Partisan gerrymandering versus geographic compactness examines the relationship between partisan fairness and district shape. It discusses how gerrymandering has traditionally been detected by analyzing district shape and compactness. However, the document notes that voting results and partisan outcomes may provide more relevant information for detecting gerrymandering than just shape alone. The author presents research questions on whether partisan unfairness can be achieved with compact districts and whether there is always a choice of compact districts that exhibit partisan fairness.

1. 1/20
Partisan gerrymandering versus
geographic compactness
Dustin G. Mixon
Workshop on Quantitative Redistricting
The Statistical and Applied Mathematical Sciences Institute
October 9, 2018

2. 1/20
Gerrymander detection by shape
Long history of detecting gerrymandering by shape
Boston Gazette, 1812 Washington Post, 2014
“bizarre shape” quantified by “geographic compactness”
constraints imposed by bipartisan redistricting commissions

3. 2/20
Gerrymander detection by shape
Does shape provide enough information?
NC-12 IL-4
What about context? Does intent matter?
images from Wikipedia

4. 2/20
Gerrymander detection by shape
Does shape provide enough information?
NC-12 IL-4
What about context? Does intent matter?
Can we gerrymander with nice shapes?
images from Wikipedia

5. 3/20
Gerrymander detection by voting results
One modern approach: compare votes vs. seats
proportionality (not a valid constitutional standard)
efficiency gap (subject of Gill v. Whitford)
images from Wikipedia

6. 3/20
Gerrymander detection by voting results
One modern approach: compare votes vs. seats
proportionality (not a valid constitutional standard)
efficiency gap (subject of Gill v. Whitford)
Do voting results provide enough information?
images from Wikipedia

7. 3/20
Gerrymander detection by voting results
One modern approach: compare votes vs. seats
proportionality (not a valid constitutional standard)
efficiency gap (subject of Gill v. Whitford)
Do voting results provide enough information?
Is partisan fairness at odds with geometry?
images from Wikipedia

8. 4/20
Basic research questions
Question 1 (see part I of this talk)
Can we achieve partisan unfairness with nice shapes?

9. 4/20
Basic research questions
Question 1 (see part I of this talk)
Can we achieve partisan unfairness with nice shapes?
Question 2 (see part II of this talk)
Is there always a choice of nicely shaped districts exhibiting partisan fairness?

10. 4/20
Basic research questions
Question 1 (see part I of this talk)
Can we achieve partisan unfairness with nice shapes?
Question 2 (see part II of this talk)
Is there always a choice of nicely shaped districts exhibiting partisan fairness?
Question 3 (see the next talk)
How hard is it to find nicely shaped districts exhibiting partisan fairness?

11. 5/20
Joint work with
Boris Alexeev
borisalexeev.com

12. 6/20
Part I
compact gerrymandering

13. 7/20
How to quantify “bizarre shape”?
image from Wikipedia

14. 8/20
Classical notions of geographic compactness
Isoperimetry. wasted perimeter
perimeter
Polsby–Popper score ∝ (area) / (perimeter)2
Duchin, sites.duke.edu/gerrymandering/files/2017/11/MD-duke.pdf

15. 8/20
Classical notions of geographic compactness
Isoperimetry. wasted perimeter
perimeter
Polsby–Popper score ∝ (area) / (perimeter)2
Convexity. wasted area, indentedness
(area) / (area of convex hull)
Reock score = (area) / (area of minimum covering disk)
Duchin, sites.duke.edu/gerrymandering/files/2017/11/MD-duke.pdf

16. 8/20
Classical notions of geographic compactness
Isoperimetry. wasted perimeter
perimeter
Polsby–Popper score ∝ (area) / (perimeter)2
Convexity. wasted area, indentedness
(area) / (area of convex hull)
Reock score = (area) / (area of minimum covering disk)
Dispersion. second moment, sprawl
average distance between pairs of points
moment of inertia
Duchin, sites.duke.edu/gerrymandering/files/2017/11/MD-duke.pdf

17. 8/20
Classical notions of geographic compactness
Isoperimetry. wasted perimeter
perimeter
Polsby–Popper score ∝ (area) / (perimeter)2
Convexity. wasted area, indentedness
(area) / (area of convex hull)
Reock score = (area) / (area of minimum covering disk)
Dispersion. second moment, sprawl
average distance between pairs of points
moment of inertia
Question: Can we gerrymander with nice shapes?
Duchin, sites.duke.edu/gerrymandering/files/2017/11/MD-duke.pdf

18. 9/20
Passing to a (cartoon) model
state = closed disk
2n voters equispaced along concentric circle
voter preferences = iid uniform from {±1}
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5

19. 10/20
Passing to a (cartoon) model
state = closed disk
2n voters equispaced along concentric circle
voter preferences = iid uniform from {±1}
Theorem
Partition a closed disk C of unit radius into two regions A, B whose closures
are homeomorphic to C. Then
max |∂A|, |∂B| ≥ π + 2, min |A|
| hull(A)|
, |B|
| hull(B)|
≤ 1, IA + IB ≥ π
2
− 16
9π
.
Equality is achieved in all three when A and B are complementary half-disks.
Proof ingredients: Jordan curve theorem, basic convexity, calculus
Alexeev, M., arXiv:1712.05390

20. 11/20
Passing to a (cartoon) model
state = closed disk
2n voters equispaced along concentric circle
voter preferences = iid uniform from {±1}
Theorem
Let Dn denote the random number of majority-positive districts in an optimal
partisan gerrymander by complementary half-disks. Then
Pr(Dn = 2) =
1
2
− Θ(
1
√
n
), lim
n→∞
Pr(Dn = 0) =
1
1 + eπ
.
Proof ingredients: IVT, Donsker’s invariance principle, Brownian bridges
Upshot: 73% of seats (on average) from 50% of votes (on average)
Alexeev, M., arXiv:1712.05390

21. 12/20Walch, thenib.com/changing-the-math-on-gerrymandering

22. 13/20
Beyond the model
Apply gerrymandered version of Smith’s split-line algorithm:
Figure 1: Partitions of Wisconsin into 8 notional voting districts. Each partition was obtained by selecting lines
(“split-lines”) to iteratively split regions into two nearly equal populations. For the left-hand partition, these lines
were selected so as to maximize the resulting number of majority-Republican districts, whereas majority-Democrat
districts were encouraged for the partition on the right. Here, we applied the 2016 presidential election returns [11]
as a proxy for the spatial distribution of Republicans and Democrats. Wisconsin was particularly competitive in this
election, with Trump and Clinton receiving 1,405,284 and 1,382,536 votes, respectively. Under this proxy, Republicans
can win all 8 of the districts using split-lines (the existence of such a unanimous districting follows from the ham
sandwich theorem [8]), while Democrats can achieve 7 out of 8 (the most possible since they lost overall). Our main
result demonstrates that for a certain model of voter locations and preferences, one may use split-lines to gerrymander
Smith, rangevoting.org/SplitLR.html
Alexeev, M., arXiv:1712.05390
Sober´on, Notices Amer. Math. Soc., 2017

23. 13/20
Beyond the model
Apply gerrymandered version of Smith’s split-line algorithm:
Figure 1: Partitions of Wisconsin into 8 notional voting districts. Each partition was obtained by selecting lines
(“split-lines”) to iteratively split regions into two nearly equal populations. For the left-hand partition, these lines
were selected so as to maximize the resulting number of majority-Republican districts, whereas majority-Democrat
districts were encouraged for the partition on the right. Here, we applied the 2016 presidential election returns [11]
as a proxy for the spatial distribution of Republicans and Democrats. Wisconsin was particularly competitive in this
election, with Trump and Clinton receiving 1,405,284 and 1,382,536 votes, respectively. Under this proxy, Republicans
can win all 8 of the districts using split-lines (the existence of such a unanimous districting follows from the ham
sandwich theorem [8]), while Democrats can achieve 7 out of 8 (the most possible since they lost overall). Our main
result demonstrates that for a certain model of voter locations and preferences, one may use split-lines to gerrymander
In general, vote majority → seat unanimity (ham sandwich theorem)
Why not impose partisan fairness?
Smith, rangevoting.org/SplitLR.html
Alexeev, M., arXiv:1712.05390
Sober´on, Notices Amer. Math. Soc., 2017

24. 14/20
Part II
an impossibility theorem

25. 15/20
How to measure partisan fairness?
Proportionality. prop votes ≈ prop seats
Stephanopoulos, McGhee, Univ. Chic. Law Rev., 2015

26. 15/20
How to measure partisan fairness?
Proportionality. prop votes ≈ prop seats
Efficiency gap. # red wasted votes ≈ # blue wasted votes
voter locations = A, B ⊆ [0, 1]2
districts = D1 · · · Dk = [0, 1]2
wasted votes by A in i ∈ [k]:
wA,i :=
|A ∩ Di | − 1
2
|(A ∪ B) ∩ Di | if |A ∩ Di | > |B ∩ Di |
|A ∩ Di | else
efficiency gap:
EG(D1, . . . , Dk ; A, B) :=
1
|A ∪ B|
k
i=1
wA,i − wB,i
Stephanopoulos, McGhee, Univ. Chic. Law Rev., 2015

27. 16/20
Technical definitions
Districting system. dist: (A, B, k) → (D1, . . . , Dk ) =: D
Alexeev, M., arXiv:1710.04193

28. 16/20
Technical definitions
Districting system. dist: (A, B, k) → (D1, . . . , Dk ) =: D
One person, one vote. ∃δ ∈ [0, 1), ∀(A, B, k), D = dist(A, B, k) satisfies
(1 − δ)
|A ∪ B|
k
≤ (A ∪ B) ∩ Di ≤ (1 + δ)
|A ∪ B|
k
∀i ∈ [k]
Alexeev, M., arXiv:1710.04193

29. 16/20
Technical definitions
Districting system. dist: (A, B, k) → (D1, . . . , Dk ) =: D
One person, one vote. ∃δ ∈ [0, 1), ∀(A, B, k), D = dist(A, B, k) satisfies
(1 − δ)
|A ∪ B|
k
≤ (A ∪ B) ∩ Di ≤ (1 + δ)
|A ∪ B|
k
∀i ∈ [k]
Polsby–Popper compactness. ∃γ > 0, ∀(A, B, k), D = dist(A, B, k) satisfies
4π ·
|Di |
|∂Di |2
≥ γ ∀i ∈ [k]
Alexeev, M., arXiv:1710.04193

30. 16/20
Technical definitions
Districting system. dist: (A, B, k) → (D1, . . . , Dk ) =: D
One person, one vote. ∃δ ∈ [0, 1), ∀(A, B, k), D = dist(A, B, k) satisfies
(1 − δ)
|A ∪ B|
k
≤ (A ∪ B) ∩ Di ≤ (1 + δ)
|A ∪ B|
k
∀i ∈ [k]
Polsby–Popper compactness. ∃γ > 0, ∀(A, B, k), D = dist(A, B, k) satisfies
4π ·
|Di |
|∂Di |2
≥ γ ∀i ∈ [k]
Bounded efficiency gap. ∃α, β > 0, ∀(A, B, k), D = dist(A, B, k) satisfies
EG(D1, . . . , Dk ; A, B) <
1
2
− α whenever |A| − |B| < β|A ∪ B|
Alexeev, M., arXiv:1710.04193

31. 17/20
Impossibility
Theorem
There is no districting system that simultaneously satisfies
one person, one vote
Polsby–Popper compactness
bounded efficiency gap
Alexeev, M., arXiv:1710.04193

32. 17/20
Impossibility
Theorem
There is no districting system that simultaneously satisfies
one person, one vote
Polsby–Popper compactness
bounded efficiency gap
Proof idea: homogeneous mixture of voters
Alexeev, M., arXiv:1710.04193

33. 18/20
Beyond the model
images from Wikipedia

34. 18/20
Beyond the model
EG ≈ 0.3 0.08
MA ∈ {MD, NC, WI, . . .}
Political geography explanation?
NEW HAMPSHIREVERMONT
CONNECTICUT
RHODE
ISLAND
NEWYORK
ME
NEW YORK
The National Atlas of the United States of America
RU.S. Department of the Interior
U.S. Geological Survey
MASSACHUSETTSWhere We Are
nationalatlas.gov TM
O
pagecgd113_ma.ai INTERIOR-GEOLOGICAL SURVEY, RESTON, VIRGINIA-2013
MILES
0 20 3010 40
Albers equal area projection
Essex
Middlesex
Worcester
Franklin
Suffolk
Berkshire
Plymouth
Norfolk
Norfolk
Norfolk
Hampshire
Bristol
Hampden
Barnstable
Nantucket
Dukes
Suffolk
ATLANTICOCEAN
Cape Cod Bay
Nantucket Sound
6
5
1
7
8
3
9
4
2
North Adams
Hyannis
Greenfield
Provincetown
Brockton
Fall River
Fitchburg Gloucester
Haverhill
Holyoke
Lawrence
New Bedford
Pittsfield
Taunton
Westfield
QuincyFramingham
Lowell
Springfield
Worcester
Boston
1
2
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The Constitution prescribes Congres-
sional apportionment based on
decennial census population data. Each
state has at least one Representative, no
matter how small its population. Since
1941, distribution of Representatives has
been based on total U.S. population, so
that the average population per
Representative has the least possible
variation between one state and any
other. Congress fixes the number of
voting Representatives at each
apportionment. States delineate the
district boundaries. The first House of
Representatives in 1789 had 65
members; currently there are 435.
There are non-voting delegates from
American Samoa, the District of
Columbia, Guam, Puerto Rico, and the
Virgin Islands.
CONGRESSIONAL DISTRICTS
113th Congress (January 2013–January 2015)
images from Wikipedia

35. 19/20
Conclusions
You can gerrymander with nice shapes
Sometimes, you need strange shapes to obtain “fairness”
There is a nontrivial tension between shape and fairness
Perhaps political geography can help clarify this tension?

36. 20/20
Questions?
Partisan gerrymandering with geographically compact districts
B. Alexeev, D. G. Mixon
arXiv:1712.05390
An impossibility theorem for gerrymandering
B. Alexeev, D. G. Mixon
arXiv:1710.04193
Also, Google short fat matrices for my research blog.