Team #1898
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Congressional Redistricting: an Unbiased Approach
Spencer Anderson
Elizabeth Huber
Michael Sass
University of San Diego
San Diego, CA 92110
Advisor: Dr. Diane Hoffoss
Summary
Using a graphical approach, our solution synthesizes the advantages of a population
density area cartogram with the simplicity of the Shortest Splitline Algorithm (Center).
Population density, at its most fundamental level on the area cartogram, conforms to
county boundaries. Consequently, we can divide a state’s total area T on the cartogram
by the number of Congressional districts O within the state and arrive at the specific area
R of each district on the cartogram. When a county’s area on the cartogram exceeds R or
it is inferior to R, the Shortest Splitline Algorithm is manipulated to provide an optimal
district boundary. When used in conjunction with each other, the area cartogram and
Shortest Splitline Algorithm provide an unbiased approach to redistricting.
When district boundaries on the cartogram are transferred back to a Mercator-like
geographic projection of the United States, it is quite clear that this method provides
exceptionally simple and elegant geometric regions that, in most instances, maintain
county boundaries. This ensures that election bureaucracy can be dealt with using current
county governments instead of a newly created layer of bureaucracy. As a result, our
method guarantees economic feasibility while providing a simple means of explaining the
redistricting scheme to the general public.
Background
The process of creating voting districts in the United States has been constantly
debated and modified for countless years. Until June 28, 2006, State legislatures redrew
districts every ten years based on census data. On that date, the Supreme Court ruled that
politicians can redistrict as often as they deem necessary. However, because of the
freedom with which the districts can be divided, politicians can easily adjust their district
composition in pursuit of political gain. As a result, another possible change to the
redistricting process has been proposed. The D.C. Fair and Equal House Voting Rights
Act was put forth for the second time in January of 2007. If this bill is approved, it
would add two more seats to the House, one for the District of Columbia (which currently
has no representation) and a second for Utah, which would politically balance out the
unquestionable Democratic vote in the Capitol. Our solution, while based upon the
current number of representatives, will undoubtedly work if these two seats are added.
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The current method of redistricting is known as the Equal Proportions Method. By
this method, the total population in the United States is taken and divided by the number
of seats. Then, the population of each state is taken individually and divided by another
number. This divisor is such a number that, when all of the equations for state population
are added together, it will exactly equal the number of seats that can be given away. The
major problem with this is that it still leaves ambiguity as to the geographic composition
of each district. Many district shapes today are quite strange and unusual. To solve any
problems of confusion regarding this, and the ability to politically manipulate voting
results, a mathematical solution that covers both the distribution and the shape of the
districts must be designed. Our solution therefore applies use of population density area
cartograms and the Shortest Splitline Algorithm (Center) to this important issue.
Assumptions
1. The population density of any area will remain constant at the time of
redistricting.
2. Racial, gender, and political issues are not factored into our solution in order to
maintain an unbiased method.
3. Geographically simply means that, on a Mercator-like projection, the overall
shape of a district shall be composed of square and rectangular sub-regions.
Solution Method
In order to develop a method to optimize the way Congressional districts are drawn,
there are certain priorities that, without a doubt, must be met. These are as follows:
1. The redistricting method must not be dependent upon any overt human biases,
either intentional or accidental.
2. Each district must theoretically have the same population at the time the districts
are determined (i.e. population shifts over time are not accounted for until further
redistricting is accomplished).
3. All districts must be composed of as geographical simple a shape as possible
without affecting any other priority.
4. The method of redistricting must be practical, economical, and easily explained to
the general population.
Our solution takes advantage of the following two unoriginal mathematical
conceptualizations:
1. A data distribution-based area cartogram.
2. The Shortest Splitline Algorithm, developed by the Center for Range Voting.
Before describing our specific solution in any detail, it is important to provide some
introductory information on these two concepts.
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The Area Cartogram
According to Christopher J. Kocmoud of the Texas Center for Applied Technology
and Donald H. House of Texas A&M University, “the area cartogram is a useful tool for
visualizing the geographic distribution of “routine” data in a variety of disciplines,
including politics, social demographics, epidemiology, and business” (Kocmoud and
House). Using this approach, a geographical visualization (i.e. a map of the United
States, for example) can be spatially transformed to emphasize a certain set of specific
data. As a result, the area of pre-determined regions within the geographical
visualization can be distorted to represent a specific data distribution. The main
advantage of this method is that it allows an individual the opportunity to visually
represent, isolate, and eventually analyze a specific variable (Kocmoud and House).
The Shortest Splitline Algorithm
This algorithm, developed by the Center for Range Voting, provides a very simple
and effective method for creating geographically simple Congressional districts that
contain the same population. There are five main steps to this process:
a. First, determine each state’s geographical boundary. If a state contains separate
geographical regions, such as Hawaii, Michigan, or New York, the separated
“pieces” of each state must be joined by line segments of a one-to-one
correspondence.
b. Set U equal to the total number of Congressional districts allocated to a specific
state.
c. Let S and D be two equal whole numbers whose sum is equal to U. Whenever it
is mathematically impossible for S and D to satisfy this condition, let S and D be
two whole numbers whose sum is equal to U and where |S – D| is as close to zero
as possible.
d. Split the specified state into two regions via a great circle (i.e. a straight line on a
map similar to a Mercator Projection) in which the two newly-created regions
have a population ratio of S:D. In the eventual case in which more than one great
circle can be drawn fitting the above constraints, choose the route whose great
circle is the shortest (i.e. the shortest distance between the great circle’s vertices).
e. Finally, using the above method for each individual region, recursively split up
regions S and D via the ratio S:D until the total number of regions created equals
U.
This is an overly-simplified explanation of the Shortest Splitline Algorithm. A more thorough analysis
of the technique will be extrapolated below.
Please Note: The above five steps were paraphrased from the Center for Range Voting.
The following two illustrations (Page 4) show the current Congressional districting
scheme for the state of Tennessee and the proposed redistricting using the Shortest
Splitline Algorithm:
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Figure 1. Graphical representation of the current geographical positions of
Tennessee’s eight Congressional districts. Courtesy of RangeVoting.org and the
U.S. Census Bureau.
Figure 2. Graphical representation of the proposed geographical positions of
Tennessee’s eight Congressional districts via the Shortest Splitline Method.
Courtesy of RangeVoting.org.
Application of Methods → Our Solution
Both of the above two methods provide a framework for which to build a practical
solution. However, when each method is isolated from the other, it does not optimize the
solution to the redistricting problem. For example, the map on the following page shows
the population density of each county within the United States (excluding Alaska, but
including Puerto Rico):
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Figure 3. Population density map by county of the United States. Courtesy of the U.S.
Census Bureau (2000 Census data).
Despite this seemingly invaluable visualization tool, it would be less than optimal to
apply the Shortest Splitline Algorithm using this data, as the population density within
each county would not be represented. Therefore, the location of the great circles would
not be accurate, and hence, the populations of each district would not be equivalent.
Our solution starts out by using the following population density-equalizing
cartogram (Page 6):
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Figure 4. Population density-equalizing cartogram. Cartogram algorithm developed by
Gastner, M.T. and M.E. J. Newman. Data developed by the U.S. Census Bureau.
Cartography developed by Willaert Didier.
Clearly, this population cartogram uses the relative populations of individual counties
and distorts the geography of the United States to adequately support this set of data.
Nonetheless, notice that the overall geometry of the U.S. and relative positions of each
state is still easily recognizable. The first part of our solution requires the following three
steps:
1. Let T be equal to the total cartogram area of each individual state.
2. Let O be equal to the total number of Congressional districts for a given state.
3. Find the total area of each Congressional district (relative to the cartogram) by
dividing T by O. Let the relative area of each district be equal to R.
The next step involves a certain level of guesswork. To accomplish simplicity, the
actual shape of our created districts shall conform to the state’s county boundaries as
much as possible. This ensures that, in most cases, districting bureaucracy can be dealt
with using current county governments instead of a newly created form of bureaucracy.
This decreases the amount of taxpayer funds needed to institute our plan, while
maintaining a reasonable level of economic feasibility at the county level. Furthermore,
there are several rules that must be following when determining the boundaries of each
Congressional district. They are as follows (Page 7):
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a. Each district must contain the same population of every other district in the state.
b. No part of a district can be surrounded by any other district on all four sides (i.e.
each district must be continuous).
c. When increasing the size of a district to conform to the demanded population
levels, no county can be “skipped” in determining the district boundary. This
ensures that no eccentric district boundaries are created. It also serves the
practical purpose of enormously reducing the prevalence of gerrymandering in the
redistricting process.
What these steps lack in mathematical rigor, they make up for in basic symmetry and
simplicity. However, there is a major complication that must be accounted for:
1. What happens when conforming to the demanded Congressional district
population level supersedes county boundaries?
For example, in the case of Los Angeles County, more than one district is required to
satisfy the population level of this county. At the other extreme, certain counties with
low to moderate population levels may not be able to be fully represented by one district
(i.e. that county would have to be divided among many districts). To sensibly solve this
issue, without adding any human bias to the solution, the Shortest Splitline Algorithm
will be applied. In this way, these special case counties can be dealt with adequately
while maintaining simple geometry.
Clearly, when used in conjunction with each other, the area cartogram and Shortest
Splitline Algorithm perfectly complement each other. When these district boundaries are
transferred back to a geographic map of the United States, it is quite clear that this
method provides an exceptionally simple and elegant solution.
However, to get a strong visual sense for the solution process, it is necessary to
demonstrate its feasibility on a single state. Although this method will theoretically work
for all states, we have chosen to analyze the State of New York in the section titled
“Solution Application”.
Mathematical Alternative Method
Another possible method would not require the use of a cartogram, so all of the states
and counties would have their actual dimensions. Given a population density map of a
given state, the state can be divided into regions of relatively equal population density.
The method is more accurate when several different population densities are used, but for
the sake of explaining simply, this example will use only two different population
densities.
Dx = first population density
Dy = second population density
Ax = Area of 1 district in the first population density zone
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Ay = Area of 1 district in the second population density zone
ADx = Area of first population density zone
ADy = Area of second population density zone
X = number of districts in first population density zone
Y = number of districts in second population density zone
N = number of districts in the state
P = population of the state
PD = population per district
PD = P / N
Ax = PD / Dx
A y = PD / D y
X = ADx/(Ax)
Y = ADy/(Ay)
Given P, Dx, Dy, N, and ADx and ADy from a surveyor, we can calculate the Area that each
district should be (Ax and Ay) and how many districts (X and Y) should be in each
population density zone. We can then draw districts of that calculated area within the
corresponding population density zones. When taking into account more than two
population densities, the only adjustment needed is to add more variables: (i.e. Adding Dz
requires the addition of the corresponding variables Az, Adz, and Z.)
Solution Application
In theory, our solution method will successfully redistrict any of the United State’s
fifty states. In fact, the methodology and logic involved will hypothetically work for any
geometrical population regions needing to be split up into concrete sub-regions, as long
as that population region is subject to the problem assumptions (i.e. U.S. Constitutional
laws pertaining to the composition of the House of Representatives, etc.). Despite this,
we have chosen to demonstrate the validity of our solution method using the State of
New York as our prime example.
During the 110th Congress, the State of New York was allocated twenty-nine
representatives for the House of Representatives. As a result, there needs to be twentynine Congressional districts within New York’s borders.
In order to demonstrate the feasibility of our method as a mere baseline exercise, we
have decided to replace Figure 4 with a more generic, and simplified population density
area cartogram. This area cartogram, taken from the U.S. Census Bureau, can be seen
below (Page 9):
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Figure 5. Simplified model of a population density area cartogram. Courtesy of
the U.S. Census Bureau.
Before continuing, it is worth noting that using such a simplified area cartogram
provides many advantages and disadvantages. The major disadvantage of basing our
solution exercise on this simplified cartogram is that it does not include county
boundaries. As a result, it is virtually impossible to ensure that Congressional districts
follow county boundaries. Nonetheless, for our demonstration, this solution avoids any
complex computations needed in determining the exact area of a more accurate, and
complex, cartogram. In reality, if we were to apply our solution to the problem of
redistricting, we would, without a doubt, use a much more mathematically rigorous
method. However, this approach would take a significant amount of non-trivial work.
For our purposes, this rigorous approach extends beyond our time limitations.
The cartogram on the following page shows the State of New York isolated from the
rest of the simplified version of the population area cartogram (Figure 5):
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Figure 6. Blown-up image of the simplified population area cartogram. Courtesy of the
U.S. Census Bureau.
As you can see, the State of New York can easily be separated into twenty-nine
Congressional districts. Then, the existing district boundaries on the cartographic
representation can be translated back to a geographic, undistorted map of the United
States due to the one-to-tone correspondence between the two projections.
References
Center for Range Voting. “Examples of our unbiased district-drawing algorithm in action
/ comparisons with districts drawn by politicians.” 9 February 2007.
<http://rangevoting.org/GerryExamples.html>.
"DC Fair and Equal House Voting Rights Act." DC Vote. 10 February 2007.
<http://www.dcvote.org/advocacy/dcvramain.cfm>.
Kocmoud, Christopher J. and House, Donald H. “A Constraint-Based Approach to
Constructing Continuous Cartograms.” 10 February 2007.
<http://www-viz.tamu.edu/faculty/house/cartograms/SDH98.PDF>.
Second Demographic Transition. University of Michigan. “Corresponding population
cartogram (2000 population).” 10 February 2007.
<http://sdt.psc.isr.umich.edu/pubs/maps/double_ref_map.pdf>.
U.S. Census Bureau. Population Density of the United States: 2000. “Population Density,
2000”. 8 February 2007.
<http://www.census.gov/population/www/censusdata/2000maps.html>.
U.S. Census Bureau. “Reading a Cartogram: U.S. Population Cartogram.” 9 February
2007.
<http://www.census.gov/dmd/www/pdf/912ch1.pdf>.