Redistricting: Drawing the Line
Sachet Bangia, Christy Vaughn Graves, Gregory Herschlag, Han Sung Kang, Justin Luo, Jonathan C. Mattingly, Robert Ravier
RREDISTRICTING: DRAWING THE LINE
SACHET BANGIA, CHRISTY VAUGHN GRAVES, GREGORY HERSCHLAG, HAN SUNG KANG,JUSTIN LUO, JONATHAN C. MATTINGLY, AND ROBERT RAVIER
Abstract.
We develop methods to evaluate whether a political districting accurately representsthe will of the people. To explore and showcase our ideas, we concentrate on the congressional dis-tricts for the U.S. House of Representatives and use the state of North Carolina and its redistrictingssince the 2010 census. Using a Monte Carlo algorithm, we randomly generate over 24,000 redis-trictings that are non-partisan and adhere to criteria from proposed legislation. Applying historicalvoting data to these random redistrictings, we find that the number of democratic and republicanrepresentatives elected varies drastically depending on how districts are drawn. Some results aremore common, and we gain a clear range of expected election outcomes. Using the statistics ofour generated redistrictings, we critique the particular congressional districtings used in the 2012and 2016 NC elections as well as a districting proposed by a bipartisan redistricting commission.We find that the 2012 and 2016 districtings are highly atypical and not representative of the willof the people. On the other hand, our results indicate that a plan produced by a bipartisan panelof retired judges is highly typical and representative. Since our analyses are based on an ensembleof reasonable redistrictings of North Carolina, they provide a baseline for a given election whichincorporates the geometry of the state’s population distribution. The Will of the People
Democracy is typically equated with expressing the will of the people through government.Perceived failures of democracy in representative governments are usually attributed to the voiceof the people being muted and obstructed by the actions of special interests or the sheer size ofgovernment. The underlying assumption is that the will of the people exists as a clear, well definedvoice which only needs to be better heard. Yet the will of the people is not monolithic. It is notalways so simple to obtain a consensus or even a clear majority opinion. We rely on our electionsas a proxy to express our collective opinions and our political will, which leads to the question, howeffective a given election is at capturing this will?In the United States, district representation schemes are used to divide the population into dis-tinct groups, each of which carries a certain amount of representation. This districting acknowledgesthat the people’s voice is geographically diverse and that we value the expression of that diversity inour government. We take election results to represent the people’s will, giving the elected officials amandate to act in the people’s name. Hence, it is reasonable to ask if and how this representationis affected by the choice of district boundaries. Just how sensitive are election results, and byextension, our impression of the people’s will, to our choices for geographic divisions? The methodwe use to reveal this will is simple. We take the actual votes cast by North Carolinians at the 2012and 2016 congressional elections and then change the boundaries of the congressional districts tosee how the partisan results of the elections change. Our results show that the will of the people isnot a single election outcome but rather a distribution of possible outcomes. The exact same votecounts can lead to drastically different outcomes depending on the choice of districts.compMany discussions of fraudulent elections emphasize voter suppression or voter fraud. How-ever if the results of an election vary so widely over different choices of redistrictings, then it isparamount that the districts employed produce results that are an accurate proxy of the “will ofthe people.” The courts have given some guidance in this direction by promoting the “one person, a r X i v : . [ s t a t . A P ] M a y ne vote” standard, requiring districts have roughly an equal number of eligible voters so thateach elected representative is a proxy for the same number of constituents. We will demonstratethat the commonly promoted criteria for the construction of redistricting (equal population appor-tionment, geographical compactness, and preservation of historical constituencies) still leave a lotof variability in the election results for a given set of precinct level votes. Given this variability, wecan ask if a given redistricting leads to a common and expected outcome, or if it gives one partyan unlikely advantage.In the 2012 congressional elections, which were based on the 2010 redistricting, four out ofthe thirteen congressional seats were filled by Democrats. However, in seeming contradiction, themajority of votes were cast for Democratic candidates on the statewide level. The election resultshinged on the geographic positioning of congressional districts. While this outcome is clearly theresult of politically drawn districts, perhaps it is not the result of excessive tampering. Our countryhas a long history of balancing the rights of urban areas with high population with those of morerural, less populated areas. Our federalist and electoral structures enshrined the idea that majorityrule must be balanced with regionalism. It might be that in North Carolina, the subversion of theresults of the global vote count would happen in any redistricting which balances the representationof the urban with the rural or the beach with the mountains, and each with the Piedmont. Maybethe vast majority of reasonable districts which one might draw would have these issues due to thegeography of the population’s distribution. We are left asking the basic question: how much doesthe outcome depend on the choice of districts? This can be further refined by asking “what are theoutcomes for a typical choice of districts,” or “when should a redistricting be considered outsidethe norm?” These last two refinements require some way of quantifying what the typical outcomesare for a given set of votes. We therefore set the vote counts based on historical data and ask howchanging the districts these votes were counted in leads to different results. Since we will explorethese questions in the context of the American political system, we will assume that people votefor parties, not people. In these polarized times this is a reasonable approximation and we find theresults extremely illuminating.In order to change the district boundaries, we use a Markov Chain Monte Carlo algorithm toproduce about 24,000 random but reasonable redistrictings. The redistrictings are constructedusing non-partisan design criteria from a proposed piece of legislation. We then re-tally the actualhistoric votes from the 2012 or 2016 elections to produce about 24,000 election outcomes, one foreach of our generated redistrictings. We observe that the number of representatives elected from aparty can vary drastically depending on the redistricting used, yet some outcomes are more frequentthan others.Once we understand the extent to which election results can vary over a collection of possibleredistrictings, we quantify how representative a particular redistricting is by observing its placein this collection of results. Similarly, with statistics of typical redistrictings in hand, we devisemeasures of gerrymandering where the effects of packing (concentrating voters to lower their po-litical power) and cracking (fragmenting voting blocks to lower their political power) can be betteridentified. Since all of our analysis is based on the interaction of actual votes with the collectionof over 24,000 reasonable redistrictings, it provides a baseline for the election which is informed byboth the geometry of the state and the distribution of the electorate though out the state. It is animportant feature of our analysis that our techniques incorporate the effect of the state’s geometrywhen developing the baseline.We apply our metrics to analyze and critique the North Carolina U.S. Congressional redistrictingsused in the 2012 and 2016 elections, as well as the redistrictings developed by a bipartisan group of Wesberry v. Sanders (1964) In [3], it was shown that redistrictings may favor a party simply due to the geography of the state. The algorithm is similar to the ones presented in [16, 1, 9]. etired Judges as part of the “Beyond Gerrymandering” project spearheaded by Thomas Ross. Werefer to these redistrictings of interest as NC2012, NC2016, and Judges respectively. See Figures 20–22 in the Appendix for visualizations of these redistrictings. Our analysis uses the actual votes castin the 2012 and 2016 N.C. congressional elections to illuminate the structure and features of aredistricting.Using a related methodology, we also assess the degree to which the three redistrictings (NC2012,NC2016, and Judges) are engineered. This is done by seeing how close their properties are to thecollection of redistrictings that can be obtained by small changes. It seems reasonable that thecharacter of an election should not be overly sensitive to small changes in the redistricting if theconcept of the “will of the people” is to have any meaning.The results of our analysis repeatedly show that the NC2012 and NC2016 redistrictings areheavily engineered and produce results that are extremely atypical and at odds with the will ofthe people. Finer analysis clearly shows that the Democratic voters are clearly packed into a fewdistricts, decreasing their power, while Republican voters are spread more evenly, thus increasingtheir power. In contrast, election results from the Judges redistricting are quite typical, producingresults consistent with what is typically seen. We emphasize that all of these conclusions come fromasking what the typical character and result of an election is if we use a “reasonable” redistrictingadhering to proposed legislation and drawn at random without any partisan input, save the possibleeffect of ensuring a few districts contain a sufficient minority population to comply with the VotingRights Act (VRA). 2. Main Results: Where do you draw the line?
We emphasize from the start that in contrast to some works (see, for example, we are not propos-ing an automated method of creating redistrictings to be used in practice. Rather, we are proposinga class of ideas for evaluating whether a redistricting is truly representative or gerrymandered. Wehope this helps draw the line between fair and biased redistricting so that the will of the peoplecan be better heard.Our analysis begins by generating over 24,000 “reasonable” redistrictings of North Carolina intothirteen U.S. House congressional districts. For each redistricting, we tabulate the votes from aprevious election, either 2012 or 2016, to calculate the number of representatives elected from boththe Democratic and Republican Parties. We emphasize that we use the actual votes from eitherthe 2012 or 2016 U.S. House of Representative elections. In using these votes, we assume that avote cast for a Republican or Democrat remains so even when district boundaries are shifted.By “reasonable,” we mean districts which are drawn in a nonpartisan fashion, guided only bythe desire to: • Divide the state population evenly between the thirteen districts. • Keep the districts geographically connected and compact. • Refrain from splitting counties as much as possible. • Ensure that African-American voters are sufficiently concentrated in two districts to givethem a reasonable chance to affect the winner.The precise meaning of “reasonable” is given in Section 3, along with the method we used togenerate the over 24,000 “reasonable” redistrictings. We construct our districts by taking VotingTabulation Districts (VTD) from NC2012 as the fundamental atomic element used as our buildingblocks. North Carolina is composed of over 2,600 VTDs.The first criterion above enforces the “one-person-one-vote” doctrine, which dictates that eachrepresentative should represent a roughly equal number of people. The second criterion reflectsthe desire to have districts represent regional interests. The third criterion embodies the ideathat districts should not fracture historical political constituencies if possible; counties provide a onvenient surrogate for these constituencies. The last criterion, which is dictated by the VotingRights Act (VRA), asks that two districts have enough African-American voters that they mightbe reasonably expected to choose the winner in that district. In particular, we emphasize that novoting or registration information is used, nor is any demographic information except for what isdictated by the VRA.The exact choice of these criteria for our study comes from House Bill 92 (HB92) of the NorthCarolina General Assembly, which passed the House during the 2015 legislative session. This billproposed establishing a bipartisan commission guided solely by these principles to create redistrict-ings. Since the companion legislation did not pass the North Carolina Senate, the provision neverbecame law. In fact, it is just the latest in a chain of bills which have been introduced over theyears with similar criteria and aims.2.1. Beyond One-Person-One-Vote.
There is a large amount of variation in the outcome of anelection depending on the districts used. The simple criteria from HB92 are not enough to producea single preferred outcome of the elections. Rather, there is a distribution of possible outcomes. Ourfindings in this direction, summarized in Figure 1, clearly show that the results generated by theredistrictings NC2012 and NC2016 are extremely biased towards the Republicans, while the Judgesredistricting produces acceptably representative results. The NC2012 and NC2016 redistrictingsproduce results that are highly atypical of the non-partisan redistrictings we have randomly drawnaccording to HB92.
JudgesNC2016NC2012 F r a c t i on o f r e s u l t JudgesNC2016NC2012 F r a c t i on o f r e s u l t Figure 1.
Probability of a given number of Democratic wins among the 13 congressionalseats using votes from the 2012 election (left) and 2016 election (right).
Over 24,000 random, but reasonable, redistrictings were used to generate the probability dis-tributions shown in Figure 1. We emphasize that the two plots use the actual votes cast by theelectorate in the 2012 and 2016 Congressional elections, respectively, to determine the outcomesfor each redistricting. For the 2012 vote counts, the NC2012 and NC2016 redistrictings both resultin four Democratic seats, a result that occurs in less than 0.3% of our collection of over 24,000redistrictings. The Judges redistricting results in the election of six Democrats, which occurs inover 39% of redistrictings. For the 2016 vote counts, the NC2012 and NC2016 redistrictings resultsin three Democratic seats, a result that occurs in less than 0.7% of redistrictings. The Judgesredistricting results in the election of six Democrats, which occurs in 28% of redistrictings.2.2.
Measuring Representativeness and Gerrymandering.
While Figure 1 is already quitecompelling, it is useful to develop quantitative measures of how representative the results of a given lection are. Gerrymandering goes beyond just affecting the results; it also makes districts so safethat representatives are less responsive to the will of the people, as their legislative choices willunlikely effect the result of an election. To measure these effects, we propose two indices. The first,which we call the Gerrymandering Index, is based on the plots used to visualize gerrymanderingintroduced in Section 2.3. It quantifies how packed or depleted the collection of districts is relativeto what is expected from the ensemble of “reasonable” redistrictings we have created. The second,which we call the Representativeness Index, is the measure of how typical the election resultsproduced by the redistricting are in the context of what is seen in the ensemble of “reasonable”redistrictings. It is based on the refinement of Figure 1 given in Figure 11 and described inSection 6.2. Later in Section 2.5, we consider a third index, the Efficiency Gap, which has recentlybeen employed in the decision Whitford Op. and Order, Dkt. 166, Nov. 21, 2016.In summary, in this section we consider • Gerrymandering Index:
Measures the degree to which the percentage of Democraticvotes in each district deviates from what is typically seen in our collection of “reasonable”redistrictings. The squareroot of the sum of the square deviations is the index. Relativelylarge scores are less balanced than the bulk of the “reasonable” redistrictings in our ensem-ble. These large indexed redistrictings typically have some districts with many more votersfrom one party than is normally seen or generally have a higher percentage of one party inmany districts than is normal, or both. How the term “normal” is understood is partiallyexplained in Section 2.3 and completely explained in Section 6.1. • Representativeness Index:
Measures how typical the results obtained by a given redis-tricting are in the context of the collection of “reasonable” redistrictings we have generated.Redistrictings with relatively large values produced an election outcome which is fartherfrom the typical election outcome in the collection of “reasonable” redistrictings. Detailsare given in Section 6.2.Both of these indices are adapted to the geometry of the votes and population density of the stateas reveiled by the ensamble of “reasonable” redistrictings. In this sense, we expect them to be morenuanced than other metrics which are not informed by the local structure of the state.As these indices are most useful when values for different redistrictings are compared, we placeeach redistricting of interest on the plot of the complementary cumulative distribution function foreach of the three above measures. This allows us to judge the relative size of each index in thecontext of our collection of “reasonable” redistrictings.In a complementary cumulative distribution function, the vertical axis shows the fraction ofrandom redistrictings which have a larger index value than a redistricting with a given index on thehorizontal axis. We plot results for the Gerrymandering Index and the Representativeness Index inFigures 2 and 3, respectively. We calculate the probability of each index obtaining a value greaterthan a given value based on our random redistrictings. We then situate each of our redistrictings ofinterest (NC2012, NC2016, and Judges) on the plot indicating the fraction of random redistrictingswhich have a larger index.Figures 2 and 3, show that the NC2012 and NC2016 redistrictings are quite atypical in both theGerrymandering Index and Representative Index, regardless if the votes from 2012 or 2016 are usedin the analysis. None of the over 24,000 reasonable redistrictings constructed had a GerrymanderingIndex bigger than NC2012, regardless whether 2012 or 2016 votes were used. Similarly, none ofthe reasonable redistrictings had a Representativeness Index greater than NC2012 when the 2012votes are used and only 172 (or 0.7%) had a greater Representativeness Index when the 2016votes are used. Again, none of the reasonable redistrictings had a Gerrymandering Index biggerthan NC2016 under both the 2012 and 2016 votes. Only 34 redistrictings (or 0.14%) and 105 C2012NC2016Judges F r a c t i on w / w o r s e i nde x NC2012NC2016Judges F r a c t i on w / w o r s e i nde x Figure 2.
Gerrymandering Index for the three districts of interest based on the congres-sional voting data from 2012 (left) and 2016 (right). No generated redistrictings had aGerrymandering Index higher than either the NC2012 or the NC2016 redistrictings. TheJudges redistricting plan was less gerrymandered than over 75% of the random districts inboth sets of voting data, meaning that it is an exceptionally non-gerrymandered redistrictingplan. redistrictings (or 0.43%) had a Representativeness Index greater than NC2016 under the 2012 and2016 votes, respectively.In stark contrast, 18,670 redistrictings (or 76.15%) and 18,891 redistrictings (or 77.05%) hadlarger Gerrymandering Index than the Judges plan under the 2012 and 2016 votes, respectively. And7,250 redistrictings (or 29.57%) and 7,625 redistrictings (or 31.1%) had larger RepresentativenessIndex than the Judges under the 2012 and 2016 votes, respectively.Our indicies indicate that the Judges plan is a very typical plan. It has a comparatively lowlevel of gerrymandering and seems to represent the will of the people. The NC2012 and NC2016are partially unrepresentative and have a high level of gerrymandering in terms of both indices.2.3.
Visualizing Gerrymandering.
While the reductive power of a single number can be quitecompelling, we have also developed a simple graphical representation to summarize the propertiesof a given redistricting relative to the collection of referenced redistrictings. The goal was to createa graphical representation which would make visible when a particular redistricting packed orfractured voters from a particular party to reduce its political power.One first needs to begin by discovering the natural structure of the geographical distribution ofvotes in the state when viewed through the lens of varying over “reasonable” redistrictings. Webegin by ordering the thirteen congressional districts which make up a redistricting from lowest tohighest based on the percentage of Democratic votes in each district. Since there are essentiallyonly two parties, nothing would change if we instead considered the percentage of Republican votes.We are interested in the random distribution of this thirteen dimensional vector. Since it is diffi-cult to visualize such a high dimensional distribution, we summarize the distribution by consideringthe marginal distribution of each position in this vector and summarize it in a classical box-plotfor each component of the thirteen dimensional vector in Figure 4. That is to say, we examinethe distribution of votes that make up the percentage of Democratic votes in the most Republicandistrict. Then we repeat the process for the second most Republican district. Continuing for eachof the rankings, we obtain thirteen box-plots which we arrange horizontally on the same plot.The box-plots are standard, meaning that within each box-plot the central line gives the medianpercentage, while the ends of the box give the location of the upper quartile and the lower quartile C2012NC2016Judges F r a c t i on w / w o r s e i nde x NC2012NC2016Judges F r a c t i on w / w o r s e i nde x Figure 3.
Representativeness Index for the three districts of interest using congressionalvoting data from 2012 (left) and 2016 (right). No redistrictings was less representative thanthe NC2012 nor NC2016 redistricting plans. Roughly 30% of redistricting plans were lessrepresentative than the Judges redistricting plan in both sets of voting data, meaning thatthe Judges plan was reasonably representative.
NC2012NC2016Judges D e m o c r a t i c v o t e f r a c t i on NC2012NC2016Judges D e m o c r a t i c v o t e f r a c t i on Figure 4.
After ordering districts from most Republican to most Democrat, these box-plots summarize the distribution of the percentage of Democratic votes for the district ineach ranking position for the votes from 2012 (left) and 2016 (right). We compare ourstatistical results with the three redistricting plans of interest. The Judges plan seemsto be typical while NC2012 and NC2016 have more Democrats than typical in the mostDemocratic districts and fewer in districts which are often switching between Democrat andRepublican, depending on the redistricting. (25% of the results exist below and above these lines). The outer bracketing line defines an intervalcontaining either the maximum and minimum values, or three halves the distance of the quantilesfrom the mean, whichever is smaller. In the interest of visual clarity, we have not plotted anyoutliers. On top of these box-plots, we have overlaid the percentages for the NC2012, NC2016, andthe Judges redistricting. In Figure 5, we also include plots that displays histograms rather than ox-plots. These plots are richer in detail. Yet, the detail makes it harder to estimate confidenceintervals. NC2012NC2016Judges D e m o c r a t i c v o t e f r a c t i on NC2012NC2016Judges D e m o c r a t i c v o t e f r a c t i on Figure 5.
We present the same data as in Figure 4, but, with histograms replacing thebox-plots. Note that the distribution of the sixth most Republican district (district withlabel 6 on the plots) is quite peaked in both the 2012 and 2016 votes, the Judges results arecentered directly on this peak while the NC2012 and NC2016 lie well outside the extent ofthe distribution.
The structure of these plots is shaped by the typical structure of the redistrictings in our ensembleand by extension, the spatial-political structure of the voters in North Carolina. It can then beused to reveal the structure of our three redistrictings of interest. Observe that for both the 2012and 2016 votes, the centers of the box-plots form a relatively straight, gradually increasing linefrom the most Republican district (labeled 1) to the most Democratic (labeled 13). The Judgesdistricts mirror this structure. Furthermore, most of the percentages from the Judges districts fallinside the box on the box-plot which marks the central 50% of the distribution. The NC2012 andNC2016 have a different structure. There is a large jump between the tenth and eleventh mostRepublican district (those with labels 10 and 11, respectively). In the NC2012 redistricting, thefifth through tenth most Republican districts have more Republicans than one would typically seein our ensemble of “reasonable” redistrictings. In the NC2016 redistricting, the shifting starts withthe sixth most Republican district and runs through the tenth most Republican district (labeled6-10). In both cases, the votes removed from the central districts have largely been added tothe three most Democratic districts (labeled 11-13). In the 2012 votes, this moved three to fourdistricts that typically would have been above the 50% line to below the 50% line, meaning thatthese districts elected the Republican rather than the Democrat. With the 2016 votes, the changesin structure only moved the tenth most Republican district across the 50% line. The geographicdistricts associated with these rankings are given in Table 1 from Section 7.2.Forgetting about the election outcomes, the structure has implications for the competitivenessof districts and likely political polarization. Rather than a gradual increase at a constant rate fromleft to right as the Judges redistricting and the ensemble of box-plots, the NC2012 and NC2016redistrictings have significantly more Democrats in the three most Democratic districts and fairlysafe Republican majorities in the first eight most Republican districts. There are establishedhypotheses that claim safe districts lead to a polarized legislative delegation with fewer centristrepresentatives on both sides of the political spectrum.Figure 4 can be used to motivate and explain the Gerrymandering Index for our redistrictings ofinterest. For example, to calculate the Gerrymandering Index for NC2012, one sums the square of he distance from the red dots to the mean in each distribution from 1 to 13. The GerrymanderingIndex is the square root of this sum. To aid with visualization, recall that the line though thecenter of each box is the median which, in these cases, is close to the mean. Clearly, this indexcaptures some of the features of Figure 4 discussed in the previous paragraph.It is remarkable how stable the structures in Figure 4 are across the 2012 and 2016 votes. The2016 plot is largely a downward shift of the 2012 plot. This stability largely explains why the twoplots in Figure 2 of the Gerrymandering Index look so similar. It also speaks to the stability ofthe Representativeness Index in Figure 3. The efficiency gap (introduced in Section 2.5) does notseem to share this stability as it changes both its values and probabilities in Figure 7 across thetwo elections.2.4. Stability of Election Results.
It would be unsettling to think that relatively small changesin a redistricting would dramatically change the results of an election or the properties of theredistricting. If this were true, it would call into question the legitimacy of the election resultsderived from the redistricting. In particular it might lead one to question the extent to which anelection captures the intent of the votes.To examine this question, we explored the degree to which the NC2012, NC2016 and Judgesredistrictings are representative of the nearby redistrictings, where we interpret nearby to meanthat roughly 10% of the VTDs are swapped between districts. (See the next paragraph for moreprecise description.) By switching nearby VTDs among districts we are able to assess whether smallchanges impact the characteristics of the districts or not. We found that the districts within theNC2012 and NC2016 redistricting plans had a Gerrymandering Index which was significantly largerthan the nearby redistrictings while the Judges plan had a Gerrymandering Index which was in themiddle of the range produced by nearby redistrictings. In other words, switching nearby districtsmade the NC2012 and NC 2016 redistrictings less partisan but did not change the characteristicsof the Judges redistricting. This suggests that the NC 2012 and NC2016 redistricting, in contrastto the Judges redistricting, were precisely engineered and tuned to achieve a partisan goal and thatthe components of the NC2012 and NC 2016 redistrictings were not randomly chosen.
NC2012 F r a c t i on w / w o r s e i nde x NC2016 F r a c t i on w / w o r s e i nde x Judges F r a c t i on w / w o r s e i nde x Figure 6.
Gerrymandering Index based on random samples drawn from nearby the threeredistrictings of interest: NC2012 (left), NC2016 (center), and Judges (right). Only for theJudges are the other points in the neighborhood similar to the redistricting of interest. Allplots use the 2012 votes.
More precisely, we randomly sample “reasonable” redistrictings which are near the NC2012,NC2016, and Judges redistrictings in the sense that no single district differs by more VTDs thana set threshold from the redistricting understudy. To set the threshold, we observe that amongthe over 24,000 redistrictings we generated, the average district size is around 210 VTDs. In aparticular typical redistricting from our ensemble, the sizes roughly varied from 140 to 280 VTDs.With these numbers in mind, we set our threshold to be 40 VTDs. Since every VTD switched is ounted twice, once for the district it is leaving and once for the district it is entering, this amountsto a total of around 10% of the VTDs switching districts.Figure 6 shows the results of these analyses applied to the NC2012, NC2016, and Judges re-districtings. The redistrictings sampled around NC2012 have markedly better GerrymanderingIndices than NC2012 itself. The results are less dramatic for NC2016, but telling nonetheless. Thisshows that a randomly chosen redistricting near NC2012 (or respectively NC2016) has very differentproperties than NC2012 (or respectively NC2016). This is convincing evidence that the NC2012and NC2016 redistrictings were deliberately constructed to have unusual properties. It would havebeen unlikely to choose such a singularly unusual redistricting by chance. In contrast, the Judgesredistricting has a Gerrymandering Index which is quite typical of its nearby redistrictings. It isworse than around 50% of those nearby it and hence better than 50% of those nearby it. Thus, itis very representative of its nearby redistrictings.2.5. Efficiency Gap and correlation between indices.
The Efficiency Gap is a third type ofindex that was used in the decision Whitford Op. and Order, Dkt. 166, Nov. 21, 2016. It quantifiesthe difference of how many “wasted votes” each party cast; a larger number means that one partywasted more votes than another. More precisely, the Efficiency Gap is the difference of the wastedvotes for the Democrats and Republicans divided by the number of the total votes in the election(for both parties). The wasted votes for each party is the sum of the fraction of votes in districtsthe party loses plus the sum over the percentage points above 50% in the districts won. NC2012NC2016Judges F r a c t i on w / w o r s e i nde x NC2012NC2016Judges F r a c t i on w / w o r s e i nde x Figure 7.
Efficiency gap for the three districts of interest based on the congressional votingdata from 2012 (left) and 2016 (right). No random redistrictings had a greater disparityin voter efficiency than the NC2012 redistricting plan. Only about 0.3% of the randomredistrictings had a greater disparity in voter efficiency than the NC2016 redistricting inboth sets of voting data. Roughly 60% and 30% of the random redistricting plans had agreater disparity in voter efficiency than the Judges redistricting plan for the 2012 and 2016election votes, respectively.
As with the other indices we are most interested in where a particular Efficiency Gap scoresits relative to Efficiency Gap scores of our ensemble of 24,000 generated redistricts. In a graphcompletely analogous to the previous plots of complementary cumulative distribution functions,Figure 7 gives the fraction of the random ensemble with an Efficiency Gap greater than a givenscore. The original used actual votes, but when the population of each district is equal then the two measures areexactly equivalent. If the actual votes is close to equal then they are almost the same. t first glance, the Efficiency Gap seems to have some similarity with the GerrymanderingIndex. Both try to detect the packing and cracking of a particular political constituency. Whilethe Efficiency Gap concentrates on the winner of a given district by using a threshold of 50% todetermine what are excess votes, the Gerrymandering Index tries to develop the unique signatureof how a given collection of votes interacts with the geographic distribution of the votes.To understand how related the Gerrymandering and Representativeness Indices are to each otherand to the Efficiency Gap, we consider the Pearson product-moment correlation coefficient. Usingthe voting data of 2012, we find that the correlation between the Gerrymandering Index and theRepresentative Index is 0.466, meaning the two indices are moderately correlated. This suggeststhat gerrymandered redistrictings tend to be less representative and vice versa. We find that theGerrymandering Index and the Efficiency Gap are less positively correlated with a correlationcoefficient of 0.333. On the other hand, the Representative Index and the efficiency gap are morehighly correlated with a correlation coefficient of 0.752. Hence, despite superficial similarity, theGerrymandering Index and the Efficiency Gap contain the most distinct information while theEfficiency Gap and Representative Index are more related. Though we only have two sets of electiondata (namely 2012 and 2016), the Gerrymandering and Representativeness Indices complementaryCDF plots are much more stable across elections than the Efficiency Gap appears to be. One cannot conclude much from two examples, but it is suggestive that the Efficiency Gap may be a lessstable index.2.6. Summary of Main Results.
By sampling over 24,000 reasonable redistrictings, we explorethe distribution of different election outcomes by estimating the probabilities of the numbers ofDemocrats elected from North Carolina to the U.S. House of Representatives. Our sampling of rea-sonable redistrictings also allows us to estimate the distribution of winning margins in each districtas well as the value of three indices representing gerrymandering, representativeness, and relativevoter efficiency. In every one of our tests, we have found that the NC2012 and NC2016 redistrictingplans are extraordinarily anomalous, suggesting that (i) these districts are heavily gerrymandered,(ii) they do not represent the will of the people and (iii) they dilute the votes of one party. We havealso uncovered evidence that these two redistricting plans employ packing and cracking. On thecontrary, the redistricting plan produced by a bipartisan redistricting commission of retired judgesfrom the Beyond Gerrymandering project produced results which were highly typical among our24,000 reasonable redistrictings. The Judges plan was exceptionally non-gerrymandered, was atypical representation of the people’s will, and does not seem to pack or crack either party.We also explored the degree to which the NC2012, NC2016 and Judges redistrictings were repre-sentative of the nearby redistrictings, where we interpret nearby to mean that roughly 10% of theVTDs are switched out of any given district. We found that the NC2012 and NC2016 redistrict-ings were significantly more gerrymandered than those around them while the Judges redistrictingwas similar to those nearby. This seems to imply that the NC2012 and NC2016 redistrictingswere carefully engineered and tuned, and not randomly chosen among those with a certain basicstructure.The remainder of the paper is organized as follows. In the remainder of this section we describethe Beyond Gerrymandering project and situate this work in previous reports produced by thisand associated teams. In Section 3, we describe how we construct our distribution of “reasonable”redistrictings and sample it using Markov chain Monte Carlo, as well as provide some brief historicalcontext concerning similar efforts. In Section 3.4, we describe how we further sub-select our samplesbased on a series of thresholds to better reflect the proposed bipartisan redistricting legislationHB92. In Section 3.5, we discuss how the parameters of our distribution are chosen to produce“reasonable” redistrictings. In Section 4 we describe characteristics of our generated “reasonable”redistrictings. In Section 5, we explore the effect of the Voting Rights Act provision in HB92 n the outcome of the elections. In Section 6, we give the missing details in the construction ofthe Representativeness and Gerrymandering Indices. In Section 7, we show that our results areinsensitive to our choice of parameters. We also provide evidence that our Markov chain MonteCarlo is running sufficiently long to produce results from the desired distribution. In Section 8,we provide some details about the data used and some of the more technical choices made in thepreceding analysis. Finally in Section 9, we make some concluding remarks and discuss futuredirections. The Appendix of the paper gives some sample maps drawn by our algorithm.2.7. The Beyond Gerrymandering Project.
The Beyond Gerrymandering project was a col-laboration between UNC system President Emeritus and Davidson College President EmeritusThomas W. Ross, Common Cause, and the POLIS center at the Sanford School at Duke Univer-sity. The project’s goal is to educate the public on how an independent, impartial redistrictingprocess would work. The project formed an independent redistricting commission made up of tenretired jurists, five Democrat and five Republican. The commission used strong, clear criteria tocreate a new North Carolina congressional map based on NC House Bill 92 from the 2015 leg-islative season. All federal rules related to the Voting Rights Act were followed but no politicaldata, election results or incumbents addresses were considered when creating new districts. Thecommission met twice over the summer of 2016 to deliberate and draw maps. The maps resultingfrom this simulated redistricting commission were released in August 2016. The Judges agreed ona redistricting at the level of Voting Tabulation Districts (VTD). This coarser redistricting wasrefined at the level of census blocks to achieve districts with less than 0.1% population deviation.The original VTD based maps are used in our study.2.8. Related Works.
Ideas to generate redistricting plans with computational algorithms havebeen being developed since the 1960’s [18, 22, 10]. There are three main classifications of redis-tricting algorithms: constructive Markov Chain Monte Carlo (MCMC) algorithms [7, 3, 4], movingboundary MCMC algorithms [15, 16, 1, 24, 9], and optimization algorithms [17, 14]. ConstructiveMCMC algorithms begin each new redistricting with an initial random seed and grow districts.Moving boundary MCMC algorithms find new redistricting plans by altering district boundaries.In [9], the authors demonstrate that moving boundary MCMC algorithms are better at samplingthe redistricting space than constructive algorithms. It is proven that the former will theoreticallysample the space with the correct probability distribution, where as the latter may construct manysimilar redistrictings of one kind while not generating as many equally likely redistricting plans,leading to a skewed distribution. Optimization algorithms are primarily concerned with generatingone or a collection of ‘elite’ districts, as opposed to sampling the space of all districtings. Recentlyan evolutionary algorithm has been proposed which begins with a constructive method, but thenuse mixing to find either elite or ‘good enough’ redistrictings [14]; it used the collection of ‘goodenough’ redistrictings to make statistical predictions, however it is still unclear how evolutionaryalgorithms compare with Monte Carlo models in terms of sampling the space properly. One advan-tage of the moving boundary MCMC approach is that we sample form a explicitly specified andconstructed probability distribution on redistrictings. We remark that all of the above works haveconsidered minimizing population deviation and compactness; a few of the works have consideredminimizing county splitting; none of these works have included Voting Rights Act requirements (seeSection 5 below). We present our algorithm for sampling the space of redistrictings in Section 3,which is a moving boundary MCMC algorithm.Once a sampling of the space redistrictings is produced, there are a variety of existing indicesthat have been used as a comparison metric for a given districting (see [6] and references thereinfor a summary and history of these indices). Such indices include competitiveness, responsiveness, For more information see https://sites.duke.edu/polis/projects/beyond-gerrymandering/ iasedness, dissimilarity, and efficiency. Similar to our work, the typical idea found in the literatureis to contextualize a given redistricting plan in the context of these indices. Each of these indicesprovides a valid method of determining if a given districting plan is typical in the context ofthe ensemble of generated redistrictings. None of these previous indices, however, account forthe underlying geography of a given state. As pointed out in [3], maps may naturally be lesscompetitive simply based on geography rather than partisan tampering. The gerrymandering andrepresentative indices we have introduced in this work are the first, to our knowledge, that accountfor the geography of a given state; in particular, the Gerrymandering Index gives the first metric ofhow gerrymandered a given redistricting plan is, in the context of drawing plans without partisandata. This coupling of indices may provide a more robust metric. We have seen some preliminaryevidence of this in Sections 2.5 and 2.2, however more study is required to test this hypothesis.2.9. Evolution of This Project.
This work originated as a PRUV project and subsequent seniorthesis of Christy Vaughn (now Christy Vaughn Graves), both of which were supervised by JonathanMattingly during the summer of 2013 and the academic year 2013-2014. This initial phase of theproject concentrated on North Carolina and was summarized in the technical report “Redistrictingand the Will of the People.” (See [16] from references.)That work grew into a summer undergraduate research project as part of the Data+ programin the Information Initiative at Duke during the summer of 2015. This work analyzed a numberof different states and introduced VRA and county fragmentation considerations. The researchteam consisted of Duke undergraduates Christy Vaughn Graves, Sachet Bangia, Bridget Dou, andSophie Guo and was again mentored by Jonathan Mattingly. The work is summarized at the onlineresource “Quantifying Gerrymandering” (See [1] from references.).During the summer of 2016, a second Data+ team was formed with the intention of analyzing theredistrictings produced by the judges in the Beyond Gerrymandering project. (See Section 2.7 fromreferences.). The Summer 2016 Data+ team consisted of Duke undergraduates Hansung Kang andJustin Luo, graduate mentor Robert Ravier, post-doc mentor Greg Herschlag, and faculty mentorJonathan Mattingly.This report is based on the new code base developed by the Summer 2016 Data+ team. It usesa refined formulation that builds on the work of the previous teams. Christy Vaughn Graves andSachet Bangia provided important continuity between the years. The 2016 Data+ team developednew analytic tools, some based on the useful visualizations provided by the box-plots developed bythe summer 2015 Data+ team. Some of the text and arguments from the 2014 report have beenintegrated into this new report.3. Random Sampling of Reasonable Redistrictings
Central to our analysis is the ability to generate a large number of different “reasonable” re-districtings. This is accomplished by sampling a probability distribution on possible redistrictingsof North Carolina. The distribution is constructed so that it is concentrated on “reasonable” re-districtings. We then filter the randomly drawn redistrictings, using only those which satisfy ourcriteria for being “reasonable.”As already mentioned, we take our definition of “reasonable” redistrictings from the unratifiedHouse Bill 92 (HB92) from the 2015 Session of the North Carolina General Assembly which statedthat a bipartisan commission should draw up redistrictings while observing the following principles: • § see http://bigdata.duke.edu/data Nonpartisan Redistricting Commission. House Bill 92. General Assembly of North Carolina Session 2015. HouseDRH10039-ST-12 (02/05) § • § • § • § • § score function which is minimized by redistrictings that aremost successful at satisfying the remaining design principles. We introduce some mathematicalformalisms in order to describe the score function.We represent the state of North Carolina as a graph G with edges E and vertices V . Each vertexrepresents a Voting Tabulation District (VTD) and an edge between two vertices exists if the twoVTDs are adjacent on the map. This graph representing the North Carolina voting landscape hasover 2500 vertices and over 8000 edges.Since North Carolina has thirteen seats in the U.S. House of Representatives, we define a redis-tricting plan to be a function from the set of vertices to the integers between one and thirteen. Moreformally, recalling that V was the set of vertices, we represent a redistricting plan by a function ξ : V → { , , . . . , } . If a VTD is represented by a vertex v ∈ V, then ξ ( v ) = i means thatthe VTD in question belongs to district i. Similarly for i ∈ { , , . . . , } and a plan ξ , the i -thdistrict, which we denote by D i ( ξ ), is given by the set { v ∈ V : ξ ( v ) = i } . We wish to only considerredistricting plans ξ such that each district D i ( ξ ) is a single connected component. We will denotethe collection of all redistricting plans with connected districts by R .3.1. The Score Function.
We now wish to define a function J that assigns a nonnegative number J ( ξ ) to every redistricting ξ ∈ R . To do this, we employ functions J p , J I , J c , and J m that measurehow well a given redistricting satisfies the individual principles outlined in HB92. The populationscore J p ( ξ ) measures how well the redistricting ξ partitions the population of North Carolina into 13equal parts. The isoperimetric score J I ( ξ ) measures how compact the districts are by returning thesum of the isoperimetric constants for each district, a quantity which is minimized by a circle. The county score J c ( ξ ) measures the number of counties split between multiple districts; the minimumis achieved when there are no split counties. Lastly, the minority score J m ( ξ ) measures the extentto which the districts with the largest percentage of African-Americans achieve stipulated targetpercentages. With these, we then define our score function J to be a weighted sum of J p , J I , J c , and J m ; we use a weighted combination so as to not give one of the above scores undue influence,since all of the score functions do not necessarily change on the same scale. Specifically, we define: J ( ξ ) = w p J p ( ξ ) + w I J I ( ξ ) + w c J c ( ξ ) + w m J m ( ξ ) , (1)where w p , w I , w c , and w m are a collection of positive weights.To describe the individual score functions, we attach to our graph G = ( V, E ) some data whichgives relevant features of each VTD. We define the positive functions pop( v ), area( v ), and AA( v )for a vertex v ∈ V as respectively the total population, geographic area, and African-Americanpopulation of the VTD associated with the vertex v . We extend these functions to a collection of ertices B ⊂ V bypop( B ) = (cid:88) v ∈ B pop( v ) , area( B ) = (cid:88) v ∈ B area( v ) , AA( B ) = (cid:88) v ∈ B AA( v ) . (2)We think of the boundary of a district D i ( ξ ) as the subset of the edges E which connect verticesinside of D i ( ξ ) to vertices outside of D i ( ξ ). We write D. i ( ξ ) for the boundary of the district D i ( ξ ).Since we want to include the exterior boundary of each district (the section bordering an adjacentstate or the ocean), we add to V the vertex o which represents the “outside” and connect it with anedge to each vertex representing a VTD which is on the boundary of the state. We always assumethat any redistricting ξ always satisfies ξ ( v ) = 0 if and only if v = o . Since ξ always satisfies ξ ( o ) = 0 and hence o (cid:54)∈ D i ( ξ ) for i ≥
1, it does not matter that we have not defined area( o ) orpop( o ), as o is never included in the districts.Given an edge e ∈ E which connects the two vertices v, ˜ v ∈ V , we define boundary( e ) to be thelength of common border of the VTDs associated with the vertex v and ˜ v . As before, we extendthe definition to the boundary of a set of edges B ⊂ E byboundary( B ) = (cid:88) e ∈ B boundary( e ) . (3)With these preliminaries out of the way, we turn to defining the first three score functions usedto assess the goodness of a redistricting.3.1.1. The population score function.
We define the population score by J p ( ξ ) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i =1 (cid:16) pop( D i ( ξ ))pop Ideal − (cid:17) , pop Ideal = N pop N pop is the total population of North Carolina, pop( D i ( ξ )) is the population of the district D i ( ξ ) as defined in (2), and pop Ideal is the population that each district should have according tothe ‘one person one vote’ standard; namely, pop
Ideal is equal to one-thirteenth of the total statepopulation.3.1.2.
The Isoperimetric score function.
The Isoperimetric score J I , which measures the compact-ness of a district, is the ratio of the perimeter to the total area of each district. The Isoperimetricscore is minimized for a circle, which is the most compact shape. Hence we define J I ( ξ ) = (cid:88) i =1 (cid:2) boundary(D. i ( ξ )) (cid:3) area( D i ( ξ )) . where D. i ( ξ ) is the set of edges which define the boundary, boundary(D. i ( ξ )) is the length of theboundary of district D i and area( D i ( ξ )) is its area.This compactness measure is one of two measures often used in the legal literature where it isreferred to as the perimeter score (See [20, 21] from references). The second measure, usually re-ferred to as the dispersion score , is more sensitive to overly elongated districts, though the perimeterscore also penalizes them. The dispersion score does not penalize undulating boundaries while theperimeter score (our J I ) does.3.1.3. The county score function.
The county score function J c ( ξ ) penalizes redistrictings whichcontain single counties contained in two or more districts. We refer to these counties as splitcounties. The score consists of the number of counties split over two different districts times a actor W ( ξ ) plus a large constant M C times the number of counties split over three of moredifferent districts times a second factor W ( ξ ). Specifically, we define: J c ( ξ ) = { } · W ( ξ )+ M C · { ≥ } · W ( ξ )where M C is a large constant and the weights W ( ξ ) and W ( ξ ) are defined by W ( ξ ) = (cid:88) countiessplit between2 districts (cid:16) Fraction of county VTDs in 2nd largestintersection of a district with the county (cid:17) W ( ξ ) = (cid:88) countiessplit between ≥ (cid:16) Fraction of county VTDs not in 1st or 2ndlargest intersection of a district with the county (cid:17) The factors W ( ξ ) and W ( ξ ) make the score function vary in a more continuous fashion, whichencourages reduction of the smaller fraction of a split county.3.1.4. The Voting Rights Act or minority score function.
It is less clear what it means for a redis-tricting to comply with the VRA. African-American voters make up approximately 20% of the eli-gible voters in North Carolina. Since 0.2 is between and , the current judicial interpretation ofthe VRA stipulates that at least two districts should have enough African-American representationso that this demographic may elect a candidate of their choice. However, the NC2012 redistrictingplan was ruled unconstitutional because two districts, each containing over 50% African-Americans,were ruled to have been packed too heavily with African-Americans, diluting their influence inother districts. The NC2016 redistricting was accepted based on racial considerations of the VRAand contained districts that held 44.48% African-Americans, and 36.20% African-Americans. Theamount of deviation constitutionally allowed from these numbers is unclear.Based on these considerations, we chose a VRA score function which awards lower scores toredistrictings which had one district with at least 44.48% African-Americans and a second districtwith at least 36.20% African-Americans. We write J m ( ξ ) = (cid:112) H (44 . − m ) + (cid:112) H (36 . − m ) , (4)where m and m represent the percentage of African-Americans in the districts with the highestand second highest percent of African-Americans, respectively. H is the function defined by H ( x ) =0 for x ≤ H ( x ) = x for x ≥
0. We chose this function to make the transition smoother,and we utilize the square root function to encourage districts that are just above the threshold tobe less probable than when no square root is included. Notice that whenever m ≥ .
84% and m ≥ .
20% we have that J m = 0.3.2. The Probability Distributions on Redistrictings.
We now use the score function J ( ξ )to assign a probability to each redistricting ξ ∈ R that makes redistrictings with lower scores morelikely. Fixing a β >
0, we define the probability of ξ , denoted by P β ( ξ ), by P β ( ξ ) = e − βJ ( ξ ) Z β (5)where Z β is the normalization constant defined so that P β ( R ) = 1. Specifically, Z β = (cid:88) ξ ∈R e − βJ ( ξ ) . he positive constant β is often called the “inverse temperature” in analogy with statistical me-chanics and gas dynamics. When β is very small (the high temperature regime), different elementsof R have close to equal probability. As β increases (“the temperature decreases”), the measureconcentrates the probability around the redistrictings ξ ∈ R which minimize J ( ξ ).3.3. Generating Random Redistrictings.
If we neglect the fact that the individual districtsin a redistricting need to be connected, then there are more than 13 ≈ . × differentredistrictings, larger than both the current estimate for the number of atoms in the universe (be-tween 10 and 10 ) and the estimated number of seconds since the Big Bang (4 . × ). Whilethere are significantly fewer redistrictings in R (the set of simply connected redistrictings), it isnot practical to enumerate all redistrictings to find those with the lowest values of J (i.e. the mostprobable ones).The standard, very effective way to escape this curse of dimensionality is to use a Markov chainMonte Carlo (MCMC) algorithm to sample from the probability distribution P β . The basic idea isto define a random walk on R which has P β as its unique, attracting stationary measure. We dothis using the standard Metropolis-Hastings algorithm.The Metropolis-Hastings algorithm is designed to use one Markov transition kernel Q (the pro-posal chain) to sample from another Markov transition kernel that has a unique stationary distri-bution µ (the target distribution). Q ( ξ, ξ (cid:48) ) gives the probability of moving from the redistricting ξ to the redistricting ξ (cid:48) in the proposal Markov chain and is readily computable. We use Q to drawa sample distributed according to µ . The algorithm proceeds as follows:(1) Choose some initial state ξ ∈ R .(2) Propose a new state ξ (cid:48) with transition probabilities given by Q ( ξ, ξ (cid:48) ).(3) Accept the proposed state with probability p = min (cid:0) , µ ( ξ (cid:48) ) q ( ξ (cid:48) ,ξ ) µ ( ξ ) Q ( ξ,ξ (cid:48) ) (cid:1) .(4) Repeat steps 2 and 3.The stationary distribution of this Markov chain matches the stationary measure µ . Thus, thestates can be treated as samples from the desired distribution. The stationary measure we wouldlike to sample is P β . We sample from three possible initial states: the NC2012, the NC2016,and the Judges redistricting. Since this algorithm is designed to converge to a unique stationarymeasure P β , any results should be independent of the initial starting point. However, this assumesthe parameters have been chosen so that the time to equilibrate is short enough to happen duringour runs. We show that the results are independent of the initial condition in Section 7.3, whichlends credence to the assertion that the algorithm is equilibrating. We define the proposal chain Q used for proposing new redistrictings in the following way:(1) Uniformly pick a conflicted edge at random. An edge, e = ( u, v ) is a conflicted edge if ξ ( u ) (cid:54) = ξ ( v ), ξ ( u ) (cid:54) = 0, ξ ( v ) (cid:54) = 0.(2) For the chosen edge e = ( u, v ), with probability , either: ξ (cid:48) ( w ) = (cid:40) ξ ( w ) w (cid:54) = uξ ( v ) u or ξ (cid:48) ( w ) = (cid:40) ξ ( w ) w (cid:54) = vξ ( u ) v Though technical, one can rigorously prove that the Markov Chain given by this algorithm converges to thedesired distribution if run long enough. One only needs to establish that the Markov Chain transition matrix isirreducible and aperiodic. Since one can evolve from any connection redistricting to another through steps of thechain, it is irreducible. Aperiodicity follows as there exist redistrictings which are connect to itself through a loopconsisting of two steps and a loop consisting of three steps. Since 2 and 3 are prime and hence have greatest commondivisor 1, the chain is aperiodic. See the Perron-Frobenius Theorem for more details. As this work was being completed an the work in [5] appeared which provides an interesting set of ideas to assessif samples being drawn are typical or outliers exactly in our context. We hope to explore these ideas in the nearfuture. et conflicted( ξ ) be the number of conflicted edges for redistricting ξ . Then we have Q ( ξ, ξ (cid:48) ) =
12 1conflicted( ξ ) . The acceptance probability is given by: p = min (cid:16) , conflicted( ξ )conflicted( ξ (cid:48) ) e − β ( J ( ξ (cid:48) ) − J ( ξ )) (cid:17) If a redistricting ξ (cid:48) is not connected, then we refuse the step, which is equivalent to setting J ( ξ (cid:48) ) = ∞ .Given a fixed set of weights ( w p , w i , w c , w m ), one still needs to determine an appropriate β so thattypical samples from the distribution are “reasonable” redistrictings. If β is chosen to be too large,the algorithm will seek out a local minimum and leave this minimum with very low probability,meaning that it may require a large amount of steps to switch between high quality redistrictings.If β is chosen to be too low, then the algorithm will never find the locally good districts as it willchoose redistrictings indiscriminately.There are several well established ideas in the literature to overcome these challenges, includingsimulated annealing (e.g. [23]), parallel tempering (e.g. [11]) and simulated tempering (e.g. [9]).In the present work, we examine simulated annealing, in which β is set to be small at first until acertain number of steps are accepted (in the sense of step (3) from the algorithm in Section 3.3).This allows the system to explore the space of redistrictings more freely. Next, β is increasedlinearly to a maximum value over the course of a defined number of steps. This slowly “cools” thesystems, hopefully relaxing it into a redistricting ξ which has a relatively low score J ( ξ ). Finally, β is kept at this fixed maximum value for a defined number of steps so that the algorithm locallysamples the measure P β sufficiently long enough to produce a good redistricting. During thesummer of 2016 Data+ Project, we explored extensively the use of parallel tempering to generateMonte Carlo sampled. We found that simulated annealing more reliably explored the state space.Parallell tempering does have the advantage of always sampling from the same target distributionwhile simulated annealing changes the target distribution to improve mixing. In theory parallelltempering should also behave well, tough we found tuning it properly more difficult.The principal results quoted in Section 2 use the low β to be zero over 40 ,
000 steps, linearlyincrease β to one over 60 ,
000 steps, and fix β to be one for 20 ,
000 steps before taking a sample.This process is repeated for each sample redistricting. One potential critique with using simulatedannealing is that the results may depend on the number of steps chosen above. We make a standardtest to confirm that we have taken an appropriate number of steps by doubling each number ofsteps and repeating our analysis. The results of this test, which are found in Section 7.3, show thatdoubling the number of steps has little effect on the results.3.4.
Thresholding the sampled redistrictings.
It is possible for the simulated annealing algo-rithm to draw a redistricting with a bad score when using the MCMC algorithm from Section 3.3combined with the probability distribution given in (5). Additionally, the use of simulated anneal-ing also increases the chance that we become stuck in a local minimum with a less than desirablescore function, as such local minimum may take longer than the time we spend at high β to escape.These local trapping events can often lead to samples with less than perfect score functions. Lastly,our score functions do not perfectly encapsulate our redistricting design aesthetic. For example,since the isoperimetric score function is the sum of the individual isoperimetric scores of each dis-trict, it is still possible to have one bad district if the rest have exceptionally small isoperimetricscores.Since we want to maximize the degree of compliance with HB92, we only use samples whichpass an additional set of thresholds, one for each of the selection criteria. This additional layerof rejection sampling was also used in reference [9], though the authors of reference [9] chose toreweigh the samples to produce the uniform distribution over the set redistrictings that satisfy the hresholds. We prefer to continue to bias our sampling according to the score function so betterredistrictings are given higher weights; we note that the idea of preferring some redistrictings toothers is consistent with the provisions HB92. We now detail our thresholding requirements.It is our experience from the Beyond Gerrymandering project that redistrictings which use VTDsas their building blocks and have less that 1% population deviation can readily be driven to 0.1%population deviation by breaking the VTDs into census tracts and performing minimal alterationsto the overall redistricting plan. We thus only accept redistrictings that have no districts above1% population deviation. Many of our samples have deviations considerably below this value. Itis important to emphasize that we require this of every district in the redistricting. In Section 7.1,we show that the results are quantitatively extremely similar, and qualitatively identical, when thepopulation threshold is decreased from 1% to 0.75 % and then to 0.5%.We have found that districts with isoperimetric scores under 60 are almost always reasonablycompact. Thus, we choose to accept a redistricting only if each district in the plan has an isoperi-metric ratio less than 60. The Judges redistricting plan would be accepted under this threshold asits least compact district has an isoperimetric score of 53.5. Neither NC2012 nor NC2016 would beaccepted with this thresholding as the least compact districts of each plan have isoperimetric scoresof 434.65 and 80.1, respectively. We also note that only two of the thirteen districts for the NC2012plan meet our isoperimetric score threshold, whereas eight of the thirteen districts of NC2016 fallbelow the threshold. Although we examine our principle results over a space of highly compactredistricting plans, we also demonstrate that our results are insensitive to lifting this restriction inSection 7.1.Though redistrictings which split a single county in three are infrequent, they do occur amongour samples. Since these are undesirable, we only accept redistrictings for which no countiesare split across three or more districts. Note that, in order to satisfy population requirements,we must allow counties to be split into two districts because of the large populations of Wakeand Mecklenburg Counties which each contain a population larger than a single Congressionaldistrict’s ideal population. We do not explicitly threshold based on number of split counties, thoughredistrictings with more split counties have a higher scores, and hence are less favored. We remarkthat none of our generated redistrictings had more county splits than the NC2012 redistrictingplan, and that the NC2012 plan was never critiqued or challenged based on the number of countysplits.To build a threshold based on minority requirements of the VRA, we note that the NC2016redistricting was deemed by the courts to satisfy the VRA. The districts in this plan with the twohighest proportion of African-Americans to total population are composed of 44.5% and 36.2%African-Americans. With this in mind, we only accept redistrictings if the districts with the twohighest percentages of African-American population have at least 40% and 33.5%, respectively.The effect of all of these thresholds was to select around 16% of the samples initially producedby our MCMC runs. Though this leads to unused samples, it ensures that all of the redistrictingsused meet certain minimal standards. This in turn allowed us to better adhere to the spirit ofHB92. The reported 24,000 samples used in our study refer to those left after thresholding. Thefull data set of samples was in excess of 150,000. That being said, we show in Section 7.1 thatresults without thresholding were quantitatively very close and qualitatively identical. As alreadymentioned, we also show that decreasing the population threshold from 1% to 0.75% and then to0.5% also has little effect on the quantitative results and no effect on the qualitative conclusions.3.5. Determining the weight parameters.
As we have mentioned above, we have four indepen-dent weights ( w p , w I , w c , w m ) used in balancing the effect of the different scores in the total score J ( ξ ). In addition to these parameters, we also have the low and high temperatures correspondingrespectively to the max and min β used in the simulated annealing. We set the minimum value of β o be zero which corresponds to infinite temperature. In this regime, no district is favored over anyother, which allows the redistricting plan freedom to explore the space of possible redistrictings.The only parameter left is to set the high value of β . Since β multiplies the weights, one of thesedegrees of freedom is redundant and can be set arbitrarily. We chose to fix the low temperature(high value of β ) to be one.To select appropriate parameters, we employ the following tuning method:(1) Set all weights to zero.(2) Find the smallest w p such that a fraction of the results are within a desired threshold(for the current work we ensured that at least 25% of the redistrictings were below 0.5%population deviation, however we typically did much better than this).(3) Using the w p from the previous step, find the smallest w I such that a fraction of theredistrictings have all districts below a given isoperimetric ratio (we ensured that at least10% of the results were below this threshold; we chose a threshold of 60 (see Section 3.4)).(4) If above criteria for population is no longer met, repeat steps 2 through 4 until both con-ditions are satisfied(5) Using the w p and w I from the previous steps, find the smallest w m such that at least 50%of all redistrictings have at least one district with more than 40% African-Americans and asecond district has at least 33.5% African-Americans.(6) If the thresholds for population were overwhelmed by increasing w m , repeat steps 2 through6. If the thresholds for compactness were overwhelmed, repeat steps 3 through 6.(7) Using the w p , w I , and w m from the previous steps, find the smallest w c such that we nearlyalways only have two county splits, and the number of two county splits are, on average,below 25 two county splits.(8) If the thresholds for population are no longer satisfied, repeat steps 2 through 8. If thecriteria for the compactness is no longer met, repeat steps 3 through 8. If the criteria forthe minority populations is not satisfied, repeat steps 5 through 8. Otherwise, finish witha good set of parameters.With this process, we settle on parameters w p = 3000, w I = 2 . w c = 0 .
4, and w m = 800 andhave used these parameters for all of the results presented in the main results above (Section 2).In Section 7.5, we show that variations of these choices have little qualitative effect on the results.4. Characteristics of “reasonable” redistrictings ensamble
We now explore the properties of the over 24,000 random restrictings we have generated using thealgorithm described in the preceding sections. All of the random redistrictings passed the thresholdtest described in Section 3.4. As such, they all have no district with population deviation above1%. However, most have a deviation much less than 1%: the mean population deviation taken overthe more-than 13 × ,
000 districts is 0.16% with a standard deviation of 0.14%. Figure 8gives a finer view for the distribution of the population deviation. We order each redistricting bythe maximum population deviation over all districts. To simultaneously give a sense of the medianpopulation deviation of the districts with a given maximum population deviation, we examine thelocal statistics of the ordered districts to find the maximum and minimum values of the local median(plotted as the blue envelope) along with the standard deviation of the median (green envelop) andexpected value of the median (dotted line). With this plot we notice that over 50% of redistrictingshave a worst case population deviation under 0.4% and many of these redistrictings have a medianpopulation deviation well below 0.2%.To compare the population deviation of our generated districts with the districtings of NC2012,NC2016 and the Judges, we note that all of these districtings all had to split VTDs in order toachieve a population deviation below 0.1%. Before splitting VTDs, NC2012, NC2016 and Judges ad a district with maximum population deviation of 0.847%, 0.683%, and 0.313% respectively,and had median population deviations of 0.234%, 0.048%, and 0.078% respectively, meaning thatthe districts sampled by our algorithm are very similar to the three districts we have compared ourresults with in terms of population deviation.Turing to the isoperimetric ratios, recall that all of the districts have an isoperimetric constantunder 60 as that was our threshold value. The mean isoperimetric ratio of the more-than 13 × ,
000 districts is 36.9 with a standard deviation of 9. Examining the second part ofFigure 8 gives an analogously finer view for the distribution of the isoperimetric ratios of all districts.The figure shows that most redistrictings have a median isoperimetric ratio in the mid-thirties andthat roughly 50% of our redistrictings have a district with isoperimetric ratio no worse than 55 foran isoperimetric ratio.When comparing our generated districts, we note that the NC2012, NC2016 and Judges redis-trictings have districts with maximum isoperimetric ratio of 434.6, 80.1, and 54.1 respectively, andhave median isoperimetric ratios of 114.4, 54.5, and 38.2 respectively. The NC2012 and NC2016districts would be rejected under our thresholding criteria. Below, we demonstrate that samplingfrom redistrictings that include less compact districts does not change our results (see section 7).
Figure 8.
The redistrictings ordered by the worst case district in terms of either popu-lation deviation (left) or isoperimetric ratio (right). The solid dark line give the worst casedistricts value while the dotted line gives the average across redistricting with a given maxvalue of median districts value. The outer shading gives the max and min value of thismedian while the inner-shading covers one standard deviation above and below the mean ofthe medians.
Next we examine the four districts with the highest minority representation in each districting.In Figure 9, we order the redistrictings in decreasing order on the over 24000 accepted redistrictings.The kink in this line at 44.46% occurs due to the minority energy function which does not favorany population above this limit (recall this number was based on NC2016). Roughly half of theredistrictings have a district with greater than 44.46% of the population as African-Americans,whereas the other half has between 40% and 44.46% in the district with the largest number ofAfrican-Americans. For the district with the second highest African-American representation, weremark that over 80% of all redistrictings have more than 35% African-American representation inthe second largest district; there is not a single redistricting that has the second largest African-American district with more than 40% of the population African-American. inally we display the histogram of the number of split counties over our generated redistrictings.We find a median of 21 split counties with a mean of 21.6, and a range from 14 to 31. We remarkthat NC2012, NC2016, and Judges districtings had 40, 13, and 12 split counties respectively. Largest AA district2nd largest AA district ( σ )3rd largest AA district ( σ ) F r a c t i on AA popu l a t i on Splits greater that 10%All splits F r a c t i on Figure 9.
The redistrictings are ordered by the district with the largest African-Americanpercentage (left). Subsequent ranges show standard deviations for districts with the secondand third largest African-American representation. We plot the histogram of the number ofcounty splits in each districting (right). The lighter histogram gives the number total splitcounties while the darker histogram gives only the number which splits the county into twoparts each containing more than 10% of the total VTDs. The effect of the Voting Rights Act
The NC2012 districts were labeled unconstitutional for over packing African-Americans anddiluting their voice in other districts. We investigate the effect of the Voting Rights Act (VRA) onelection outcomes by considering samples taken from simulations that do not take the VRA intoaccount, which is to say that we set w m = 0. We examine the distribution of elected Democratsalong with the histogram box-plots in Figure 10. We find that the VRA, even with the moremodest thresholds of 40% and 33% required African-Americans, significantly favors the Republicanparty. Without the VRA, there is roughly a 65% chance that 7 or more Democrats will be elected,with a 20% chance that 8 Democrats will be elected; in contrast, with the VRA considered, thereis a 50% chance that 7 or more Democrats are elected, with a 10% chance that 8 Democrats areelected. Without commenting specifically on the VRA, we point out that these results highlight theimportance of § Details of the Indices
We begin by expounding and clarifying how we compute the Gerrymandering Index and theRepresentativeness Index. We have thoroughly explained the efficiency gap above so omit furtherdiscussion in the current section.6.1.
Details of Gerrymandering Index.
To compute the Gerrymandering Index, we examinethe mean percentage of Democratic votes in each of the thirteen districts when the districts areordered from most to least Republican (see Figure 4). To calculate the Gerrymandering Index forany given redistricting plan, we take the Democratic votes for each district when the districts areagain ordered from most to least Republican. The differences between the mean and the observed RA consideredVRA not considered P r obab ili t y o f r e s u l t ( % ) VRA consideredVRA not considered D e m o c r a t i c v o t e f r a c t i on Figure 10.
We display changes of the distribution of election results when the VRA isnot taken into consideration (left). The histogram formed from the distribution of our mainresults overlays this image with the gray shaded histogram. We display changes to thehistogram of the box-plot when comparing the results when VRA is considered or not (right). democratic percentage are taken for each district using a given set of votes. These differences arethen each squared and summed over the 13 districts. The square root of this sum of squares is ourGerrymandering Index.The Gerrymandering Index is smallest when all of the ordered Democratic vote percentages areprecisely the mean values. However, this is likely not possible as the percentages in the differentdistricts are highly correlated. To understand the range of possible values, we plot the complemen-tary cumulative distribution function of the Gerrymandering Index of our ensemble of randomlygenerated reasonable redistrictings (see Figure 2). This gives a context in which to interpret anyone score.The mean percentages for the collection of redistricting we generated is(0 . , . , . , . , . , . , . , . , . , . , . , . , . . If a given redistricting is associated with the sorted winning Democratic percentages(0 . , . , . , . , . , . , . , . , . , . , . , . , . . then the Gerrymandering Index for the redistricting is the square root of(0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . + (0 . − . = 0 . √ . . Details of Representativeness Index.
To calculate the Representativeness Index, we firstconstruct a modified histogram of election results that captures how close an election was to swap-ping results. To do this for a given redistricting plan, we examine the least Republican district inwhich a Republican won, and the least Democratic district in which a Democrat won. We thenlinearly interpolate between these districts and find where the interpolated line intersects with the50% line. For example, in the 2012 election, the 9th most Republican district elected a Republican ith 53.3% of the vote, and the fourth most Democratic district won their district with 50.1% ofthe vote. We would then calculate where these two vote counts cross the 50% line, which will be50 − (100 − . . − (100 − . ≈ . , (6)and add this to the number of Democratic seats won to arrive at the continuous value of 4.03.This index allows us to construct a continuous variable that contains information on the numberof Democrats elected, and also demonstrates how much safety there is in the victory.Fractional parts close to zero suggest that the most competitive Democratic race is less likely togo Democratic than the most competitive Republican race is to go Republican. On the other hand,fractional parts close to one suggest that the most competitive Republican race is less likely togo Republican than the most competitive Democratic race is to go Democratic. Instead of simplycreating a histogram of the number of seats won by the Democrats, in Figure 11 we construct ahistogram of our new interpolated value. We define the representativeness as the distance fromthe interpolated value to the mean value of this histogram (shown in the dashed line). These arethe values we report in Figure 3. For the 2012 vote data, we find that the mean interpolatedDemocratic seats won is 7.01, and the Judges plan yields a value of 6.28, giving a RepresentativeIndex of | . − . | = 0 .
73. The NC2012 and NC2016 plans both have representative indicesgreater than two.
JudgesNC2016NC2012 P r obab ili t y o f r e s u l t JudgesNC2016NC2012 P r obab ili t y o f r e s u l t Figure 11.
For the 2012 votes (left) and the 2016 votes (right), we plot the interpolatedwinning margins, which give the number of seats won by the Democrats in finer detail.We determine the mean of this new histogram and display it with the dashed line. TheRepresentativeness Index is defined to be the distance from this mean value. The histogrampresented in Figure 1 is overlaid on this plot for reference. Testing the Sensitivity of Results
We wish to ensure that our algorithm has sampled the space of redistrictings in a robust way.We use this section to carefully study the effect of changing the number of samples used, changingthe set of threshold values, changing the weights in our distribution, changing the type of energyfunction used for compactness, changing simulated annealing parameters on election results, anddetermining the possible effect of splitting VTDs to achieve zero population deviation. We alsoverify that the choice of the initial district does not influence our results and that this informationis lost as the algorithm updates the redistrictings. .1. Varying thresholds.
Achieving a 0.1% population deviation is the only statute of HB92 thatwe violate. Although we have noted above that the Judges original redistrictings in the ‘BeyondGerrymandering’ project were all slightly over 1% population deviation, and splitting VTDs to fallbelow this threshold had little impact on the election results. We test this for our own redistrictingsby changing the population threshold to 0.75% and 0.5%. The results are shown in Figure 12, forwhich we have used the 2012 vote data. We find that tightening the population threshold hasnegligible impact on the number of Democrats elected, and that the variation in the histogrambox-plots is barely perceptible. In the 0.5% population deviation threshold plots, we have discardedover half of our results and we still do not see any significant changes. These results support ourclaim that splitting VTDs to achieve a less than 0.1% deviation will have a negligible effect on ourconclusions.
Population threshold at 1%Population threshold at 0.75%Population threshold at 0.5% F r a c t i on o f r e s u l t Population Threshold 1%Population Threshold 0.5% D e m o c r a t i c v o t e f r a c t i on Figure 12.
We display changes of the distribution of election results with changes to thepopulation threshold (left). The histogram formed with 1% population deviation overlaysthis image with the gray shaded histogram. We display changes to the histogram of thebox-plot when comparing 1% population deviation threshold with 0.5% (right).
Next, we note that there is no corresponding law to dicate a choice of compactness threshold.The NC2016 districts have a maximum isoperimetric ratio of around 80, and the NC2012 districtshave a maximum of over 400. The Judges redistricting has a district with maximum isoperimetricratio of around 54. To test the effect of setting different compactness thresholds, we repeat ouranalysis by choosing 54, 80 and no threshold for the maximum isoperimetric ratio of all districtswithin a redistricting. We find that relaxing the compactness threshold minimally changes theelection results as demonstrated in Figure 13. We note that having no threshold does not meanthat we have arbitrarily large compactness values. This is because of the cooling process in thesimulated annealing algorithm and the fact that we continue to penalize large compactness scores.We find that we have an average maximum isoperimetric ratio of around 75 and that we rarely seeredistrictings with maximal ratio larger than 120.7.2.
Sample Maps, Details of Districts and Raw Data.
Table 1 gives the percentage ofDemocratic votes in NC2012, NC 2016, and the Judges redistrictings. The numbers in parenthesesgive the numerical label of the individual districts as identified in the maps in the Appendix.The last set of columns contains the mean values for each positon in the ranked ordered vectorcontaining the percentage of Democratic votes for each redistricting. As already described, theGerrymandering Index is Euclidean distance from this vector of marginal means. One can easilyidentify the particular districts which have likely been packed or cracked by comparing the values ompactness threshold at 54Compactness threshold at 60Compactness threshold at 80No compactness threshold F r a c t i on o f r e s u l t Compactness threshold at 60No compactness threshold D e m o c r a t i c v o t e f r a c t i on Figure 13.
We display changes of the distribution of election results with changes tothe compactness threshold (left). The histogram formed with a maximum of 60 for theisoperimetric ratio overlays this image with the gray shaded histogram. We display changesto the histogram of the box-plot when comparing a maximum of 60 in the isoperimetricratio without any thresholding on compactness (right). for a given district to this vector of means. In particular, the three most Democratic districtsNC2012 NC2016 Judges MeanRank 2012 2016 2012 2016 2012 2016 2012 20161 37.5 (3) 34.2 (3) 38.7 (3) 32.8 (3) 35.5 (10) 28.9 (10) 37.0 30.62 39.0 (6) 34.6 (11) 42.5 (10) 35.8 (11) 40.0 (2) 33.6 (2) 39.1 33.03 42.4 (5) 36.2 (7) 43.7 (6) 36.8 (10) 42.6 (12) 36.3 (7) 41.0 35.34 42.5 (11) 36.6 (8) 43.9 (11) 39.0 (7) 42.7 (7) 37.6 (12) 43.7 38.55 42.6 (2) 37.4 (10) 44.0 (2) 40.7 (6) 44.5 (9) 40.0 (9) 46.4 40.66 43.1 (10) 38.9 (5) 45.1 (5) 41.2 (8) 48.5 (8) 41.9 (3) 48.4 42.27 43.5 (13) 40.8 (6) 46.3 (13) 41.6 (5) 48.8 (11) 42.7 (11) 50.2 44.38 46.2 (8) 41.2 (2) 47.3 (8) 41.8 (9) 50.5 (4) 45.7 (4) 52.3 47.79 46.7 (9) 44.0 (9) 49.4 (9) 43.3 (2) 57.0 (3) 48.1 (8) 55.1 51.210 50.1 (7) 45.8 (13) 51.6 (7) 43.9 (13) 57.5 (5) 55.9 (1) 57.2 54.611 74.4 (4) 71.5 (1) 66.1 (4) 66.6 (12) 59.2 (1) 59.7 (5) 59.5 57.512 76.0 (1) 73.0 (4) 69.8 (12) 68.2 (4) 64.6 (6) 63.3 (13) 62.6 61.413 79.3 (12) 75.3 (12) 70.9 (1) 70.3 (1) 66.0 (13) 65.3 (6) 67.5 65.1
Table 1.
Percentage of Democratic votes in each district when districts are rankedfrom most Republican to most Democratic. Number in parentheses give label ofactual district using the numbering convention from maps in the Appendix. Thisdata is plotted in Figure 4 and 5 on top of the summary box-plots.labeled 1, 4 and 12 in both the NC2012 and NC2014 plan have significantly more Democraticvotes. Districts 9 and 13 both show evidence of having less Democrats than one would expect fromtheir rankings. These conclusions are consistant across the 2012 and 2016 votes.The raw data used to produce Figure 1 is given in Table 2. It underscores how atypical theresults produced by the NC2012 and NC2016 redistrictings are. If one is ready to accept four seatsin the 2012 vote then one should equally accept nine. Similarly in the 2016 votes, if one accepts hree as a legitimate outcome then one should also be willing to accept seven seats. None of theseresults seem particularly representative of the votes cast. Table 2.
Among the 24,518 random redistrictings generated, the number whichproduced the indicated number of Democratic seats in the Congressional Delegation.7.3.
Independence of initial conditions and simulated annealing parameters.
There isa possible pitfall of using simulated annealing: we may become trapped in local regions, leavingus unable to explore the entire space of redistrictings. This may be because we have cooled thesystem down too quickly, keeping it trapped in a local region, or it may be because the likelihoodof finding a path out of one local region of redistrictings and into another is small. We note that wehave animated our algorithm and have found that districts may travel from one end of the state toanother; such motion suggests that many types of redistrictings are sampled, and it is reasonableto hypothesize that as districts exchange locations, they lose information on past configurations.To more fully vet this idea, we examine the effect of (i) choosing a different initial redistricting inour algorithm, and (ii) doubling the simulated annealing parameters, thus cooling the system downtwice as slowly. To clarify the point (ii), instead of remaining hot ( β = 0) for 40,000 steps, coolinglinearly for 60,000 steps, and remaining cold ( β = 1) for 20,000 steps, we instead remain hot for80,000 steps, cool linearly for 120,000 steps, and remain cold for 40,000 steps. We then check to seeif the election results are altered by changing these conditions and display our results in Figure 14.We find that the changes with respect to both initial conditions and the slowdown of the annealingprocess have little effect on the election results. There are slight effects; for example, the initialcondition for the NC2012 redistricting has a 15% chance of electing five Democrats rather than the12% chance we have seen before. We note that these are exploratory runs, so we have less than1000 accepted districtings for the NC2012 and NC2016 initial conditions (each has close to 1000)and less than 2500 runs for the increased cooling times. These sample sizes are robust enough toprovide a general trend but are subject to statistical variations. Hence the small sample sizes area possible and likely culprit of these variations.7.4. Evidence of proper sampling.
The above test gives strong evidence that we have properlysampled the probability distribution of redistrictings. To strengthen this claim, we also continueto allow the algorithm to sample the space until we have sampled roughly 120 thousand acceptableredistrictings as defined by the original thresholding criteria. We then compare the results of theelections along with the box plots and histogram plots. We find that there is negligible changein the distribution of outcome both for the overall number of elected representatives and for eachordered district from most to least republican. We display our results in Figure 15. The stabilityof these results together with those presented in Section 7.3 provides robust evidence that we havecorrectly recovered the underlying probability distribution of redistrictings.7.5.
Varying weights.
We have proposed a methodology for determining the weights in the scorefunction that is primarily concerned with obtaining a high percent of redistrictings below ourchosen threshold values (see Section 3.5). We note that other parameters may be chosen, andhere we test whether making a different choice will affect the statistics on the election outcomes.We are in a four dimensional space, meaning that the parameter space is very large. Exploring udges (initial)NC2012 (initial)NC2016 (initial) F r a c t i on o f r e s u l t Judges (initial)NC2012 (initial)NC2016 (initial) D e m o c r a t i c v o t e f r a c t i on Reported S.A. parametersDoubled S.A. parameters F r a c t i on o f r e s u l t Reported S.A. parametersDoubled S.A. parameters D e m o c r a t i c v o t e f r a c t i on Figure 14.
We display the probability distribution of elected Democrats with respect toinitial conditions (top left) and the original versus doubled simulated annealing parameters(bottom left). The histogram formed with the Judges as an initial condition and the pre-viously reported simulated annealing parameters overlays this image with the gray shadedhistogram. We display our standard box-plots for the three initial conditions as we needto compare three results rather than two (top right) along with the histogram box-plots tocompare the effect of changing the simulated annealing parameters (bottom right). this space exhaustively would come at an large computational cost. We instead perform a simplesensitivity test on our current location in the parameter space by exploring the four dimensionalspace in four linearly independent directions. We explore over three directions by significantlyincreasing and decreasing w p , w I , and w m . For the fourth direction, we note that we could simplyincrease or decrease w c ; however, we thought it might be interesting to increase and decrease β instead. Because changing β is equivalent to changing all parameters, this forms a fourth linearlyindependent search direction, and provides us with information similar to changing w c . This leadsus to examine eight different parameter sets, which still requires a large number of runs. To cutdown on the computational cost, we take advantage of the result presented in section 7.1 above,where we conclude that ignoring the compactness threshold has a minimal effect on our results.The compactness threshold is by far the most restrictive, so omitting it will allow us to samplemore redistrictings with fewer runs.We present our results in Figure 16, and find that the results are very robust in all examineddirections of changing parameters. We note, however, that the percentage of redistrictings that falls × samples119.3 × samples F r a c t i on o f r e s u l t × samples119.3 × samples F r a c t i on o f r e s u l t × samples119.3 × samples D e m o c r a t i c v o t e f r a c t i on × samples119.3 × samples D e m o c r a t i c v o t e f r a c t i on Figure 15.
We extend the samples from the main text by allowing the sampling algorithmto continue until we have sampled roughly 120 thousand districts that fall below the thresh-old. We find almost no difference between the distributions in the original and extendedsamples. below our compactness acceptance threshold does change with varying parameters. Based on ourresult that election results are robust with respect to large changes in the compactness threshold,we conclude that significant changes in the parameters will have little effect on the statistical resultsof the election data.7.6.
Different weights for lower county splits.
In the above analysis we have prioritized com-pact districts over those with low two county splits. The result of this is presented in Figure 9and Section 4 above, and we note again that the number of two county splits in our samples isfar less than the NC2012 plan, but generally greater than that of NC2016 or the Judges. In thissection we determine the sensitivity of our results when we prioritize keeping a low number ofcounty splits. To make this examination, we double the county weight ( w c = 0 .
8) and reduce thecompactness weight ( w I = 2). By resetting the compactness threshold to be 80, we obtain justunder 15 thousand redistrictings and note that all of them have a worst district better than theworst NC2016 district. Keeping the threshold at 60 only yields a couple thousand samples, thus inorder to obtain more samples and because we have found that compactness does not have a largeeffect on the results, we select the higher threshold. We find that despite changing the weights inthis severe way, the over all election results, in addition to the distribution of results per ordereddistrict remains remarkably stable (see Figure 17). We also remark that we now have a median of o change β =0.8 β =1.2 w I =2w I =3w m =700 w m =900w p =2500w p =3000 D e m o c r a t i c v o t e f r a c t i on Figure 16.
We display standard box-plots and demonstrate how the election resultschange with respect to changing the values of the weights.
16 two county splits with a mean of 16.5, in contrast with a median of 21 and mean of 21.6 fromthe main results presented above.7.7.
Using a different compactness energy.
We have used the isoparametric ratio for thecompactness energy however there are other possible choices. Dispersion, mentioned in Section 3,measures how spread a district is. Typically it is thought of as the ratio between the area ofthe minimal bounding circle and the districts area. Although useful as a metric to compare twodistrictings, the dispersion score does not minimized for jagged perimeters and cannot be used as asufficient criteria to draw reasonable districts. Never-the-less, we examine the space of districts inwhich we replace the isoparametric ratios with the dispersion ratio. Given § Technical Discussions
Data sources and extraction.
The VTD geographic data were taken from the NCGAwebsite (see [12] from references) and the United States Census Bureau website (see [2] fromreferences), which provide for each VTD its area, population count of the 2010 census, the countyin which the VTD lies, its shape and location. Perimeter lengths shared by VTDs were extractedin ArcMap from this data. Minority voting age population was found on the NCGA website using2010 census data (see [13] from references). Data for the vote counts in each VTD for the 2012House elections was taken from Harvard’s Election Data Archive Dataverse (see [8] from references). ain runsLow county splits F r a c t i on o f r e s u l t F r a c t i on Main resultsLow county splits D e m o c r a t i c v o t e f r a c t i on Main resultsLow county splits D e m o c r a t i c v o t e f r a c t i on Figure 17.
By changing the weights on the energy function we alter the distribution oftwo county splits (top right). Despite these changes, the over all election results (top left)and box and box histogram plots by district (bottom) remain stable.
Main resultsDispersion ratio for compactness F r a c t i on o f r e s u l t Main resultsDispersion ratio for compactness D e m o c r a t i c v o t e f r a c t i on Main resultsDispersion ratio for compactness D e m o c r a t i c v o t e f r a c t i on Figure 18.
We change the compactness energy from the isoparametric ratio to a type ofdispersion score. Despite this drastic change in energy definition, we find the results to beremarkably similar.
Vote count data for the 2016 House elections was provided by NCSBE Public Data (See [19] fromreferences). We note that for the 2016 election, VTD data was not reported for all VTDs, but ratherfor each precinct; 2447 of the precincts are VTDs, meaning that we have data for the majority ofthe 2692 VTDs. However 172 precincts contain multiple VTDs, 66 VTDs were reported with splitdata, and 7 VTDs were reported with complex relationships. To extrapolate VTD data on those ontained in the 172 precincts containing multiple VTDs, we split the votes for a precinct amongthe VTDs it contained proportional to the population of each VTD. For the split VTDs, thosecontaining multiple precincts, we simply added up the votes among the precincts it contained.There was no extrapolation for these VTDs. For the VTDs with complex relationships, we dividedup the votes using estimates based on the geography and population of the VTDs. We note thatroughly 10% of the population lies in the VTDs with imperfect data, and that we do not expectsignificant deviation in our results based on the above approximations.In using 2012 and 2016 data we have only used presidential election year data. Unfortunately,the 2014 U.S. congressional election in North Carolina contained an unopposed race which preventsthe support for both parties being expressed in the VTDs contained in that district. In reference[1], the missing votes were replaced with votes from the Senate race. However, since we had twofull elections, namely 2012 and 2016, which needed little to no alterations, we chose not to includethe 2014 votes in our study.8.2. Examining nearby redistrictings within a distance.
The random sampling of the nearbydistricts is accomplished by running the same MCMC algorithm described in Section 3.3 with thesmall modification that if a proposed step ever tries to increase the deviation between any of thedistricts from the original redistricting in question (either NC2012, NC2016, or the Judges) above40 VTDs, then the step is rejected and the chain does not move on that round. Alternatively, onecan think of J ( ξ ) = ∞ for any ξ ∈ R which has a district that differs from the original redistrictingby more than 40 VTDs. As before, we then threshold the results for NC2016 and the Judges onthe Population Score, the County Score, and the Minority Score as described in Section 3.4. Wedo not threshold on the Isoperimetric Score as (i) keeping the redistricting near the original islikely sufficient and (ii) would be too severe for the NC2016 redistricting. We do not threshold theNC2012 at all since most of the redistrictings close to NC2012 would fail the Population thresholdsince the compactness energy is so large in this region that it overwhelms population considerations.We examine the difference in the local complementary cumulative distribution function (thresh-olded and not) to get a sense of how accurate the NC2012 local complementary cumulative distri-bution function is without thresholding. We find that there is only a modest difference betweenthe thresheld and non-thresheld results from the Judges which provides evidence that using thenon-thresheld results for NC2012 is unimportant for obtaining a representative space of nearbydistricts. Judges Without thresholdingWith thresholding F r a c t i on w / w o r s e i nde x Figure 19.
There is not a large difference between the thresholded and unthresholded results. . Discussion
We have provided a prototype probability distribution on the space of congressional redistrictingsof North Carolina. This distribution is non-partisan in that it considers no information beyond thetotal population, shape of the districts, and relevant demographic information for the VRA. Theprobability model was then calibrated to produce redistrictings which are comparable to the currentdistrict to the extent they partition the population equally and produce compact districts. Then,effectively independent draws were made from this probability distribution using the Metropolis-Hastings variant of Markov chain Monte Carlo. For each redistricting drawn, the 2012 and 2016U.S. House of Representatives election was retabulated using the actual vote counts to determinethe party affiliation of the winner in each district. The statistics of the number of Democraticwinners gave a portrait of the range of outcomes possible for the given set of votes cast. Thisdistribution could be viewed as the true will of the people.Redistricting’s reach is beyond simply the election outcome. By packing and cracking groups,districts can be made safe, which arguably can shift the ideological center of the candidate electedaway from the most representative positions. Our work also helps to identify when Gerrymanderinghas produced districts with unusually large concentrations of one party.Redistrictings producing outcomes which are significantly different than the typical results ob-tained from randomly sampled redistrictings are arguably at odds with the will of the peopleexpressed in the record of their votes. The fact that the election outcomes are so dependent on thechoice of redistrictings demonstrates the need for checks and balances to ensure that democracy isserved when redistrictings are drawn and the election outcome is representative of the votes cast.It seems unreasonable to expect that politics would not enter into the process of redistricting.Since the legislators represent the people and presumably express their will, restricting their abilityto express that will seems contrary to the very idea of democracy. Yet the work in this note couldlikely be developed into a criteria to decide when a redistricting fails to be sufficiently representativeof the will of the people. It would perhaps be reasonable to only allow redistrictings which yield themore typical results, eschewing the most atypical as a subversion of the people’s will. This wouldstill leave plenty of room for politics, but add a counter-weight to balance that role of partisanshipwhen it acts against the democratic ideals of a republic governed by the people.The most basic critiques of this work is that we have assumed that the candidate does not matter,that a vote for the Democrat or Republican will not change, even after the districts are rearranged.Furthermore, as districts become more polarized and many elections result in a forgone conclusion,voter turnout is likely suppressed. While we could try to correct for these effects, we find thesimplicity and power of using the actual votes very compelling. The results are both striking andilluminating.
Acknowledgments
We would like to thank the Duke Math Department, the Information Initiative at Duke (iID), thePRUV and Data+ undergraduate research programs for financial and material support. BridgetDou and Sophie Guo were integral members of the Quantifying Gerrymandering Team. Their workwas central in the first Data+ summer team which greatly influenced and informed the world ofthe second Data+ team on which this note is largely based. In particular, Bridget Dou wrotemuch of the code used that summer and Sophie Guo headed the GIS efforts. We would also liketo thank Mark Thomas, and the rest of the Duke Library GIS staff for help with extracting theneeded data from the congressional maps. We are also indebted to Robert Calderbank, GalenReeves, Henry Pfister, Scott de Marchi, and Sayan Mukherjee for help during the many phasesof this project. John O’Hale helped in procuring the 2016 election Data. We are also extremelygrateful to Tom Ross, Fritz Mayer, Land Douglas Elliott, and B.J. Rudell for letting us observe the eyond Gerrymandering Project. It was very educational and inspirational. We also appreciatedall of their help, encouragement, and insightful discussions. References [1] Sachet Bangia, Bridget Dou, Jonathan C. Mattingly, Sophie Guo, and Christy Vaughn. Quantifying gerryman-dering. https://services.math.duke.edu/projects/gerrymandering/ , 2015.[2] U.S. Census Borough. Shapefiles for congressional districts. , 2010.[3] Jowei Chen and Jonathan Rodden. Unintentional gerrymandering: Political geography and electoral bias inlegislatures.
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We display the various energies for each of the districtings that we present inthe appendix. Note that reported numbers for districtings are before VTD splits.
Departments of Economics and Computer Science, Duke University, Durham NC 27708
E-mail address : [email protected] Christy Vaughn Graves, Program in Applied and Computational Mathematics, Princeton Univer-sity, Princeton NJ
E-mail address : [email protected] Gregory Herschlag, Departments of Mathematics and Biomedical Engineering, Duke University,Durham NC 27708
E-mail address : [email protected] Han Sung Kang, Departments of Electrical and Computer Engineering, Computer Science, andMathematics, Duke University, Durham NC 27708
E-mail address : [email protected] Justin Luo, Departments of Electrical and Computer Engineering and Mathematics, Duke Univer-sity, Durham NC 27708
E-mail address : [email protected] Jonathan Mattingly, Departments of Mathematics and Statistical Science, Duke University, DurhamNC 27708
E-mail address : [email protected] Robert Ravier, Department of Mathematics, Duke University, Durham NC 27708