Andrew Gelman, David Epstein, Sharyn O’Halloran and Jared Lander

Departments of Statistics and Political Science

Columbia University8 Jan 2008

Defining and Measuring the Partisan Fairness of Districting

Plans

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

thinkGelman 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 standardPopular in Europe, via PR electoral systems“Fairness” is . . . If your party receives x% of the

vote, it should receive x% of the seatsThis 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 curveThis describes the function S(V), the seats won S for a

given percentage V of the voteFor 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 curveTraditionally (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 delegationsFor each state and each election, extrapolations +/-

5% using uniform partisan swingCreate 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

Overall, get something that looks like a confidence band

Can use this to judge proposed districting plans

Texas

DiscussionTraditional methods of analysis are not well-

designed to 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…