Ensembles and Outliers

The dictionary definition of gerrymandering, “to manipulate or change the boundaries of (an electoral constituency) so as to favor one party or class,” presupposes some conception of the expected outcome and structure of an election. Implicitly a complainant challenges a map as gerrymandered because it fails to align with this expectation in some critical way.

Measures like the efficiency gap or partisan symmetry offer conceptual, idealized alternatives against which a given map’s compliance is measured. The efficiency gap values a rather arbitrary relationship between the number of seats and the overall vote fractions. (See Bernstein and Duchin or Tapp). Partisan symmetry posits the superficially appealing idea that both parties should be treated symmetrically when each have the same global vote fraction, ignoring if symmetry is truly reflected in a geopolitical landscape. Neither takes into account the heterogeneity of population densities, the shapes of states or differences in where various political constituencies live. Requirements imposed by the Voting Rights Act or statutes requiring that counties or municipalities be left whole where possible, are not directly incorporated into the partisan symmetry measure.

There is another path, which builds on well- established methods of modern computational statistics: constructing a representative ensemble which can be used normatively to establish a baseline against which deviations can be measured.

It does not dictate political outcomes based on some idea of fairness where it is difficult to incorporate demographic ground truths, the effects of geography, or the preservation of municipal entities and communities of interest. Rather one encodes the design principles of a desirable map in an ensemble of maps and then lets the typical electoral behavior reveal itself.

The design criteria are typically nonpartisan criteria such as spatial compactness, contiguity, equal population division and the minimization of fragmented counties and municipalities. It can also include items such as compliance with the Voting Rights Act (VRA). With the possible exception of VRA compliance, these criteria do not overtly create any new partisan bias, leaving (rightly) only those inherent in our political system and the distribution of voters about a particular geographically distinct state.

It is important to separate (i) the use of such an ensemble, (ii) its generation and (iii) the choice of the design criteria. Each is intellectually independent and, as such, can be critiqued separately. Here we mainly wish to concentrate on the use of an ensemble. Though we will comment briefly on the other elements, we will largely leave the discussion of the other elements to future works or posts.

Using Normative Ensembles

Once one has fixed a collection of representative maps, they can be used to probe the structure of a given election. Among other things, the ensemble reveals typical outcomes of an election and the typical over all structure of the election. The ensemble encodes the geopolitical structure of the state as it includes the population distribution as well as the spatial alignment and connectivity of the state. If desired, it can also encode the need to keep cities or counties whole or to satisfy the Votings Rights Act. Hence any conclusions drawn on the typical structure and outcome of a collection of votes is inherently informed by the geopolitical makeup of the region and the other desired criterion encoded in the construction of the ensemble. Continue reading “Ensembles and Outliers”

The Signature of Gerrymandering

Our Quantifying Gerrymandering group at Duke generated an ensemble of over 24,000 redistricting plans, sampled from a probability distribution placed on the collection of redistricting plans.  The ensemble was used to evaluate the 2012 and 2016 congressional district plans enacted by the NC General Assembly.  The two enacted plans were both found to be statistical outliers in the context of the ensemble of 24,000 plans; this outlier analysis formed the central argument of Jonathan Mattingly’s testimony in Common Cause v. Rucho.

In the outlier analysis, the most obvious statistic to consider is the partisan makeup of the congressional delegation each map produces.  The following histograms show that the 2012 maps (NC2012) and 2016 maps (NC2016) produce unlikely results.  In contrast, a map produced by a bipartisan panel of retired judges (Judges) produces typical results.

However, this simple analysis does not tell a complete story: In particular, as shown in the discussion of Firewalls,  a map can produce quite typical results for some elections and outlier results for other elections.

When analyzing the ensembles of predicted election results, different elections probe different elements of a redistricting plan’s structure.  A redistricting plan yields atypical election results only when the plan’s overall structure is anomalous in a way that is relevant to a particular election.   In short, the same plan can yield both anomalous and typical results for different elections, however some plans always give typical, expected results. Continue reading “The Signature of Gerrymandering”

Marginal Box-Plots: Summarizing what is Typical

One of the principle visualizations we have used to explore and and communicate our results is the Marginal Box-plot. Marginal box plots were one of the principle graphics presented in Redistricting: Drawing the line Evaluating Partisan Gerrymandering in Wisconsin, and the  group’s testimony in Common Cause v. Rucho.

The box-plots give a way to visually spot anomalous properties in a given redistricting plan by summarizing the structure of a typical plan, drawn without overt partisan considerations. For example, they can help identify what districts have been packed or cracked, showing which districts have many more or many less votes for a certain party than expected. The marginal box-plot give a baseline with which a given map should be compared.

Two prototypical examples of marginal box-plots are giving below. They summarize what we would expect  from redistricting of  North Carolina in to 13 Congressional districts and viewed through the lens of the actual votes cast in the 2012 and 2016 congressional elections.

Box-plot summary of districts ordered from most Republican to most Democratic, for the congressional voting data from 2012 (left) and 2016 (right).

Continue reading “Marginal Box-Plots: Summarizing what is Typical”

Hearing the Will of the People

Democracy is typically equated with expressing the will of the people through government. In a Republic, the people elect representatives who then act on their behalf and derive their political mandate from having won the election.

Possible corruption of the  electoral results is often framed in terms of voter suppression, voter fraud, or the undo sway of money on people’s votes. Once the votes are collected, once the access to information and the ballot box is unfettered, all that remains to register the will of the people is to count each vote once and only once.

Yet,  by varying how districts are drawn one can cause tremendous variation  in the outcome of the elections without changing a single vote. There is so much variability, that one might wonder if the effect is greater all the previously mentioned effects combined.  Continue reading “Hearing the Will of the People”

Gerrymandering is Not about Oddly Shaped Districts

It is tempting to assume that gerrymadnering requires the presence of oddly shaped districts.  After all, the term gerrymandering derives from the salamander-shaped maps produced by Massachusetts’s 1812 Governor Elbridge Gerry, and pictures of that meandering district are practically required in any discussion of gerrymandering.

The story goes that the irregularly shaped boundary attempts to include or exclude particular voters from the district.  Although irregular boundaries might well cause one to suspect gerrymandering, it is quite possible for a gerrymandered map to have quite regular districts. Oddly shaped districts, such as like North Carolina’s 12th congressional district and Pennsylvania’s 7th congressional district, make provocative tee-shirts and posters, but they are  only a symptom and not the point of Gerrymandering . Continue reading “Gerrymandering is Not about Oddly Shaped Districts”


A Firewall is a buffer used to block unauthorized access.  We adapt the term ‘Firewall,’ in the context of gerrymandering, to describe a districting plan that artificially protects the power of a political party.  What follows is an exposition on how we discovered a Firewall in the enacted districting plan for the Wisconsin General Assembly  that protects the Republican Party from losing the majority of the seats.

For districting plans of the Wisconsin General Assembly, we generate thousands of compliant redistricting plans.  To evaluate the enacted districting plan (the Act 43 plan), we ask if how many officials are elected by each party for each plan for a given a set of votes: Each plan in the ensemble will generate a certain number of Democrats and a complementary number of Republicans (ignoring independent candidates), and we can construct a histogram that measures the number of representatives from each party, out of the 99 available seats.

For example, when looking at the Wisconsin General Assembly districts, we construct histograms of the projected number of Republican elected officials that would have won based on 2012, 2014, and 2016 voting data.

Continue reading “Firewalls”

Drawing New Maps for Pennsylvania

The  recent court ruling in Pennsylvania invited  “all parties and interveners” to submit their own maps to replace the maps which have been ruled unconstitutional.  (See also here.)

In the interest of facilitating such maps and their analysis, the Duke Quantitative Gerrymandering group is making  available our processed and cleaned map files and associated data at the following public git repository.


A local copy can be downloaded with

git clone https://git.math.duke.edu/gitlab/gjh/PennsylvaniaRedistrictingData.git

We encourage those who produce maps to share them with us.