Political Redistricting
By Saad Padela
The American Political System

Legislative bicameralism

Number of seats in lower house is proportional to
population

Single-member districts

First-past-the-post (or plurality) voting

“One man, one vote”
The Case for Redistricting

New Census data every 10 years

# of Representatives = α * Population

0<α<1

# of Representatives = # of Districts

Population rises => More seats

Districts must be redrawn
Gerrymandering
Types of Gerrymandering

Partisan


Bipartisan


Incumbents vs. Challengers
Racial and ethnic


Democrats vs. Republicans
Majority vs. Minority groups
“Benign”

In favor of minority groups
Gerrymandering Strategies


Different election objectives

To win a single district

To win a majority of many districts
Partisan

Own votes



Win districts by the smallest margin possible
Minimize wasted votes in losing districts
Opponent's votes


Fragment them into different districts
Concentrate them into a single district
Gerrymandering Strategies

Bipartisan


Racial and ethnic


Maximize number of “safe” districts
Fragment supporters of minority candidates
“Benign”

Maximize chances of minority representation by
concentrating them into single districts
A Linear Programming Formulation?

Easy to see

Small scholarly literature

Those who are involved in it like to keep their work
secret
Detection of Gerrymandering


A rich literature
Hess, S.W. 1965. “Nonpartisan Political
Redistricting by Computer.” Operations
Research, 13 (6), 998-1006.
Good Districts are...

Equally populous

Contiguous

Compact
Equal population

Easy to write as a constraint
Contiguity

Highly intuitive

Sometimes tedious to code
Compactness

Ambiguous

Difficult to measure

Niemi et al. 1990. “Measuring Compactness
and the Role of a Compactness Standard in a
Test for Partisan and Racial Gerrymandering.”
The Journal of Politics, 52 (4), 1155-1181.

“A Typology of Compactness Measures” (Table 1)



Dispersion
Perimeter
Population
A Typology of Compactness
Measures: Dispersion

District Area Compared with Area of Compact
Figure

Dis7 = ratio of the district area to the area of the
minimum circumscribing circle

Dis8 = ratio of the district area to the area of the
minimum circumscribing regular hexagon

Dis9 = ratio of the district area to the area of the
minimum convex figure that completely contains the
district

Dis10 = ratio of the district area to the area of the
circle with diameter equal to the district's longest
axis
A Typology of Compactness
Measures: Dispersion

District Area Compared with Area of Compact
Figure

Dis7 = ratio of the district area to the area of the
minimum circumscribing circle

Dis8 = ratio of the district area to the area of the
minimum circumscribing regular hexagon

Dis9 = ratio of the district area to the area of the
minimum convex figure that completely contains the
district

Dis10 = ratio of the district area to the area of the
circle with diameter equal to the district's longest
axis
A Typology of Compactness
Measures: Dispersion

Moment-of-inertia

Dis11 = the variance of the distances from all points
in the district to the district's areal center for gravity,
adjusted to range from 0 to 1

Dis12 = average distance from the district's areal
center to the point on the district perimeter reached
by a set of equally spaced radial lines
A Typology of Compactness
Measures: Perimeter

Perimeter-only


Per1 = sum of the district perimeters
Perimeter-Area Comparisons

Per2 = ratio of the district area to the area of a circle
with the same perimeter

Per4 = ratio of the perimeter of the district to the
perimeter of a circle with an equal area

Per5 = perimeter of a district as a percentage of the
minimum perimeter enclosing that area
A Typology of Compactness
Measures: Population


District Population Compared with Population of
Compact Figure

Pop1 = ratio of the district population to the
population of the minimum convex figure that
completely contains the district

Pop2 = ratio of the district population to the
population in the minimum circumscribing circle
Moment-of-inertia

Pop3 = population moment of inertia, normalized
from 0 to 1
Warehouse Location model




Hess, S.W. 1965. “Nonpartisan Political
Redistricting by Computer.” Operations
Research, 13 (6), 998-1006.
Garfinkel, R.S. And G.L. Nemhauser. 1970.
“Optimal Political Districting By Implicit
Enumeration techniques.” Management
Science, 16 (8).
Hojati, Mehran. 1996. “Optimal Political
Districting.” Computers and Operations
Research, 23 (12), 1147-1161.
All these formulations have class NP
Heuristic Methods

Hess, S.W. 1965.

Garfinkel, R.S. And G.L. Nemhauser. 1970.

Hojati, Mehran. 1996.

Bozkaya, B., Erkut, E., and G. Laporte. 2003. “A
tabu search heuristic and adaptive memory
procedure for political districting.” European
Journal of Operational Research, 144, 12-26.
Statistical physics?

Chou, C. and S.P. Li. 2006. “Taming the
Gerrymander – Statistical physics approach to
Political Districting Problem.”
Criticisms of Compactness

Altman, Micah. 1998. “Modeling the effect of
mandatory district compactness on partisan
gerrymanders.” Political Geography, 17 (8),
989-1012.

Nonlinear effects – “electoral manipulation is much
more severely constrained by high compactness
than by moderate compactness”

Context-dependent, and purely relative

Asymmetrical effects on different political groups

Compactness can also disadvantage
geographically concentrated minorities
More Sophisticated Measures

Niemi, R. and J. Deegan. 1978. “A Theory of Political
Districting.” American Political Science Review, 72 (4),
1304-1323.

Neutrality


Range of Responsiveness


% range of the total popular vote over which seats
change from one party to the other
Constant Swing Ratio


v% of the popular vote results in s% of the seats
rate at which a party gains seats per increment in votes
Competitiveness

% of districts in which the “normal” vote is close to 50%

Balinksi, Michel. 2008. “Fair Majority Voting (or
How to Eliminate Gerrymandering).” The
American Mathematical Monthly, 115 (2), 97114.