One of the strengths of sampling the space of redistricting plans is that it makes no assumptions about the relationship between the statewide votes cast and the seats won. The collection of plans do not consider proportionality between seats and votes or symmetry in electoral outcomes.
These methods reveal the expected election results under our election system; they assume only the redistricting criteria (usually non-partisan). The sampled maps provide a null hypothesis. Any enacted map can be compared to this null hypothesis to understand the extent of any possible gerrymandering.
The collection of maps, and the expected election outcome they reveal, automatically includes the effect of cities, the distribution of voters and the shape of the region in a simple and principled way. We have looked at North Carolina, Wisconsin and more informally Maryland and Pennsylvania. We have never seen proportionality between the votes and typical seats given by an ensemble.
For example, consider the election of the US House of representatives for North Carolina and the election of the General Assembly of delegates for Wisconsin. In the first, we compare the 2012 enacted plan against the ensemble using the 2012 Congressional votes; in the second, we compare the 2011 enacted Wisconsin plan against the ensemble using the 2014 General Assembly votes.
NC House 2012
Vote
Seats
Democratic
50.65%
4 (31%)
Republican
48.80%
9 (69%)
Votes and Seats for N.C. House in 2012 using 2012 enacted map.
Wisconsin 2014
Votes
Seats
Democratic
51.28%
36 (36%)
Republican
48.72%
63 (64%)
Votes and Seats for Wisconsin General Assembly using votes and enacted plan from 2014.
Clearly neither seat outcome of the enacted plan would be call proportional representation. However, the large ensemble of non-partisan maps reveals that the results in Wisconsin are, in fact, typical for this set of votes whereas the North Carolina outcome is not.
Although the WI enacted plan is representative for this particular set of votes, when the Democratic vote fraction drops below 50%, the enacted plan acts as an extreme outlier producing highly atypical results. (This effect is well understood: See Firewall and The Signature of Gerrymandering.) This effect is shown below.
Enacted plan is shown with read dot. Histogram of results from ensemble using different sets of votes. WSA= WI State Assemble, GOV= Governor , SOS=Secretary of State, USS= US Senate, PRE=President). Proportionality shown with dashed line.
Notice that the trend shown by the ensemble histograms in blue do not follow the vote-seat proportionality line marked on the plot. However the Enacted map is still shown to be an outlier as the vote fraction drops below 50%. It shows a “baked in” result which is do not move as the people’s will expressed in their votes changes.
This “baked in” election outcome is even more pronounced in N.C. where the enacted maps (NC 2012 & NC 2016) produce the same result over a large range of elections. These elections have varying statewide vote fractions; the ensemble shows that the number of elected Democrats change as the elections change, whereas the enacted plans nearly always give the same result.
Collection of histograms plotted in blue for various elections (PRE=president, USH= US House, GOV=Governor). Three plans are show: 2012 Enacted (NC2012), 2016 Enacted (NC2016) and the Panel from the Beyond Gerrymandering Project (Judges) . Proportionality shown with dashed line.
Again the proportionally line on the plot does not track the shift in the blue histograms produced by the ensembles as the election used varies. The ensemble reveals the natural baseline without assuming proportionality.
When discussing gerrymandering, there is an intuitive drive to discuss how many seats were won by a given party, and how egregious this result may be. The measure of egregiousness may come from an assumed ideal of proportionality or symmetry, or may come from a comparison with an ensemble of alternative plans.
But as we have shown, the number of elected party officials in a gerrymandered plan may be entirely typical of the ensemble; it may, for example, only be when a party is in danger of losing the majority of seats that a plan becomes a typical. In this previous post, we have made efforts to visualize how changes in the Democratic or Republican statewide vote fraction change the number of elected Democrats and Republicans in both the ensemble and an enacted plan. In this post, we create an animation of this effect.
In North Carolina, state house Republican Representative David R. Lewis famously stated that the 2016 North Carolina congressional maps were drawn “to give a partisan advantage to 10 Republicans and three Democrats because I do not believe it’s possible to draw a map with 11 Republicans and two Democrats.” We ask how robust this effect is by considering the 2016 United States Congressional voting data in North Carolina and then using the uniform swing hypothesis to vary the statewide Democratic vote percentage from 42.5% to 57.5%. We render an animation that demonstrates how the number of elected Democrats changes in the ensemble and the 2016 enacted plan with respect to the statewide Democratic vote fraction.
When the statewide Democratic vote is between 42.5% and 52.25%, the enacted plan consistently elects 3 of 13 Democrats. From 47% to 53.25%, the enacted plan is an extreme outlier with respect to the ensemble. Even as the Democrats pick up a fourth seat in the enacted plan at 52.37%, nearly all plans in the ensemble elect 5 or more Democrats. In short, under this election structure and swing assumptions, the Democrats would need nearly 53% of the statewide vote in order to gain a number of seats that are even some what typical of the ensemble of plans.
We repeat this animation in the Wisconsin general assembly, examining the 2012 United States Senate vote in Wisconsin and using the uniform swing hypothesis to vary the Democratic statewide vote percentage from 44% to 56%.
The firewall is now animated: Again, when the statewide Democratic vote is less than 50%, the enacted plan and the ensemble lead to a Republican majority and the enacted plan is typical of the ensemble; the statewide Democratic vote rises to 51% , the enacted plan becomes atypical of the ensemble, however both the ensemble and the enacted plan still yield a Republican majority; as the Democratic vote fraction continues to increase, the enacted plan becomes more atypical yielding far fewer Democrats than is expected by the ensemble. In a large range the ensemble predicts the that the Democrats should expect to receive a majority, however they do not under the enacted plan; at the higher end, the Democrats begin to expect a supermajority from the ensemble, but they do not achieve this in the enacted plan.
This post continues the theme of localized analysis begun in the post Towards a Localized Analysis. The analysis presented in our articles and blog post such as The Signature of Gerrymandering , Firewalls, and Hearing the Will of the People are largely based on a global, statewide analyses. In many ways, Gerrymandering is a global phenomenon as one can not change one district with out effecting its neighbors which often causes a cascade of changes across a state. Packing a particular group in one district is done to dilute the groups effect in other districts.
The most basic measure of a districting plan’s character is the partisan make up of its elected representatives. When comparing a plan’s partisan make up with a set of comparison maps, we obtain a view that is fundamentally global in measuring gerrymandering. Marginal box-plots can be used to identify particular districts as unusual. This is a step towards a more localized analysis. As described in The Signature of Gerrymandering post the marginal box-plots of the ranked vote curve were used to identify specific districts, in the legislatures 2016 maps, which were arguably cracked and packed in court case Common Cause v. Rucho concerning the 2016 NC Congressional maps.
In the Marginal box-plots, a particular set of votes is used to order the districts from most to least Republican in the ensemble of districting plans; to examine a particular district in a given plan, we first consider the district’s rank in the order (from most to least Republican) and then compare that district’s partisan make up with the partisan make up of the districts in other plans with the same rank in the order. Such analysis examines deviations in the the overall statewide structure of votes along with with where this deviation may occur, but it does not geographically constrain the districts with which to compare the district of interest.
We wish, however, to perform a more geographically localized analysis. Such analysis still contains the fundamental framework of examining outliers within the context of an ensemble of nonpartisan maps. (Outlier analysis using an ensemble of maps is generally discussed here. The basic idea of the localized analysis was previously presenter here.)
The Typical Local Political Environment of a Precinct
Localized analysis begins by choosing a particular precinct of interest and collecting the districts containing this precinct from an ensemble of state wide maps. The resulting collection of districts, all which contain the precinct of interest, can then be used to characterize the typical district-level political environment that voters in this precinct could expect to experience.
By contextualizing a precinct within the collection of districts containing it, we may relate the partisan preferences of the precinct’s voters to the preferences of their typical district. For example, a precinct’s voters may prefer one party, but typically find themselves in a district that votes for the opposing party; in this sense we may determine whether or not it is natural for the typical voter within a given precinct to be able to elect a candidate associated with the party they have voted for.
We contextualize the typical district containing a precinct by constructing a histogram over the ensemble of districts (this process was described in detail the previous post Towards a Local Analysis.) This distribution can then be used to deicide if the district containing the precinct in a given plan is atypical. If a precinct does find itself in an atypical district, then its residents may have cause to object. In this way, the ensemble of districts can be used in a normative way to identify outliers.
Extreme Outliers in the NC Congressional Races
Similarly to our previous blog post (Towards a Local Analysis) we again apply the above analysis the North Carolina Congressional Map from the 2016 and 2012 elections which we will abbreviate respectively NC2106 and NC2012. We will also consider the map produce by the retired judges from Beyond Gerrymandering Project lead by Tom Ross. This map will be denoted “Judges”. Previously we used a 5% outlier condition which many may classify as rare, but is more arguably extreme. We adapt the current analysis to examine more extreme outliers (those that occur less than 1% of the time, rather than 5%).
We begin by analyzing each precinct to determine if the district counting the precinct in the map of interest has an unusual partisan make up from that precincts perspective. Precincts which find themselves in a district whose partizan make up is in the in tails corresponding less than 1% of probability are labeled as extreme (at p=99%)
In the table below, we show the number of precincts which are extreme outliers from 2,692 predicts in different NC Congressional maps for the using the 2012 and 2016 US Congressional elections, denoted by USH12 and USH16 respectively. For comparison purposes, we then tabulate the number of maps in our ensemble which have that many or more precincts which are extreme outliers.
Map
2012 Votes
2016 Votes
# Outliers
# in ensemble
w/ more outliers
# Outliers
# in ensemble
w/ more outliers
NC2012
796
0 (0%)
655
0 (0%)
NC2016
154
336 (1.4%)
233
76 (0.31%)
Judges
2
15022 (61%)
3
13448 (55%)
The histograms below give the full histograms of the number of maps with different numbers of outlier precincts using the two sets of election data mentioned above.
Number of Outlier Precincts (USH12)
The spatial location of each of the outlier districts is show on the following maps. The first three us the US12 election data
The second three use the USH16 election data.
Precinct by Precinct Log Likelihood
Instead of just using the using the precinct localized distribution constructed above to flag districts individual as outliers, one can calculate the likelihood of partizan vote fraction being as far or farther in the tail then that each precinct in a given map. Averaging this across the state gives a measure of how typical the precinct by precinct are across the whole map. To contextualize this average spatial log likelihood calculate the value for each map in our ensemble. The following two histograms summarize the results.
Again to show the spatial structure better, we plot the log likelihood across the state, precinct by precinct. So that the direction of the swing is visible, we label democratic with positive values and republican with negative.
The above maps use USH12 election results while the maps below use UH16 election results.
Up until now, this blog has investigated whether gerrymandering has occurred. In this post we begin to investigate where gerrymandering has occurred. The question of ‘where’ is interesting for both scientific and legal reasons. Scientifically, one may want to determine which precincts were atypically manipulated to achieve a political goal. Legally, an argument for individual harm is needed to pursue suits based on 14th amendment claims: Whitford vs Gill was decided on standing and dismissed because the plaintiffs did not establish that they had been individually harmed by the redistricting process.
In this post we explore the merits of a particular measure of localized gerrymandering. The idea is a simple one and continues to rely on the ensemble analysis that we have already employed. Any given precinct will always lie within some district; given historical vote counts this district will have a democratic vote fraction. For a given precinct we can construct a histogram of all democratic vote fractions over all maps within the ensemble for a fixed historical election. Then, given some reference map, we can ask how atypical the precinct’s observed margin is, given this distribution. If a precinct lies within a district that has a much higher democratic vote fraction than expected, we will color it blue; if it lies in a district that has a much higher republican vote fraction than expected, we will color it red. We color based on the log-likelihood of of the cumulative distribution functions (e.g. more red means there is a low probability that the likelihood of finding a more Republican district is small). We first give an example distribution of the vote fractions of a precinct’s districts along with how we color them
A histogram of vote fractions over a collection of districts all containing a given precinct.
Using the votes cast in the 2016 congressional election, we color the district maps for NC2012, NC2016 and the Judges plan, respectively:
A new Motion to Affirm was filed recently discussing, among other things, the Quantifying Gerrymandering group’s work in the case. It particularly highlighted the box plot showing the “Signature of Gerrymandering” and how the analysis indicated that some districts were packed while other cracked.
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”
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).
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”
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.
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.
