REVEALING THE GEOPOLITICAL GEOMETRY THROUGH SAMPLING JONATHAN MATTINGLY (+ THE TEAM) DUKE MATH

REVEALING THE GEOPOLITICAL GEOMETRY THROUGH SAMPLING JONATHAN MATTINGLY (+ THE TEAM) DUKE MATH

gerrymander manipulate the boundaries of an electoral constituency to favor one party or class. achieve (a result) by manipulating the boundaries of an electoral constituency. "a total freedom to gerrymander the results they want" racial vs partisan gerrymander

North Carolina 13 Congressional Representatives

NC has around 10.2 million people Every decade, required to redo the (13) congressional districts Population Density Presidential Election 2016 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

U.S. House congressional districts for 2012 election

DEMOCRACY

DEMOCRACY The will of the people is expressed every person can VOTE every vote is COUNTED (once)

one person one vote principle By ensuring that each representative is subject to requests and suggestions from the same number of constituents, total population apportionment promotes equitable and effective representation. Justice Ruth Bader Ginsburg Evenwel v. Abbott, April 2016

2012 North Carolina Elections for U.S. House VOTES PERCENTAGE SEATS Democratic 2,218,357 50.65% 4 Republican 2,137,167 48.80% 9 Libertarian 24,142 0.55% 0 -The most Democratic district had 79.63% Democratic votes. -The most Republican district had 63.11% Republican votes.

Are these results due to political gerrymandering? or Are these results natural outcomes of NC's geopolitical structure of the spatial distribution of partisan votes?

40% Blue Red wins 3 Red wins 5 Red wins 2 60% Red Blue wins 2 Blue wins 0 Blue wins 3 from Wikipedia after an image by Steven Nass

2012 2016 Judges

How to quantify how gerrymandered How to quantify gerrymandering? or unrepresentative a redistricting is? How to reveal a state s geopolitical structure?

How does one find the true message in an election?

What if we drew the districts randomly? with no regard for party registration or most demographics reveal the geopolitical structure encoded in the votes

Many Groups using algorithmic generated maps to benchmark Jowei Chen (Univ Michigan) Wendy Cho (UIUC) Samuel Wang (Princeton) Kosuke Imai, Benjamin Fifield (Princeton) Alan Frieze, Wesley Pegden, Maria Chikina (CMU) Not all the same. Not all Random. Some generating alternative maps. Some Sampling a defined distribution. some using actual surrogate districts. Focus on our group at Duke

Impact of Duke Team work Gill v. Whitford (WI State Assembly) : Oral argument held in Supreme Court (SCOTUS) October 2 Provide report supporting Amicus Brief by Eric S. Lander Common Cause v. Rucho (N.C. Congressional): 3 judge conditional panel. Direct appeal to SCOTUS. Provide expert testimony and report in lawsuit. Closing arguments on October 16 North Carolina v. Covington (N.C. State Assembly): 3 judge conditional panel rule racial gerrymander. Affirmed by SCOTUS in June. Provide expert testimony on new maps produces at courts order Preparing for partisan gerrymander

The Recipe 1. Make a good random redistricting of N.C. into 13 U.S. house districts. 2. Count number of Democratic and Republican votes in each of the new districts using the actual 2012 votes. 3. Determine winner in each district of the random redistricting. 4. Return to step 1. Use Markov Chain Monte Carlo to sample a distribution on redistrictings

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

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 n=1 h i 2 Ideal (Pop in district n) 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.

Election Results for Ensemble of Redistricting Plans 0.6 0.4 Judges 0.5 Fraction of result 0.3 0.2 0.4 0.3 0.2 Judges 0.1 0 NC2012 NC2016 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) 0.1 0 NC2012 NC2016 2 3 4 5 6 7 8 Number of Democrats Elected (2016 votes)

Historical and Shifted Elections 0.58 0.57 0.56 GOV12 0.56 6 0.55 Republican Vote Fraction 0.54 0.53 0.52 0.51 USH16 PRE16 PRE12 GOV16 0.54 0.52 0.50 4 2 0 0.50 USH12 0.48 8 0.49 0.48 NCSS16 0.46 6 0.47 4 6 8 10 12 0.44 0 5 10 4 0 5 10 Elected Republicans Elected Republicans (2012 Votes) Elected Republicans (2016 Votes)

2012 2016 Judges

Other states?

Wisconsin General Assembly Fraction of result Fraction of result 0.2 0.1 0 0.2 0.1 50 55 60 65 70 75 WSA12 WI WSA14 WI 0 WSA16 WI (int) Fraction of result 0.2 0.1 0 WI (act) 50 55 60 65 70 75 Elected Republicans

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

Shift the global percentages 0.56 Majority Super Majority Majority Super Majority 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

seats vs global vote Number of Republican seats 90 80 70 60 50 40 30 Expected seats WI (contested) Standard Deviation 90% of ensemble Bound Super Majority Majority 20 10 WSA12 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

100 Structural advantage doesn t explain Probability 80 60 40 20 WI 0 0.46 0.48 0.50 Republican vote needed for parity in election (2012) 00 80 50 Probability 60 40 WI 20 WI 0 0.44 0.46 0.48 0.50 Republican vote needed for parity in election (2014) 0 0.44 0.46 0.48 Republican vote needed for parity in election (2016)

Shift to Median Parity % of maps 20 15 10 5 WI 0 40 45 50 55 WSA12 Interpolated Votes (shifted to parity) 20 20 % of maps 15 10 % of maps 15 10 5 WI 5 WI 0 40 45 50 55 WSA14 Interpolated Votes (shifted to parity) 0 40 45 50 55 WSA16 Interpolated Votes (shifted to parity)

What produces these effects? What is the signature of gerrymandering?

Red wins 2 districts by 8 votes each Blue wins 3 districts by 2 votes each Percentage of Democrats from lowest to highest 10% 10% 60% 60% 60% Red wins 2 Blue wins 3

Democratic Winning Percentages (House 2012) Democratic Winning Percentages 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican to Most Democratic Districts

Democratic Winning Percentages (House 2012) Democratic Winning Percentages 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican to Most Democratic Districts

Democratic Winning Percentages (House 2012) Democratic Winning Percentages 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1 2 3 4 5 6 7 8 9 10 11 12 Most Republican to Most Democratic Districts 13

Democratic Winning Percentages (House 2012) Democratic Winning Percentages 0.8 NC2012 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1 2 3 4 5 6 7 8 9 10 11 12 Most Republican to Most Democratic Districts 13

Democratic Winning Percentages (House 2012) Judges NC2012 NC2016 Democratic Winning Percentages 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1 2 3 4 5 6 7 8 9 10 11 12 Most Republican to Most Democratic Districts 13

Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 NC2012 NC2016 Judges 0.3 2 4 6 8 10 12 Most Republican To Most Democratic Districs (2012 votes)

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

NC Congressional Delegation

Wisconsin General Assembly 1.0 0.55 40 50 60 Fraction of Democratic vote 0.8 0.6 0.4 0.50 0.45 0.40 0.2 WSA14 0 10 20 30 40 50 60 70 80 90 100 District from most to least Republican

Metrics

Probability density Gerrymander Index Measure deviation for expected district structure 30 Judges Judges 25 20 15 10 5 NC2016 NC2012 NC2016 NC2012 0 0 0.1 0.2 0.30 0.1 0.2 (2012 votes) Gerrymandering index (2016 votes) Fraction w/ worse index 1.0 0.8 0.6 0.4 0.2 Judges NC2016 NC2012 Judges NC2016 NC2012 0 0 0.1 0.2 0.30 0.1 0.2 (2012 votes) (2016 votes)

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 `(map) = log Prob(outcome map produces) 0.48 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 Probability 0.5 WI 0 1 2 3 4 5 6 7 H 2.0 WSA14 1.5 1.0 The Wisconsin plans are clearly an outlier for the average log likelihood over shifts 45%-55% 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

Efficiency Gap Situate EG inside the ensemble 1.0 1.0 Fraction w/ worse index 0.8 0.6 0.4 0.2 0 Judges NC2016 NC2012 0.1 0.2 0.3 0.4 0.5 Efficiency gap (2012 votes) Fraction w/ worse index 0.8 0.6 0.4 0.2 0 Judges NC2016 NC2012 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Efficiency gap (2016 votes)

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

Judges NC2012 NC2016 NC 2012 NC 2016 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)

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 0 Main results Dispersion ratio for compactness 3 4 5 6 7 8 9 10 Number of Democrats Elected (2012 votes) Democratic vote fraction 0.8 0.7 0.6 0.5 0.4 0.3 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 1 2 3 4 5 6 7 8 9 10 11 12 13 Most Republican To Most Democratic Districs (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

Accelerate the sampling parallel tempering accelerated sampling hierarchical sampling parallel algorithms

The Team Jonathan Mattingly Christy Graves Sachet Bangia Sophie Guo Bridget Dou Justin Luo Hansung Kang Robert Ravier Greg Herschlag Michael Bell MATH arxiv:1410.8796 arxiv:1709.01596 arxiv:1704.03360

Conclusions Identify the background NULL hypothesis Identify the background geopolitical structure of state. Identify the outliers, the unreasonable maps. Benchmark different proposed metrics Interaction of geopolitical structure and metrics Considered other states and effect of VRA Lots of interesting math questions