Newly Proposed NC Maps are more gerrymandered and less responsive than maps struck down in 2021

Note: The original post analyzed the congressional plan proposed in Senate Bill 756; we have updated the analysis to include the plan proposed in Senate Bill 757.

We have completed an initial analysis of the redistricting plans recently proposed by the NC Legislature on Wednesday, October 18. The legislature has proposed changing the maps used to elect the North Carolina congressional delegation to the US House of Representatives and the maps used to elect the NC House and the NC Senate members of the NC Legislature.

We use the same methodology as described in our report here. The nonpartisan ensembles of maps used in the below analysis are the primary ensembles previously presented to the court in Harper v Hall/Moore in 2021.

A first indication of gerrymandering in the new Congressional maps comes from comparing it with the remedial map from 2022.  We do this by using the votes from the 2022 US Senate race which has a statewide partisan vote share of 48.35% preferring the Democrats. Under these votes, both proposed maps (from SB756 and SB757) would elect 11 Republicans and 3 Democrats to the US House of Representatives while the remedial map it replaces would have elected 7 Republicans and 7 Democrats.

One of the basic principles of a functioning democracy is that when the electorate changes its political opinion this change should be reflected in the elections outcome. The proposed congressional maps are both highly non-responsive to changes in the opinion of the electorate. We find that over a number of recent historical elections and other possible political environments, the newly proposed congressional map from SB756 reliably elected 11 Republicans and 3 Democrats despite the elections having wildly different votes.  Similarly, the map proposed in SB757 reliably elected between 10 and 11 Republicans and between 3 and 4 Democrats. Furthermore, even under dramatic voting swings, the newly proposed congressional plans consistently elect more Republicans than would be expected from a non-partisan process.

To see this, we examine voting patterns between 45% and 55% Democratic statewide vote shares based on the 2022 US Senate election and the 2020 Presidential election and reproduce the movies described here.  We see that the newly proposed maps are significantly less responsive to the changing will of the people as expressed in their votes than the 2022 remedial map they would replace. Over all of the historical elections considered in the plot below, the proposed map from SB756 always elects 11 Republicans and 3 Democrats and the proposed map from SB757 elects 10 or 11 Republicans and 3 or 4 Democrats. In contrast, the remedial map that is likely to be replaced and which was used in the 2022 elections varies from electing 9 Republicans and 5 Democrats to electing 7 Republicans and 7 Democrats.

Furthermore, we observe that the newly proposed plans are even less responsive than the originally overturned plan from 2021 and lean more Republican than either the overturned plan drawn by the legislature in  2021 and the remedial plan that will likely be replaced from 2022.  The collection of histograms over historic elections displays the rigidity of both plans.  It also displays how the plan from SB756 leads to an extra Republican seat when comparing to the overturned 2021 map.  The difference between the overturned 2021 map and the plan proposed in SB757 is not evident in the collection of histograms, however, the difference can be observed in both the videos and also in the rank-ordered districts that we present at the end of this post and which are explained in this past post.  The key take away is that there are three Democrats residing in packed districts that will reliably vote Democratic, whereas the 4th most Democratic seat is even less reliably Democratic in the newly proposed plan from SB757 than it was under the overturned 2021 plan.

The two videos below show how the remedial map that was used in the 2022 election responds to elections swinging from preferring Republicans to elections preferring Democrats while the newly proposed maps fail to change the seats elected as the election votes swing from  45% to 53% Democratic vote share.

We display the election results in the ensemble and the two most recently enacted congressional maps in North Carolina along with the newly proposed maps. We see the rigidity in both the 2021 and newly proposed 2023 plans over a wide range of historic voting patterns and the extreme Republican bias that they show relative to the ensemble of plans.

In addition to the analysis in the Congressional maps, we also present an analysis of the state house and state senate maps.  We reproduce the animated effect using the Presidential 2020 results.  In both chambers, we find that the proposed plans are even more extreme than the originally enacted 2021 maps. Additionally, we find that new proposed maps are less responsive to changes in the votes than the remedial maps used in the 2022 elections.

Both the Senate and House maps under-elect Democrats as one moves to more balanced elections with Republican statewide vote fractions near 50%. This has important implications for the preservation of the super-majority in the chamber.  Under the newly proposed Senate maps, the Republicans may reasonably expect to obtain a super majority, even when the statewide Democratic vote share is over 50%.


The figures below show the number of Democrats elected to the NC Senate and House using a number of different historical elections which have a wide range of state-wide Democratic vote share. As one moves into the elections the ensemble shows that the Republicans would typically fail to obtain a super majority, the proposed map significantly under-elects Democrats.

In the Senate, using the AG16, the AG20, the GV16, the USS20, the CL20, and the PR20 elections, Democrats would typically break the supermajority using the maps in the ensemble or using the remedial map from 2022. Yet the newly proposed map would preserve the Republican super-majority. In these elections, the statewide Democratic vote share ranges from 48.9% to 50.2%.

In the House, using CL20 and  USS20 election data, the Democrats would typically break the supermajority using the maps in the ensemble or using the remedial map from 2022. Yet the newly proposed map preserves the super-majority. In the more democratic-leaning elections the ensemble and the remedial map from 2022 would typically give control of the chamber to the Democrats but the newly proposed map leaves the Republicans with a sizable majority.


We display the election results in the ensemble and the two most recently enacted Senate maps in North Carolina along with the newly proposed map.
We display the election results in the ensemble and the two most recently enacted House maps in North Carolina along with the newly proposed map.

As stated above, we conclude by returning to the congressional maps and compare the 4th most Democratic seat between the plan from SB757 and the overturned plan from 2021.  We observe that under both the Presidential 2020 votes and the US Senate 2022 votes, the 4th most Democratic seat is more Republican under the plan from SB757 than it is under the overturned 2021 plan.  We have also repeated the analysis for the votes under the 2020 Attorney General election and the 2016 Presidential election and find the same conclusion:  Under the 2020 Attorney General election, the fourth most Democratic seat would yield a 53.6% of the vote going to the Democratic candidate under the overturned results, while giving a narrower margin of 52.8% to the Democratic candidate under the plan proposed in SB757.  Similarly, under the 2016 Presidential election, the fourth most Democratic seat would yield a 53.0% of the vote going to the Democratic candidate under the overturned results, while giving only a narrower margin of 52.4% to the Democratic candidate under the plan proposed in SB757.

We generate the ranked-order district statistics for the ensemble along with the 4 plans under the votes from the 2022 US Senate election . Both the plan from SB757 and the overturned plan from 2021 elect the three Democrats, however the 4th most Democratic seat (labelled “4th D”) is more Republican leaning in the newly proposed map than in the overturned map.
We generate the ranked-order district statistics for the ensemble along with the 4 plans under the votes from the 2020 Presidential election . Both the plan from SB757 and the overturned plan from 2021 elect the four Democrats, however the 4th most Democratic seat (labelled “4th D”) is more Republican leaning in the newly proposed map than in the overturned map.


Jonathan Mattingly
Greg Herschlag


The data used in this analysis can be found here. In addition to this, we have imputed the 2022 voting data down to the census block level and report those numbers here.

Comparing Algorithms for Generating Ensembles to Detecting Gerrymandering

Jonas Eichenlaub, 2022, Bowdoin College
Faculty Mentors: Jack O’Brien (Bowdoin), Gregory Herschlag (Duke), Jonathan Mattingly (Duke)

[What follows is a short summary of work done as a summer research project in 2021. The longer report can be found here.]

Gerrymandering is a pervasive issue in American politics and over the last decade mathematicians have made notable contributions towards its detection and avoidance. One focus of these efforts has been using computers to generate an ensemble of potential legislative maps and then comparing a real map against this distribution of maps. Vote counts from recent elections can then be plugged into both the real and simulated maps to understand whether the number of districts a given party wins under the real map is statistically likely under the distribution of generated maps. In other words, if one party wins far more seats in the real map than the mean number of seats they win in the simulations, it is likely the real map is the result of gerrymandering. Since gerrymandering is often hard to precisely define, it is easy for policymakers to circumvent any one metric of how gerrymandered a district is; the strength of this technique is that it does not rely on a single metric, instead wholistically comparing the real map with the ensemble of possible maps.

A prevalent way of creating such an ensemble of simulated maps is with a recombination (or ReCom) algorithm, which involves encoding a state as a graph with voting precincts as nodes. The graph is partitioned into districts according to constraints and preferences within the algorithm to make sure that the edges that are cut to create connected components (partitions or districts) with roughly equal populations, minimal municipality splitting, and whatever other legal criteria legislative maps must meet. By cutting different edges of the original graph, different maps are formed, which collectively create an ensemble of possible maps. Duke Professors Jonathan Mattingly and Gregory Herschlag have proposed formalizing such an approach by placing a recombination algorithm into a Metropolis-Hastings Markov Chain, where the proposal of a new map is separated from the acceptance of that proposal into the ensemble. Instead of making decisions about how maps are added to the ensemble based on the structure of the algorithm, Mattingly and Herschlag’s work applies the existing mathematical theory on Metropolis-Hastings chains to justify their approach. Each map represents a stage in the chain, and new maps are proposed with the merge-split technique described above. Maps are then added to the chain based on an acceptance probability that quantifies how well the proposed map fits within the legal criteria, which is designed to be changeable to meet the specific policy parameters of different states. By making the choices within the algorithm explicit, Mattingly and Herschlag offer an approach to detecting gerrymandering that is more defensible when used as evidence in legal challenges to gerrymandered maps.
Part of their contribution is to sample maps from possible partitions of a state rather than from possible spanning trees of the graph of the state, since each partition of a state can be formed by a different number of spanning tree configurations. My research this summer focused on determining whether sampling from partitions generates different ensembles of maps than that from the spanning tree method; more specifically, I investigated whether algorithms that sample from the space of spanning trees are less likely to generate maps that draw district lines along the border between areas with high and low population density (i.e., along the borders of cities). I ran a version of Prof. Mattingly and Herschlag’s algorithm that sampled from the partition space and a version that sampled from the spanning tree space on a variety of “test states” that I constructed. I represented each test state simply as a lattice graph, with a few nodes on the lattice replaced with smaller, denser lattice graphs representing cities. The “city” nodes along the edge of the denser lattice all had edges connecting to the same neighboring “noncity” node, which meant that drawing a district line between these two areas required cutting far more edges than drawing it between other sections of the graph (which normally would only require one cut); the motivating idea behind my investigation was that the spanning tree algorithm views this as costlier and is less likely to select maps with district lines the run between the city and noncity nodes. I varied the placement, size, and number of these “cities” to gain insight into how often each of the two versions of the algorithm cut the edges between the city and noncity nodes.

Notably, I created a test city that was a 9×9 grid, with the center node replaced by a 4×4 city. I ran both versions of the algorithm on this test state with 3, 4, 6, 8, and 12 districts. For each of the two ensembles, I then count the number of times that edges of the graph that connect a node in a city to another node fell on a legislative district border across all maps in that ensemble; this number is then divided by a tally of the total number of edges that fell on district borders. This statistic is the fraction of edges that cross district borders that go from city to anywhere that looks at the proportion of edges that are cut which connect one city node to either a noncity node or another city node. The difference between the results of this statistic from running the two algorithms on each scenario is reported in the table below, with positive results indicating it is higher under the gamma equals one algorithm.

Number of Districts346812
Ideal District Pop320002400016000120008000

I found that the algorithm that samples uniformly from spanning trees is less likely to make districts that have edges along the border of cities, but only under certain situations. The two main factors that determine whether such bias appears in an ensemble are the district population size and the city population size. When the city population fits perfectly within a single or multiple districts, as is the case when there are 4 districts in the above test state, the spanning tree algorithm is significantly more likely to create maps that do not break up cities compared to the partition-space algorithm; hence, a hidden bias in the original merge-split approach is evident. However, if the city population does not fit evenly into one or more districts, as is the case when there are 6, 8, or 12 districts, both algorithms are forced to break up the city and there is not a notable difference between the ensembles they generate. These results reinforce why it is important to use Mattingly and Herschlag’s approach, as their formalized algorithm does not automatically keep municipalities within the same district in a hidden and uncontrollable manner. Instead, it allows for policymakers to explicitly decide how much they want to preserve cities by incorporating this preference into the acceptance probability. Given that the redistricting process from the 2020 Census is currently underway, this insight could find application in the upcoming legal challenges to the inevitably gerrymandered maps that politicians propose.

Evaluating the Enacted North Carolina Redistricting Plans

We analyze the enacted North Carolina congressional plan (CST-13), the enacted North Carolina House General Assembly plan (SL 2021-175), and the currently proposed North Carolina Senate General Assembly plan (SST-13).

We compare these three districting plans with the ensembles we generated from non-partisan redistricting policies. See here for the congressional analysis and here for the general assembly analysis; these documents contain information on the non-partisan criteria considered along with the resulting ensemble of maps.

A comparison of the enacted CST-13 congressional map with a non-partisan ensemble of maps under a range of historic elections.

As an example, we look at the number of elected Democrats that would have occurred under a range of historic elections.  Under each election, we compare the ensemble of non-partisan maps with the enacted map.  The enacted congressional plan elects 4 Democratic representatives under a large range of statewide Democratic vote fractions.  This level of non-responsiveness is not seen in the ensemble of plans.

For detailed analysis, see our reports:

Nonpartisan Distribution on NC Congressional Plan using 2020 Census.

We have performed an analysis of the geopolitical landscape of the North Carolina State using the 2020 Census data. We provide a
summary plot derived from an explicit distribution on redistricting
plans. The distribution favors plans with compact districts which keep counties intact.

As the character of the votes considered swing from more Republican to more Democrat, we see a gradual increase in the number of seats won by the democratic party. This responsiveness to the changing opinion of the electorate is consistent with what has been observed when maps are drawn without partisan considerations; either by sampling or by bipartisan committees.

A more detailed analysis can be found in “Analysis of the Geopolitical Landscape for the 2020 North Carolina Congressional Districts.

The details of the distributions are given in the accompanying
document “Methods used to Analyze 2020 North Carolina State Congressional Redistricting Landscape.”

County Clustering: Looking towards the 2020 Census


The decennial redistricting cycle is an important moment in our democracy. Drawing districts can have a dramatic effect on how our elections are interpreted and the people’s ability to hold their representatives accountable.

Today, roughly 1 month before the 2020 Census is release, we and 3 other North Carolinians are releasing an analysis around the coming 2020 Census and how it will affect the county clusters used to draw the legislative districts for the N.C. State Legislature.

The study can be found HERE

For more information on what county clusters are and how they affect districting, see here and here.

We do not intend our analysis to have much predictive value.  A main takeaway from the study is that the clusters are very sensitive to fluctuations in the county populations; the official county populations will not be released until mid-august.  Nonetheless, it is clear that the current county clusters will change in the coming redistricting cycle and that there will likely be multiple options for which clusters we implement. We hope that our note will inform public discussion around this process.

Quoting from the study, the main takeaways are:

    1. We expect that county clusters based on the 2020 Census population will be very different from those used in the last decade.
    2. There is significant variation in the county clusters across the different population estimates for 2020. Hence, caution should be exercised when drawing conclusions from the specifics of the maps we have included. We note the few clusters which seem to be stable across the population estimates.
    3. Using the 2010 population figures we found 2 possible clusterings for the N.C. Senate. The number of clusterings for the 2020 population estimates ranged from 12 to 33.
    4. Using the 2010 population figures we found 4 possible clusterings for the N.C. House. The number of clusterings for the 2020 population estimates ranged from 2  to 16.

Some members of the study team will be releasing further commentary on what we have seen here during the coming month.  We expect to release a similar discussion once the official 2020 Census population figures are released.  Please watch this site and the following venues for further commentary:

Jonathan Mattingly
Gregory Herschlag

Guidance on the Ensemble Method for Analysing Redistricting Plans

[TLDR : See this PDF for guidance:  Ensemble Method Guidance ]

As the country gears up for the next redistricting cycle we have attempted to collect our thoughts and guidance on using the ensemble method to critique a particular redistricting plan. This summarizes our current thinking stretching back to 2013-2014 as one of the originators of the Ensemble Method; it reflects our experience gained through providing expert guidance, including live testimony, to the court in a number of cases.

The document begins with the following summary:

The ensemble of maps translates the specific redistricting design criteria into an understanding of what a redistricting plan should look like if the specified criteria are followed without ulterior motive. The criteria are formalized in a distribution on redistricting plans from which a sufficiently rich ensemble is drawn to capture the properties of the distribution. If the criteria are non-partisan then the ensemble gives a non-partisan normative collection against which other plans can be compared for their partisan and non-partisan properties.

The rest of the text can be found in this PDF document: Ensemble Method Guidance.

Court Chooses to Stick with Relatively Unresponsive Maps

Today the state court chose to accept the Remedial Map which the legislature produced in response to the maps we have been using since 2016 being declared an illegal gerrymander in Harper v. Lewis.

Sadly this new map still has districts containing an abnormally large number of Democrats and others with comparably few. The result is a map which elects the same number of Democrats and Republicans over an abnormally large range of partisan election outcomes.

My analysis shows that the Remedial Map was much less sensitive to swings in the partisan vote fractions than the vast majority of the maps in the ensemble. The plots below, where the Remedial Plan is labeled HB 1029,  show that under a uniform swing analysis the nonpartisan maps in the ensemble often produce 6 and sometimes 7 Democratic seats in election environments when the Democrats perform well (a statewide vote fraction in the low 50%) for many sets of votes, while the 2019 Remedial Map reliably produces 5 Democratic seats in most instances. Each of the different plots uses a different set of historical votes (Labels: USS16- US Senate 2016, AG16 – Attorney General 2016, USH16 – US House 2016, GOV16 – Governor 2016).

The important feature is the atypically large jump between the 6th and 5th most Democratic district. This is the same “signature of gerrymandering” jump which we saw in Common Cause v. Rucho.  It makes the maps much less responsive to the  votes cast. They produce the same results over a large number of election scenarios.

The fact that the court felt it did not have time to investigate this fully shows how the judicial pathway of preventing gerrymandering is limited. We need legislative reform. It seems the only way to ensure that the maps used in elections are responsive to shifts in the political sentiment of the electorate.

More details can be found in the report I submitted to the court which can be found here: Mattingly Nov. 26 Declaration.

Jonathan Mattingly

Quantifying Gerrymandering in North Carolina – Revisited

We have revised our work quantifying gerrymandering in the N.C. Congressional Districts. The original work was presented in Common Cause  v Rucho. We have taken the opportunity to  use the improved code base which performs more sophisticated  convergence test and update our discussion to better reflect our current perspective. We have also corrected a few inaccuracies and small errors. That being said, the results are fully in line with those from the original version.

The new text can be found HERE.

A collection of related animations can be found “The Animated Firewall.” Two related discussions can be in It’s Not about Proportional Representation and The Fix is in: The votes don’t matter.

The data and code for this work can be found in this git repository:


Majority and Supermajority Firewalls in the N.C. General Assembly

[Note: there is a known bug between WordPress and Safari; if the videos below do not render on Safari try switching to a different browser]

In our analysis of the 2017 North Carolina  General Assembly redistricting plan, one of our central findings was that the NC Legislature’s  2017 Redistricting Plan implement a firewall protecting Republican majorities and supermajorities.

In trial, for the House districts, we showed animated bar-graphs that demonstrated how the range of democratic seat counts shifted with the statewide fraction of Democratic votes, under various shifts to historical elections.  For example, under the United States Senate vote in 2016, the enacted plan elects a typical number of democrats when compared to the ensemble when the statewide Democratic vote fraction is below 49%.  As the Democratic vote fraction rises to roughly 50.5% to over 52%, nearly all plans in the ensemble break the Republican supermajority, but the enacted plan remains stuck electing fewer than 48 Democrats to the state House.  As the Democratic vote fraction continues to rise, the enacted plan consistently elects fewer Democrats than the ensemble; at a Democratic vote fraction of 54.5%, nearly all plans in the ensemble predict a Democratic majority in the House, yet the ensemble retains a Republican majority.  The Republicans retain their majority in the enacted plan even when the Democratic vote fraction surpasses 55%; plans in the ensemble yield a strong majority to the democrats at this point.


The story is NOT about proportional representation, rather about how the #ncga 2017 maps (represented by arrow) systematically under-elect Democrats to a shocking degree.

The story is similar when using vote counts from the 2012 presidential race. Notice, in both videos, as the Democratic vote fraction rises to break the Republican supermajority in the ensemble, the enacted plan dramatically remains to the left of the majority/supermajority line.


And as we examine more elections, the story still remains the same.  With Commissioner of Insurance votes from 2012, notice how Democrats are systematically under-elected by the #ncga 2017 Redistricting Plan when compared to our ensemble of thousands and thousands of non-partisan maps


Now with 2008 United States Senate votes. Getting worried about our democracy yet? Across many different vote patterns, same exceptionally atypical under-electing of Democrats persists. The effect is very robust across different offices, across different years.


Same story, now using votes from 2012 Governor election. Each election has a different spatial vote pattern, yet story persists. 2017 #ncga map under-elects Dems when they would typically gain more power. Notice map keeps a Republican majority even when some maps give Democrats a supermajority.


Finally, using 2016 Lt. Governor votes. Again same story.  The ensemble accounts for natural packing and the voting geography of North Carolina.  But natural packing is not enough to explain the enacted plan’s extreme Republican bias.  By comparing the enacted plan to the ensemble of non partisan maps, we separate the effects of natural packing from partisan gerrymandering.


Although the above maps will be remedied by the decision of Lewis v. Common Cause, many states are still highly vulnerable to the effects and consequences of partisan gerrymandering with no hope for remedy from the federal courts.  Map makers have inserted themselves in the electoral process and suppressed the Will of the People.  If you are worried about the state of our democracy, you should be.

Jonathan Mattingly
Gregory Herschlag

Optimal County Clustering in NC

North Carolina’s constitution requires that state legislative districts should not split counties. However, counties must be split to comply with the “one person, one vote” mandate of the U.S. Supreme Court. Given that counties must be split, the North Carolina legislature and courts have provided guidelines that seek to reduce counties split across districts while also complying with the “one person, one vote” criteria. Under these guidelines, the counties are separated into clusters. For a great explainer about the County Clustering problem see this blog entry on Districks.

A group of high school students from the NC School of Science and Mathematics worked with us over the last academic year and summer to develop computer algorithms to optimally cluster counties according to the guidelines set by the court in 2015. We recently released an article that presents the algorithm along with publicly accessible code which anyone can use.

Additionally, in our article, we use this to investigate the optimality and uniqueness of the enacted clusters under the 2017 redistricting process. We verify that the enacted clusters are optimal, but find other optimal choices. We emphasize that the tool we provide lists all possible optimal county clusterings. We also explore the stability of clustering under changing statewide populations and project what the county clusters may look like in the next redistricting cycle beginning in 2020/2021.

The posting to the ArXiv can be found here.
The article is now published and can be found here.
The code is referenced in the ArXiv post, and may also be accessed here.

[Edits: Added like to Districks  explainer (9/4/2019)]