1. 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

2. 2003 Texas Redistricting
Texas House delegation went from Democrat in 2002 to Republican in 2004 (while voting 61%-38% for Bush)
Is this an unfair partisan gerrymander?
Supreme Court (Kennedy) said there is no workable standard

3. 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

4. Some historical background

5. Some historical background

6. 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

7. 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?

8. 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

9. The seats-votes curve

10. 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)

11. 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

12. 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 mark
But each election gives us 50 data points, not just one…

13. 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

14. 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.

16. 1900
1920
1940
1960
1980
2008

17. Overall, get something that looks like a confidence band
Can use this to judge proposed districting plans

18. Overall, get something that looks like a confidence band
Texas
Overall, get something that looks like a confidence band
Can use this to judge proposed districting plans

19. Discussion
Traditional methods of analysis are not well-designed to assess the fairness of districting plans for states that are far from a 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…