Defining and Measuring the Partisan
Fairness of Districting Plans
Andrew Gelman, David Epstein,
Sharyn O’Halloran and Jared Lander
Departments of Statistics and Political Science
Columbia University
8 Jan 2008
2003 Texas Redistricting
• Texas House delegation went from 17-15 Democrat in 2002 to
21-11 Republican in 2004 (while voting 61%-38% for Bush)
• Is this an unfair partisan gerrymander?
– Supreme Court (Kennedy) said there is no workable standard
Outline: Standards of fairness
• Some historical background
• The proportionality standard and its problems
• The seats-votes curve
• The symmetry standard and its problems
• Toward a comparative standard
• “Fairness” matters
– For the courts
– For democracy
– Need fairness standard to determine what’s unfair
Some historical background
Some historical background
Some historical background
 “Gerrymandering” isn’t as bad as people think
 Gelman and King (1994b)
– Empirically, redistricting decreases partisan bias and increases
competitiveness
– Why? Because redistricters work under many constraints
 But fairness is still a concern
The proportionality standard
 Popular in Europe, via PR electoral systems
 “Fairness” is . . . If your party receives x% of the vote, it
should receive x% of the seats
 This does not work, in general, with first-past-the-post
systems such as the U.S.
– Can win 55% of the vote in every district,100% of the seats.
– In fact, can win a majority with ~25% of the votes
– In general, bonus for majority party (e.g., cube law)
 So how do we describe the relation between voter behavior
and electoral outcomes?
The seats-votes curve
 This describes the function S(V), the seats won S for a given
percentage V of the vote
 For a single election, calculate this as follows:
– Take the vector of votes V = (V1, V2, …, V435), where Vi is the
percentage of Democratic votes in district i
– From this get the average Democratic vote and percentage of seats
won by the Democrats – this is the actual electoral outcome
– Now consider the vector V+1% = (V1+1, V2+1, …, V435+1)
– I.e., a uniform partisan swing of 1% for the Democrats
– Perform the same calculations for V+ x% for all values of x
– This will fill out the range, yielding a nondecreasing function S(V)
 This is the seats-votes curve
The seats-votes curve
The seats-votes curve
 Traditionally (since Edgeworth, 1898) thought of as a
deterministic function: S(V)
 Actually it’s probabilistic: p(S|V)
 Usually summarized by its expectation: E(S|V)
The symmetry standard
 “Fairness” is . . . E(S|V) = 100 – E(S|1-V)
– For example, in 2008 the Democrats averaged 56% of the vote
in U.S. House races and received 59% of the seats.
– This is symmetric (i.e., “fair”) if the Republicans would have
received 59% of seats had they won 56% of the vote
 In particular, symmetry requires that E(S|V=0.5) = 0.5
 King and Browning (1987): partisan bias defined as deviation
from symmetry
 Gelman and King (1990, 1994a): empirical estimate of
partisan bias by extrapolation
Problems with symmetry standard
• Problem 1: Need to extrapolate to 50%
– Consider a state such as Massachusetts
– It will never be 50-50, so how can we tell what’s fair?
• Problem 2: Mixing apples and oranges
– Seats-votes calculations use all districts at all points along the
curve to estimate the relationship
– So we use Montana to estimate Massachusetts, and vice-versa
• Real problem is that the S(V) curve is designed to answer
questions about the electoral system as a whole
– E.g., bias (intercept at V=.5) and responsiveness (slope at V=.5)
– Less useful when we’re interested in behavior away from the
50-50 mark
– But each election gives us 50 data points, not just one…
Toward a comparative standard
• Goal: to solve the “Massachusetts problem”
• Not merely an academic exercise!
– Consider the 2003 Texas redistricting
– Availability of computer programs will make this worse
• Method of overlap
– For any state, extrapolate a bit in either direction (based on
historical levels of variation)
– Compare a state to similar historical cases
– A chain of extrapolations gets you to 50% (and symmetry)
• Symmetry is thus a baseline but not always a direct standard
Seats-votes curves from state
congressional delegations
 For each state and each election, extrapolations +/- 5%
using uniform partisan swing
 Create hypothetical elections, adding x% to Dem. share in
each district, with x = -5.0, -4.9, -4.8, . . . , +4.9, +5.0
– Full implementation would also add noise (“JudgeIt”)
 These will overleaf with each other, creating an overall seats-
votes curve with a range of variation at each point
– Variation is within states with similar partisan makeups
 Then can obtain semi-parametric confidence intervals, taking
into account state size, incumbency, etc.
1900
1920
1940
1960
1980
2008
 Overall, get something that looks like a confidence band
 Can use this to judge proposed districting plans
Texas
 Overall, get something that looks like a confidence band
 Can use this to judge proposed districting plans
Discussion
 Traditional methods of analysis are not well-designed to
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assess the fairness of districting plans for states that are far
from a 50-50 partisan split
We propose instead the aggregation of local seats-votes
curves to provide variation across states and over time
These can be used to estimate normal seats-votes
relationships for states with high levels of partisanship
Then, define unfair districting relative to this standard
See if Kennedy goes for it…