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Can Math Help Save Democracy?

December 5th, 2015, 10:41am by Sam Wang

In the New York Times I have a piece in the Sunday Review, “Let Math Save Our Democracy.” It describes some simple statistical standards for partisan gerrymandering, and how they might resolve an upcoming Supreme Court case (Harris v. Arizona Independent Redistricting Commission) quickly and definitively. It’s based on my law article, “A Three-Prong Standard for Practical Evaluation of Partisan Gerrymandering,” which you can read here. My amicus brief in Harris is here.

In a short piece like that, some key points had to be glossed over. An extremely important concept is the “zone of chance.” That is basically a catchy phrase to describe a statistical significance test. When comparing lopsided winning margins (for example, let’s say we have a state where Democratic wins average 80%-20%, whereas Republican wins average 55%-45%), one should compare the Democratic winning percentages (80%) with the Republican winning percentages (55%). This is done by a grouped t-test, “probably the most widely used statistical test of all time,” as VassarStats puts it. The average-median difference can also be evaluated with a version of the t-test.

Other stuff: University of Chicago law professor Nicholas Stephanopoulos and his collaborators have developed a measure of gerrymandered voters that they call the efficiency gap. In the Harris case, they found that small variations in district population were not enough to skew the overall outcome.

Also, there is the analysis of individual districts. This approach has a long history, dating to the work of Gary King and Andrew Gelman’s JudgetIt program. Recently, Jowei Chen and John Rodden have developed automated districting procedures to probe the whole range of possible districting schemes that obey a few basic principles. And of course there is my own “fantasy delegations” approach (PEC essays here and here), which can help estimate the right number of seats for a given statewide vote.

Finally, there is a common objection to claims of partisan gerrymandering: namely that population clustering is the true culprit. Population clustering does indeed have a substantial effect. However, post-2010, partisan gerrymandering in a handful of states shifted more Congressional seats than population-clustering effects in all 50 states combined. See the calculation on pages 36-37 of my law article.

Tags: Redistricting

15 Comments so far ↓

  • Olav Grinde

    So glad this is getting well-deserved attention. Gerrymandering deserves to be high up on the judicial and political agendas — but it needs to be dealt with correctly. In other words, on the basis of solid math. And it’s optimal if the core of the argument is simple and intuitive.

    Kudos to Dr Wang for this work!

  • Amitabh Lath

    Brilliant article Sam. The difference between the mean and median is simple and intuitive. And yes, it uses grade school math that hopefully the justices on the court can understand.

    To visualize the question of population clustering vs. gerrymandering, would it possible to compare states that have equal amounts of population clustering, but some gerrymandered and others not?

    • Sam Wang

      Thank you!

      The issue of population density is probably the one I will hear the most about. With no natural clustering, the bias should be zero. My resampling approach gives an overall net bias (see the SSRN article at page 36), which quantifies the net total population effect — averaged nationwide. Turning it around the other way, I have simulated individual states using national data with urbanized states removed (or using only urbanized states); the same states are flagged as being asymmetric.

      More to your point, you are asking about how a single state is affected – Pennsylvania and Michigan being examples of places with big cities. A good counterexample is New York, whose outcome does not show asymmetry. I believe that is kind of control addresses your concern directly.

      At some level, I believe there is a deeper point. Redistricting is a human activity. Inevitably, both structural biases (i.e. population density) and political biases (i.e. partisanship or incumbent protection) creep in. But asymmetry is asymmetry. If partisan symmetry is an ideal then legislators and commissions should, at a minimum, not magnify the structural biases.

  • Stephen Marmon

    I do hope that you file this and all related research as part of an amicus brief with the Court in regards to Harris v. Arizona.

  • Juan Maldacena

    A simple way to limit gerrymandering is to demand that voting districts should have a convex shape.

    • Sam Wang

      This is conceptually appealing, but is not consistent with existing federal voting law. This would have to be done at the state level.

      Also, note that sometimes it is desirable to connect communities of shared interest, and the Voting Rights Act in fact requires this. How much it requires this is unsettled.

      Finally…your fix would leave almost half the asymmetry in the system. See pp.35-37 of my SSRN paper. Apologies for the lack of heavy math there.

      Oh, and this is an important one…packing a bunch of convex shapes leaves lots of holes! Oops. :-P

  • Leading Edge Boomer

    Great article. I hope the Supremes (or their more numerate clerks) take notice.

    I found reader comments in the NYT that suggest alternatives to be impractical and downright goofy. Comments are closed now.

  • Francois Christen

    Great idea to do a stat test to confirm or refute gerrymandering. But couldn’t we go one step further and use math (perhaps some type of cluster analysis) to do the redistricting and remove bias altogether? I’m new to this topic, so the idea may have been considered and rejected in the past.

    • Sam Wang

      Jowei Chen and John Rodden have come up with automated procedures. My piece links to their work.

      Legislatures won’t give up that power willingly. Also, I am not entirely sure we want such a process to be automated. There are so many interests to balance.

  • Lee

    Thank you for an excellent article. Forgive me, but I can’t help but thinking what could be done even better. While I can see advantages to putting voters into groups, I find that the use of geography for this purpose is rather dated and it wouldn’t be on my top three list of grouping criteria. Instead of patching the districting process, would you consider advocating for some sort of party-based proportional representation system?

    • Sam Wang

      There is no obvious political or judicial path to that endpoint. My thought was to push for something that could be achieved under current conditions.

  • Noah Kennedy

    Dr. Wang- thanks for your very thoughtful article and the great work you have been doing!

    In response to your comment above (“sometimes it is desirable to connect communities of shared interest”), I wonder if we should look harder at that. The unintended consequence of creating districts that are “safe” for communities of interest is that it becomes safe for politicians outside those districts to flout the desires of people who have been excluded from their district, which leads to more heightened partisan behavior.

    I am very much in favor of a geometrical means of assuring districts are as compact/convex as possible, and letting the chips fall where they may from there. Once _anyone_ starts “drawing” districts for partisan reasons– which includes favoring one political party over another but also includes favoring one ‘community of interest’ over another– the door is open for mischief. Please take a look at for more information on how that might be achieved.

  • Art

    Looks like both sides agree that the election results did not harm Republicans, so this case will be decided on other grounds. You will have to keep on plugging… Suggest more discussion on where to draw the line.

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