QUANTIFYING GERRYMANDERING REVEALING GEOPOLITICAL STRUCTURE THROUGH SAMPLING GEOMETRY OF REDISTRICTING WORKSHOP CALIFORNIA GREG HERSCHLAG, JONATHAN MATTINGLY + THE TEAM @ DUKE MATH
Impact of Duke Team s work Gill v. Whitford (WI State Assembly) : Oral argument held in Supreme Court (SCOTUS) October 2, 2016 Undergraduate research 2014, 2015, 2016, 2018 Provide report supporting Amicus Brief by Eric S. Lander Common Cause v. Rucho (N.C. Congressional): As seen in 3 judge conditional panel. Direct appeal to SCOTUS. Nov 2016 Provided expert testimony and report in lawsuit Heavily cited in court judgment 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
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
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
One Step of MCMC Proposal Then accept/reject according to score function
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
Does Gerrymandering mean skewed election results? NC : US House 2012 WI : Gen Assembly 2014 Vote Seats Vote Seats Democratic 50.65% 4 (31%) Democratic 51.28% 36 (36%) Republican 48.80% 9 (69%) Republican 48.72% 63 (64%)
Situate enacted maps in respective ensembles 0.4 Fraction of result 0.3 0.2 0.1 NC2012 Act43 0 3 4 5 6 7 8 9 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 2012 NC congressional votes 2014 WI state assembly votes Number of Democrats Elected
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
Skewed election results do not necessarily imply Gerrymandering
District compactness and the skewness of election results are not enough to detect Gerrymandering How can we use ensembles to robustly detect Gerrymandering?
Order the districts by the Democratic vote fraction Percentage of Democrats from lowest to highest Most Republican Most Democratic 10% 10% 60% 60% 60% Red
Order the districts by the Democratic vote fraction Percentage of Democrats from lowest to highest Most Republican Most Democratic 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 Identify Cracked and Packed 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 Two principle plots presented in Common Cause v. Rucho
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
Wisconsin General Assembly 0.9 0.8 0.55 0.50 40 45 50 55 60 65 % of Dem. Vote 0.7 0.6 0.5 0.4 0.45 0.40 0.3 0.2 WSA16 0 10 20 30 40 50 60 70 80 90 100 District from most to least Republican
Structural advantage exists; sampling 100 80 decouples geopolitical effects from Gerrymandered effects Probability 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 0.44 0.46 0.48 0.50 0.44 0.46 0.48 Republican vote needed for parity in election (2014) Republican vote needed for parity in election (2016) Chen and Rodden. Quarterly Journal of Political Science. (2013) 8:239 269
Stagnating 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
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
Compact districts do not preclude gerrymandering Skewed vote counts do not necessarily indicate gerrymandering Sampling techniques can detect gerrymandering Local analyses can help determine 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)
The Team Jonathan Mattingly Christy Graves Justin Luo Sachet Bangia Sophie Guo Hansung Kang Robert Ravier Greg Herschlag Bridget Dou Michael Bell MATH https://sites.duke.edu/quantifyinggerrymandering/
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.
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. 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
N.C. Precincts around 3,000
Score function P ( ) = 1 Z e J( ) : {Precincts} 7! {1,...,13} J( ) =w p J pop ( )+w I J compact ( )+w c J county ( )+w m J mino ( ) (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
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
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
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)
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