# How to Avoid Gerrymandering ANew Algorithmic Solution by lhh12385

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```									                                    Re-Districting: A . . .
How to Avoid . . .
Toward Formulation of . . .

How to Avoid                      Relation to Clustering
Towards a More . . .

Gerrymandering:                    Example
Resulting Iterative . . .
Conclusion

A New Algorithmic                   Acknowledgments

Solution                                Title Page

Gregory B. Lush
Department of Electrical and
Computer Engineering
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Esteban Gamez

Department of Computer Science            Full Screen

University of Texas at El Paso              Close
El Paso, TX 79968, USA
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Re-Districting: A . . .

1.     Re-Districting: A Practical Problem                         How to Avoid . . .

Toward Formulation of . . .
• The notion of electoral districts: in the USA and in
Relation to Clustering
other countries, voting is done by electoral districts.
Towards a More . . .

• First objective: equal representation.                      Example

• How: all voting districts of the same level (federal,       Resulting Iterative . . .

state, city, etc.) contain same number of voters.           Conclusion

Acknowledgments
• Problem: in time, demography changes – some districts
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lose voters, some gain them.
• Solution: re-districting.
• Main objective of re-districting: best represent geo-
graphic regions.                                                   Page 2 of 16

• Additional objective: represent rural and city areas, mi-            Go Back

norities, border areas, ﬂooded areas, etc.                          Full Screen

• Open problem: how to take all these factors into ac-                   Close

count?
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Re-Districting: A . . .

2.     Gerrymandering: A Problem                                     How to Avoid . . .

Toward Formulation of . . .
• How re-districting is done: in most states, it is voted
Relation to Clustering
upon by the legislature.
Towards a More . . .

• Drawback: in the next elections, the representation           Example

may be unfairly biased towards a party in power.              Resulting Iterative . . .

• Phenomenon behind bias: for parties A and B, A votes          Conclusion

in B-majority district are “lost” – this district votes for   Acknowledgments

B anyway.                                                             Title Page

• Gerrymandering – how: a party in power (A)
– divides all the A voters into A-majority districts,
and                                                               Page 3 of 16

– attaches, to each A-district, many B-voters (w/out                  Go Back

violating A-majority).                                             Full Screen

• Result: many B votes are lost, while no A votes are                      Close
lost.
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Re-Districting: A . . .

3.     Gerrymandering: Example                                                  How to Avoid . . .

Toward Formulation of . . .
• Example: 18 towns on similar size:
Relation to Clustering

– A1 , . . . , A10 vote for A,                                          Towards a More . . .

– B1 , . . . , B8 vote for B.                                           Example

Resulting Iterative . . .
• Objective: form 6 electoral districts.
Conclusion
10        10
• Fair representation: A gets    ·6 =      = 3.33 . . . votes              Acknowledgments
18         3                                         Title Page
– i.e., 3 or 4.
• Example of gerrymandering: divide 18 towns into the
following 6 districts:
– {A1 , A2 , B1 }, {A3 , A4 , B2 }, {A5 , A6 , B3 }, {A7 , A8 , B4 },          Page 4 of 16

and {A9 , A10 , B5 } vote for A;                                               Go Back

– only 1 district out of 6, {B6 , B7 , B8 } votes for B.                        Full Screen

• Result: A gets 5 votes out of 6.                                                    Close

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Re-Districting: A . . .

4.     How to Avoid Gerrymandering: Known Approaches                 How to Avoid . . .

Toward Formulation of . . .
• Typical idea: limit the “weirdness” of the district shapes.
Relation to Clustering

• Example: make them as round as possible, with the             Towards a More . . .

smallest possible length of the separation lines.             Example

• Limitations:                                                  Resulting Iterative . . .

Conclusion
– these approaches only take into account geograph-          Acknowledgments
ical closeness;                                                    Title Page

– they may not give adequate representation to mi-
norities or rural population.
• Remaining problem: take into account
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– not only geographical closeness and diﬀerences,
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– but also other types of closeness and diﬀerence.
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Re-Districting: A . . .

5.      Toward Formulation of the Problem in Precise Math-                  How to Avoid . . .

ematical Terms                                                      Toward Formulation of . . .

Relation to Clustering
• Representing voters: by parameters x1 , . . . , xn :
Towards a More . . .

–   geographic location (xi are geographic coordinates),           Example

–   income,                                                        Resulting Iterative . . .

–   rural vs. urban status,                                        Conclusion

–   number of children, etc.                                       Acknowledgments

Title Page
• Representatives are described by v1 , . . . , vn .
• Utility approach: in decision making, preferences are
represented by utility functions:
u = u(x1 − v1 , . . . , xn − vn ).                          Page 6 of 16

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• Diﬀerent characteristics are usually independent: it is
known that then u(d1 , . . . , dn ) = u1 (d1 ) + . . . + un (dn ).           Full Screen

• Diﬀerences are small: representatives are close to the                          Close

def
voters, so diﬀerences di = ri − vi are small.                                    Quit
Re-Districting: A . . .

6.     Toward Formulation of the Problem in Precise Math-                              How to Avoid . . .

ematical Terms (cont-d)                                                         Toward Formulation of . . .

Relation to Clustering
• Taylor expansion: since diﬀerences di = ri − vi are
Towards a More . . .
small,
Example
u (0) 2
ui (di ) = ui (0) + ui (0) · di +      · di + . . .                      Resulting Iterative . . .
2
Conclusion
• Analysis: the largest possible utility is when a repre-
Acknowledgments
sentative is a perfect match, i.e., di = 0.                                             Title Page

• Conclusion: ui (di ) ≈ ui (0) − wi ·               d2
i   and
n                  n                n
u(d1 , . . . , dn ) =         ui (di ) =         ui (0) −         wi · d2 .
i
i=1                i=1              i=1                      Page 7 of 16

• Thus, maximizing utility is equivalent to minimizing                                     Go Back
the disutility
n                                                    Full Screen

ρ(x, v) =             wi · (ri − vi )2 .                                 Close
i=1
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Re-Districting: A . . .

7.     Resulting Formulation of the Problem                        How to Avoid . . .

Toward Formulation of . . .
• We have:
Relation to Clustering

– an integer n (number of characteristics),                Towards a More . . .

– positive real numbers w1 , . . . , wn (weights);         Example

(k)      (k)
– n-dimensional vectors x(k) = (x1 , . . . , xn )          Resulting Iterative . . .

(1 ≤ k ≤ N ) describing voters;                          Conclusion

Acknowledgments
– number c of voting districts.
Title Page

• Objective:
– subdivide N voters into c groups D1 , . . . , Dc , and
– select a vector v(1), . . . , v(c) within each group            Page 8 of 16

– so as to minimize the overall disutility
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c
ρ(x(k) , v(j)).                      Full Screen

j=1 k∈Dj                                           Close

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Re-Districting: A . . .

8.     Relation to Clustering                                        How to Avoid . . .

Toward Formulation of . . .
• Informal objective: a voter is closer to other voters from
Relation to Clustering
his/her district than to voters from other districts.
Towards a More . . .

• Relation to clustering: such groups are called clusters.      Example

• Iterative clustering: we start with some representations      Resulting Iterative . . .

v(1), . . . , v(c), then repeat the following 2 steps:        Conclusion

Acknowledgments
– each voter x(k) is assigned to the group Dj for which
Title Page
the disutility ρ(x(k) , v(j)) is the smallest;
– after that, for each group Dj , we re-calculate v(j)
as the average of all x(k) (k ∈ Dj ).
• Limitation of this approach: unequal clusters.                       Page 9 of 16

• Example: a population consisting of a (larger) city and                Go Back

a (smaller) rural area.                                               Full Screen

• Resulting clusters: a larger all-city cluster and a smaller              Close

all-rural cluster.
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Re-Districting: A . . .

9.     Towards a More Adequate Solution                                How to Avoid . . .

Toward Formulation of . . .
N
• Main idea: groups D1 , . . . , Dc must have equal size .        Relation to Clustering
c
Towards a More . . .
• Technical challenge: when we have the representatives
Example
v(j), how can we get districts?
Resulting Iterative . . .
• Technical result: in the optimal districting, the division      Conclusion
between Di and Dj is determined by a threshold tij for          Acknowledgments
def ρ(x, v(i))
the ratio rij (x) =              :                                      Title Page
ρ(x, v(j))
– if rij (x) < tij , then x ∈ Di ;
– if rij (x) > tij , then x ∈ Dj .
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• Conclusion: for some weights αj , a voter x is assigned
to the class Dj for which αj · ρ(x(k) , v(j)) is the largest.            Go Back

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• Remaining problem: ﬁnd the weights αj for which the
resulting districts Dj are of equal size.                                  Close

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Re-Districting: A . . .

10.    Towards an Algorithm                                    How to Avoid . . .

(0)           (0)                       Toward Formulation of . . .
• We start: with α1 = . . . = αc = 1.
Relation to Clustering
(p)          (p)
• Start iteration: with some values       α1 , . . . , αc .   Towards a More . . .

• Main part of the iteration: for each j, we ﬁnd βj for       Example

which there are exactly N/c points x for which              Resulting Iterative . . .

(p)                        Conclusion
βj · ρ(x, v(j)) ≤ αk · ρ(x, v(k))
Acknowledgments
for all k = j.                                                   Title Page

• How to ﬁnd βj : bisection
– if we get < N/c points, we decrease βj ;
– if we get > N/c points, we increase βj .                    Page 11 of 16

• Observation: if we multiply all the values αj by the                 Go Back
same constant, we get the same classes.
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αj   βj
• Last part of the iteration: ﬁnd αj for which    ≈ (p)
αk  αk
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for all j = k.                                                          Quit
Re-Districting: A . . .

11.   Analysis of the Auxiliary Problem                    How to Avoid . . .

Toward Formulation of . . .
αj   βj
• Problem – reminder: ﬁnd αj for which    ≈ (p) for       Relation to Clustering
αk  αk
Towards a More . . .
all j = k.
Example
• Diﬃculty: this problem is non-linear in unknowns αj .   Resulting Iterative . . .
def               def
• Idea: turn to logarithms Aj = ln(αj ), Bj = ln(βj ),    Conclusion

(p) def   (p)
and Aj = ln(αj ).                                       Acknowledgments

Title Page
• New problem: ﬁnd Aj for which
(p)
Aj − Ak ≈ Bj − Ak .
• Least Squares solution:
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1       (p)
Aj = · (Bj + Aj ) + const.                          Go Back
2
• Resulting value of αj :                                         Full Screen

(p)                             Close
αj =   βj · αj .
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Re-Districting: A . . .

12.   Example                                                 How to Avoid . . .

Toward Formulation of . . .
• Situation: uniform population distribution on [0, 1].
Relation to Clustering

• Starting reps: v(1) = 0, v(2) = 0.5, and v(3) = 1.0.       Towards a More . . .

• Simple clustering: D1 = [0, 0.25], D2 = [0.25, 0.75],      Example

D3 = [0.75, 1].                                            Resulting Iterative . . .

Conclusion
• Problem: D2 is twice larger than D1 or D3 .                Acknowledgments

• 1st iteration: ﬁnd β1 for which β1 · x2 ≤ (0.5 − x)2 for           Title Page

exactly 1/3 of points.
• Computing βj : β1 = 0.25, similarly β2 = 4, β3 = 0.25.
(2)            (1)   √                          Page 13 of 16
• Computing αj : α1 = β1 · α1 = 0.25 · 1 = 0.5;
(2)        (1)   √             (2)                      Go Back
similarly, α2 = β2 · α2 = 4 · 1 = 2, α3 = 0.5.
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• Result: 1/3 of voters in the ﬁrst district, for which
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α1 · ρ(x, v(1)) → min, and 1/3 each in D2 and D3 .
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Re-Districting: A . . .

13.   Resulting Iterative Algorithm                               How to Avoid . . .

Toward Formulation of . . .
• Start: representations v(1), . . . , v(c) (e.g., representa-
Relation to Clustering
tives of the existing districts).
Towards a More . . .

• Main idea: iterations of the 2-step process:                   Example

– subdivide voters into c equal groups corr. to v(j);       Resulting Iterative . . .

– re-calculate v(j) as the average of the j-th group.       Conclusion

Acknowledgments
• 1st step starts with an iterative process:                             Title Page
(1)            (1)
α1            = 1;
αc
– use bisection to ﬁnd βj for which there are exactly
N/c points x(k) for which, for all l = j:
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βj · ρ(x(k) , v(j)) ≤ α(p) · ρ(x(k) , v(l));
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(p+1)                (p)
– compute   αj       =    βj ·   αj .                               Full Screen

• 1st step ends: assign each voter x to to the group Dj                     Close
for which αj · ρ(x(k) , v(j)) is the smallest.
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Re-Districting: A . . .

14.   Conclusion                                               How to Avoid . . .

Toward Formulation of . . .
• What we propose: a new algorithm for dividing an area
Relation to Clustering
into voting districts.
Towards a More . . .

• New features: we take take into account not only geo-       Example

graphic closeness, but also common interests of voters.     Resulting Iterative . . .

• Necessary input: weights wi of diﬀerent factors:            Conclusion

Acknowledgments
– geographic location (xi are geographic coordinates),           Title Page

– income,
– rural vs. urban status,
– number of children,
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– etc.
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15.   Acknowledgments                                      Re-Districting: A . . .
How to Avoid . . .

This work was supported in part by:                        Toward Formulation of . . .
Relation to Clustering

• by National Science Foundation grants HRD-0734825,      Towards a More . . .
Example
EAR-0225670, and EIA-0080940,                           Resulting Iterative . . .
Conclusion
• by Texas Department of Transportation grant             Acknowledgments
No. 0-5453,
• by the Japan Advanced Institute of Science & Technol-            Title Page

ogy (JAIST) Int’l Joint Research Grant 2006-08, and
• by the Max Planck Institut f¨r Mathematik.
u
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