. This is quantitative evidence that a good lawyer could use in court to prove that there is a gerrymander.

This thought experiment shows that the sampling algorithm is, simultaneously, a tool for designing a gerrymander that might pass judicial muster and a tool for documenting just how far out of line a districting plan is. Math is a two edged sword; Occupy Math prefers to think that if these tools are available, citizens will come down on the side of fairness and democracy. This is why the good guys need to know math!

The topic of gerrymandering, especially in light of the 2004 Supreme Court declaration that detecting gerrymanders was an intractable problem, has turned into a real success story for math. Mathematics cuts the Gordian knot in several ways and with only moderate effort. Overall, this is a brilliant illustration of Occupy Math’s motto Math is the Right of all Free People. Occupy Math loves quantitative social justice. In that light, please suggest topics: comment or tweet!

I hope to see you here again
Daniel Ashlock
University of Guelph
Department of Mathematics and Statistics


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