QUANTIFYING GERRYMANDERING REVEALING GEOPOLITICAL STRUCTURE THROUGH SAMPLING

QUANTIFYING GERRYMANDERING REVEALING GEOPOLITICAL STRUCTURE THROUGH SAMPLING TURING INSTITUTE LONDON, UK JONATHAN MATTINGLY, GREG HERSCHLAG + THE TEAM @ DUKE MATH

Impact of Duke Team s work Common Cause v. Rucho (N.C. Congressional): 3 judge conditional panel. Direct appeal to SCOTUS. Nov 2017 Undergraduate research 2014, 2015, 2016, 2018 Provided expert testimony and report in lawsuit Heavily cited in court judgment As seen in Gill v. Whitford (WI State Assembly) : Oral argument held in Supreme Court (SCOTUS) October, 2017 Provide report supporting Amicus Brief by Eric S. Lander North Carolina v. Covington (N.C. State Assembly): 3 judge panel rule racial gerrymander. Affirmed by SCOTUS in June Provide expert testimony on new maps produces at courts order Preparing for partisan gerrymander 2014 - present (arxiv:1410.8796 - arxiv:1801.03783) sites.duke.edu/quantifyinggerrymandering

No. 17-1295 din THE ROBERT A. RUCHO, et al., Supreme Court of the United States v. Appellants, Last Month submitted to the US Supreme court 14 packed into overwhelmingly Democratic districts at the top of the S or dispersed within safe Republican districts at the bottom of the S. App.102-103. 13 COMMON CAUSE, et al., Appellees. ON APPEAL FROM THE UNITED STATES DISTRICT COURT FOR THE MIDDLE DISTRICT OF NORTH CAROLINA MOTION TO AFFIRM BY THE COMMON CAUSE APPELLEES GREGORY L. DISKANT JONAH M. KNOBLER PETER A. NELSON ELENA STEIGER REICH PATTERSON BELKNAP WEBB & TYLER LLP 1133 Avenue of the Americas New York, New York 10036 (212) 336-2000 gldiskant@pbwt.com EMMET J. BONDURANT Counsel of Record BENJAMIN W. THORPE BONDURANT MIXSON & ELMORE LLP 3900 One Atlantic Center 1201 West Peachtree Street Atlanta, Georgia 30309 (404) 881-4100 bondurant@bmelaw.com Counsel for the Common Cause Appellees (Counsel continued on inside cover) On January 9, 2018, the District Court issued an opinion holding unanimously that Appellants had standing to challenge the 2016 Plan on a statewide and district-by-district basis. The court unanimously found that the Plan violates the Equal Protection Clause and Art. I, 2 and 4. A two-judge majority also held that the Plan violates the First Amendment. On January 18, 2018, this Court stayed the judgment pending appeal. Dr. Mattingly, meanwhile, generated over 24,000 alternative maps using only nonpartisan districting criteria. Fewer than 0.7% of them resulted in a Republican advantage as lopsided as 10-3. App.101. Dr. Mattingly s simulations also confirmed that the 2016 Plan both packed and cracked Democratic voters. As he explained, this can be shown by plotting the Democratic vote share of each district on a graph, with the most Republican districts on the left and the most Democratic on the right. As the diagram below reflects, with no packing or cracking, the median map in Dr. Mattingly s simulation set (shown in yellow) yields a straight line. The actual results for the 2016 Plan (shown in blue) are quite different. They resemble an S curve, with Democratic voters either

Gerrymander Manipulate district boundaries to favor one party (partisan) or class (racial) Change the outcome of an election "gerrymander the results Boston Gazette 26 March, 1812 racial vs partisan gerrymander

North Carolina 13 Congressional Representatives

NC 2012 US Congressional Districts Is Gerrymandering Oddly Shape Districts?

Which Doesn t Belong?

Same NC 2012 Same NC 2016 Different Beyond Gerrymandering Judges

Gerrymandering as Startling Election Results NC : US House 2012 WI : Gen Assembly 2014 Vote Seats Vote Seats Democratic 50.65% 4 (31%) Republican 48.80% 9 (69%) MD : US House 2012 Democratic 51.28% 36 (36%) Republican 48.72% 63 (64%) USA : US House 2012 Vote Seats Vote Seats Democratic 63% 7 (87.5%) Republican 33% 1 (12.5%) Democratic 50.65% 4 (31%) Republican 48.80% 9 (69%) The most Democratic district had 79.63% Democratic votes The most Republican district had 63.11% Republican votes.

How to quantify how gerrymandered or U.S. Not a Proportional Representation System \ unrepresentative a redistricting is? Geographically Localize Representation

Geographically Diverse Population Density Presidential Election 2016 Ideal Congressional District (1/13 of population) : 733,499 people Charlotte Area: Charlotte-Gastonia-Salisbury- population 2,402,623 The Triangle: Raleigh-Durham-Cary-Chapel Hill- population 1,749,525 The Piedmont Triad: Greensboro Winston-Salem High Point- population 1,589.200

How to quantify gerrymandering? maybe we should understand this as How to reveal a state s natural geopolitical structure?

60% Red Red wins 3 Red wins 5 Red wins 2 40% Blue Blue wins 2 Blue wins 0 Blue wins 3 Wikipedia; image by Steven Nass

How to quantify how gerrymandered or When is a map fair? unrepresentative a redistricting is? When is a map typical?

How to quantify how gerrymandered or What if we drew the districts randomly? unrepresentative a redistricting is? with no regard for party registration or most demographics Look for the likely behavior of an ensemble of districting plans create a null-hypothesis without partisan bias

Groups using algorithmic generated maps to benchmark Jowei Chen (Michigan), Jonathan Rodden (Stanford) Wendy Cho (UIUC) Kosuke Imai, Benjamin Fifield (Princeton) Alan Frieze, Wesley Pegden, Maria Chikina (CMU,Pitt) All generating alternative maps. Some sampling a defined distribution. Some using actual surrogate districts. Focus on our group at Duke is based on principled, explicit distribution on redistricting plans

The Recipe 1. Determine a compliant random redistricting plan (equal population, compact, VRA compliant, communities of interest kept intact) 2. Count number of Democratic and Republican votes in each of the new districts using actual votes 3. Determine winner in each district of the random redistricting plan 4. Return to step 1 Use Markov Chain Monte Carlo to sample a distribution on redistricting plans

Criteria for Sampling

non-partisan design criteria (HB 92) 1. districts have equal population 2. the districts are connected and compact, 3. splitting counties is minimized, and 4. African American voters are sufficiently concentrated in 2 districts to affect the winner.

N.C. HOUSE BILL 92 REDISTRICTING STANDARDS Districts within 0.1% of equal population Districts shall be reasonably compact Contiguous territory, attempting not to split cities or counties Comply with the Voting Rights Act of 1965 Ignore: Incumbency, party affiliation, demographics

N.C. HOUSE BILL 92 REDISTRICTING STANDARDS Districts within 0.1% of equal population (we get close) Districts shall be reasonably compact Contiguous territory, attempting not to split cities or counties Comply with the Voting Rights Act of 1965 Ignore: Incumbency, party affiliation, demographics

Use Markov Chain Monte Carlo to sample from redistricting with good scores. Sample: (density) / e (score of redistricting) Know what distribution we are sampling from. Not just generating a large number of alternatives.

N.C. Precincts: around 3,000 : {Precincts} 7! {1,..., 13} 1 P ( ) = e Z J( ) (a 13 color Potts Model with an unusual energy)

Score function J( ) =w p J pop ( )+w I J compact ( )+w c J county ( )+w m J mino ( ) : {Precincts} 7! {1,...,13} P ( ) = 1 Z e J( ) (a 13 color Potts Model with an unusual energy)

Population Score Sum of square deviation from ideal district population 13 X h i 2 Ideal (Pop in district n) n=1 Ideal = Population of N.C. 13 733, 499

Compactness score (Perimeter) 2 Area 4 12.5 Minimized for a circle Also considered the ratio of district s area to the smallest circumscribing rectangle

Also include score terms for Voting Rights Act and Preserving County Boundaries Soft penalization : for number of split counties of different sizes redistricting plans without two districts meeting minimal voting age black population.

Use Markov Chain Monte Carlo to sample Sample: (density) / e (score of redistricting)

One Step of MCMC Proposal Then accept/reject according to score function

N.C. Precincts around 3,000

Ensemble of ~24,000 NC redistricting plans 0.6 0.5 Fraction of result 0.4 0.3 0.2 0.1 0 3 4 5 6 7 8 9 3 4 5 6 7 8 9 2012 congressional votes 2016 congressional votes Number of Democrats Elected

NC 2012 NC 2016 Panel of Judges

Situate maps in ensemble of 24,000 redistricting plans 0.6 0.5 Fraction of result 0.4 0.3 0.2 0.1 NC2016 NC2012 Judges NC2016 NC2012 Judges 0 3 4 5 6 7 8 9 3 4 5 6 7 8 9 2012 congressional votes 2016 congressional votes Number of Democrats Elected

Across many elections NC2012 Statewide Democratic Vote Fraction 0.52 0.50 0.48 0.46 0.44 NCSS16 USH12 GOV16 USS14 PRE12 PRE16 USH16 GOV12 NC2016 Judges 0 2 4 6 8 10 Democrats Elected

Atypical NC 2012 Atypical NC 2016 Typical Panel of Judges

Gerrymandering can occur in the absence of oddly shaped districts

What about Startling results?

Order the districts by the Blue vote fraction Percentage of Blue from lowest to highest Most Red Most Blue 10% 10% 60% 60% 60% Red

Order the districts by the Blue vote fraction Percentage of Blue from lowest to highest Most Red Most Blue 10% 10% 60% 60% 60% Red Red Red 0% 0% 0% 100% 100% 40% 40% 40% 40% 40% 20% 30% 40% 50% 60%

NC Congressional Delegation 0.8 Democratic vote fraction 0.7 0.6 0.5 0.4 0.3 0.2 2 4 6 8 10 12 2 4 6 8 10 12 (2012 congressional votes) (2016 congressional votes). Most Republican To Most Democratic Districts

Judges Comp/County a 538 plan Compact a 538 plan

Are we sampling the space in a reasonable way? 0.8 Democratic vote fraction 0.7 0.6 0.5 0.4 0.3 Judges 538 - Comp/Cnty 538 - Compact Medians 0.2 2 4 6 8 10 12 2 4 6 8 10 12 (2012 congressional votes) (2016 congressional votes). Most Republican To Most Democratic Districts

Gerrymandering Index Probability distribution 30 25 20 15 10 5 Judges 538-Comp 538-Comp/Cnty Judges 538-Comp/Cnty 538-Comp 0 0 0.1 0 0.1 (2012 congressional votes) (2016 congressional votes) Gerrymandering index

NC2012 NC2016

NC Congressional Delegation Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 0.3 Judges NC2012 NC2016 Medians 0.2 2 4 6 8 10 12 2 4 6 8 10 12 (2012 congressional votes) (2016 congressional votes). Most Republican To Most Democratic Districts

Gerrymandering Index Probability distribution 30 25 20 15 10 5 Judges NC2016 NC2012 Judges NC2016 NC2012 0 0 0.1 0.2 0.30 0.1 0.2 (2012 congressional votes) (2016 congressional votes) Gerrymandering index Outlier analysis Eric Lander s Amicus Brief in Gill v. Whitford

Rep (538) Dem (538)

Signature of Gerrymandering Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 0.3 538 - GOP 538 - Dem NC2012 NC2016 Judges Medians 0.2 2 4 6 8 10 12 2 4 6 8 10 12 (2012 votes) (2016 votes). Most Republican To Most Democratic Districts

Signature of Gerrymandering 49%-51% 53%-47% -6% -8% -7% +15% +14% +11% -6% -8% -10% +10% +6% +5% D3 D6 D5 D11 D2 D10 D13 D8 D9 D7 D4 D1 D12 D3 D11 D10 D7 D6 D8 D5 D9 D2 D13 D12 D4 D1 Two principle plots presented in Common Cause v. Rucho Identify Cracked and Packed districts

The Signature of Gerrymandering and its effects

Ensemble of around 19,000 districting plans Wisconsin General Assembly 0.2 50 60 70 WSA 2012 0.1 Act 43 0 Fraction of result 0.2 0.1 0 WSA 2014 Act43 WSA 2016 0.2 Act43 0.1 0 Act 43 (act.) 50 60 70 Elected Republicans

Wisconsin historical elections 0.54 WSA16 GOV12 Fraction of Republican vote 0.53 0.52 0.51 0.50 USH14 USH12 GOV14 WSA14 PRE16 WSA12 0.49 0.48 45 50 55 60 65 70 75 Number of Republican seats

Wisconsin historical elections 0.54 WSA16 GOV12 GOV14 Fraction of Republican vote 0.52 0.50 0.48 USH14 USH12 WSA12 WSA14 PRE16 0.46 40 50 60 70 Number of Republican seats

Wisconsin historical elections 0.54 WSA16 GOV12 GOV14 Fraction of Republican vote 0.52 0.50 0.48 0.46 USH14 USH12 SOS14 USS12 PRE12 WSA12 WSA14 PRE16 Firewall 40 50 60 70 Number of Republican seats

Stagnating NC election results due to Gerrymandering Statewide Democratic Vote Fraction 0.52 0.50 0.48 0.46 0.44 NCSS16 USH12 GOV16 USS14 PRE12 PRE16 USH16 GOV12 0 2 4 6 8 10 Democrats Elected NC2012 NC2016 Judges Democratic vote fraction Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 0.3 0.8 0.7 0.6 0.5 0.4 0.3 0.2 NC2012 NC2016 Judges Medians 5 10 USH 2012 USH 2016

Back to WI 100 Structural advantage exists; sampling decouples geopolitical effects from Gerrymandered effects Probability 80 60 40 20 WI 0 0.46 0.48 0.50 Republican vote needed for parity in election (2012) 100 80 60 Probability 50 Probability 40 WI 20 WI 0 0.44 0.46 0.48 0.50 0 0.44 0.46 0.48 Republican vote needed for parity in election (2014) Republican vote needed for parity in election (2016)

Where is Gerrymandering occurring? Localized analysis

Precinct Level Analysis vote fraction at predict level is this precinct gerrymandered?

Precinct Level Analysis vote fraction at predict level pick a districting plan

Precinct Level Analysis vote fraction at predict level the district has a partisan vote fraction

Precinct Level Analysis vote fraction at predict level 10 Example District Map Probability 5 0 0.3 0.4 0.5 0.6 0.7 Democratic vote fraction

NC 2012 Red = more Republican than expected Blue = more Democratic than expected vote fraction at predict level

NC 2016 average (signed) log likelihood of NC2016 district level results relative to ensemble vote fraction at predict level

NC 2016 - Triangle average (signed) log likelihood of NC2016 district level results relative to ensemble vote fraction at predict level

NC Beyond Gerrymandering Judges average (signed) log likelihood of Judges district level results relative to ensemble vote fraction at predict level

Local analysis can detect which districts have been Gerrymandered

Stability of Conclusions

Fraction of result 0.6 0.5 0.4 0.3 0.2 24.5 10 3 samples 119.3 10 3 samples Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 24.5 10 3 samples 119.3 10 3 samples Fraction of result 0.6 0.5 0.4 0.3 0.2 Reported S.A. parameters Doubled S.A. parameters Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 Reported S.A. parameters Doubled S.A. parameters 0.1 0.1 0.3 0 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) 0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (2012 votes) 0 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (2016 votes) Fraction of result 0.6 0.5 0.4 0.3 0.2 Judges (initial) NC2012 (initial) NC2016 (initial) Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 Judges (initial) NC2012 (initial) NC2016 (initial) Fraction of result 0.6 0.5 0.4 0.3 0.2 Population threshold at 1% Population threshold at 0.75% Population threshold at 0.5% Democratic vote fraction 0.7 0.6 0.5 Population Threshold 1% Population Threshold 0.5% 0.1 0.3 0.1 0.4 0 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (2016 votes) 0 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) 0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (2012 votes) Fraction of result 0.6 0.5 0.4 0.3 0.2 0.1 Main results Dispersion ratio for compactness Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 No change β=0.8 β=1.2 w I =2 w I =3 w m =700 w m =900 w p =2500 w p =3000 0 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) 0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (2012 votes)

Common Metrics

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sha1_base64="npr2df2vqwbg1fere+du8lcgv4q=">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</latexit> Common Metrics Efficiency Gap: (McGhee & Stephanopoulos) X X Waste = (vote fraction 0.5) + vote fraction districts won districts lost EG = Waste(Democrat) Waste(Republican) [Vote(Dem) Vote(Rep)] 1 2 [Seats(Dem) Seats(Rep)] Bernstein & Duchin 16 Partisan Bias: (Gelman and King) difference in seat fraction won by the Republicans if they receive 55% of the vote and the seat fraction won by the Democrats if they receive 55% of the vote (under partisan swing assumption).

Efficiency Gap 120 More Wasted R Votes More Wasted D Votes More Wasted R Votes More Wasted D Votes Probability distribution 100 80 60 40 20 Judges NC2012 NC2016 Judges NC2012 NC2016 0 0.1 0 0.1 0.2 0.1 0 0.1 0.2 (2012 votes) (2016 votes) Efficiency gap

Efficiency Gap 0.58 Statewide Republican Vote Fraction 0.56 0.54 0.52 0.50 0.48 GOV12 USH16 USS16 PRE16 PRE12 USS14 GOV16 ATT16 USH12 NCSS16 0.2 0 0.2 0.4 Efficiency Gap

Partisan Bias 0.4 Dem. Bias Rep. Bias Dem. Bias Rep. Bias Fraction w/ result 0.3 0.2 0.1 Judges NC2016 NC2012 Judges NC2012 NC2016 0 0.2 0 0.2 0.2 0 0.2 (2012 votes) (2016 votes) Partisan bias

0.58 Statewide Republican Vote Fraction 0.56 0.54 0.52 0.50 0.48 GOV12 USH16 USS16 PRE16 PRE12 USS14 GOV16 ATT16 USH12 NCSS16 0.2 0 0.2 0.4 Partisan Bias Percent of result 40 30 20 10 Partisan Bias 0 0.2 0 0.2 0.4 Partisan bias over all elections and plans

The Team PRUV 2014 Jonathan Mattingly Christy Graves Sachet Bangia Sophie Guo Bridget Dou Data+ 2015 Data+ 2016 Data+ 2018 Bass Connection 2018-2019 Justin Luo Hansung Kang Robert Ravier Greg Herschlag Michael Bell MATH https://sites.duke.edu/quantifyinggerrymandering/

Conclusions Compact districts do not preclude gerrymandering Sampling techniques can detect gerrymandering Provide a Null model against which to compare Local analyses indicate which districts have been gerrymandered

Math Questions?

Assume the population is uniform model a random distribution of political parties Q: Find null distribution of order statistics for district make up

Q: Give some form of stability of plots over a class of energy functions which have certain marginal statistics. Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 No change β=0.8 β=1.2 w I =2 w I =3 w m =700 w m =900 w p =2500 w p =3000 0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (2012 votes)

Q: Characterize the structure of the energy landscape Even with just population and compactness some evidence of phase transitions, and shattering of phase space

Accelerate the sampling parallel tempering accelerated sampling hierarchical sampling parallel algorithms

One Step of MCMC Proposal Then accept/reject according to score function

More details on Wisconsin

Shift the global percentages 0.56 Majority Super Majority Majority Super Majority Under Uniform Partisan Swing Assumption 0.54 WSA16 Fraction of Republican vote 0.52 0.50 0.48 WSA14 0.46 0.44 40 60 80 40 60 80 Republicans elected

90 seats vs global vote (Wisconsin) Number of Republican seats 80 70 60 50 40 30 20 10 Expected seats WI (contested) Standard Deviation 90% of ensemble Bound WSA12 Super Majority Majority 45 50 55 60 % Vote to the Republicans Number of Republican seats 90 80 70 60 50 40 30 20 Expected seats WI (contested) Standard Deviation 90% of ensemble Bound WSA14 % Vote to the Republicans Super Majority 45 50 55 60 Majority Number of Republican seats 90 80 70 60 50 40 30 Expected seats WI (contested) Standard Deviation 90% of ensemble Bound WSA16 Super Majority 45 50 55 60 % Vote to the Republicans Majority

Measuring Representativeness 0.66 0.64 0.62 WSA16 WI (int) 0.60 Fraction of result 0.2 0.1 0 WI (act) 50 55 60 65 70 75 Elected Republicans Frac Republican Vote 0.58 0.56 0.54 0.52 0.50 0.48 `(map) = log Prob(outcome map produces) 0.46 0.44 Average `(map) over shift 0.42 40 60 80 Republicans Elected

2.0 WSA12 Probability 1.5 1.0 Measuring Representativeness 0.5 WI 0 1 2 3 4 5 6 7 H 2.0 WSA14 1.5 The Wisconsin plans are clearly an outlier for the average log likelihood over shifts 45%-55% Probability 1.0 0.5 2.0 0 WI 1 2 3 4 5 6 7 H WSA16 1.5 Probability 1.0 0.5 0 WI 1 2 3 4 5 6 7

Engineered? results should be stable under small changes to districts sample near by districts and observe changes

NC 2012 NC 2016 Judges NC2012 NC2016 Judges NC2012 NC2016 Judges Judge s districts resemble near by districts NC 2012 and NC 2016 do not

Gerrymander Index Local Perturbations Fraction w/ worse index 1.0 0.8 0.6 0.4 0.2 0 Judges NC2016 NC2012 0.20 0.22 0.24 0.26 0.28 0.12 0.13 0.14 0.15 0.16 0.04 0.05 0.06 0.07 Gerrymandering index (2012 votes)