Jamie Saxon at the Harris School

Gerrymandering and Compactness

This page provides a basic overview of gerrymandering. It explains the project methodology.
The interactive map puts the results in context.

After each decennial Census, Congress apportions representatives among the fifty states. The states in turn draw single-member districts for each representative. In most states, that districting process is controlled by the majority party in the state legislature. Too often, that party engages in "gerrymandering" -- manipulating district boundaries to their own advantage.

Compactness has long been proffered as a constraint on gerrymandering. From 1901 to 1929 it was required by Congress. The Supreme Court has regularly enumerated it as a "traditional districting principle." It is consistent with the American tradition of geographic representation. The political science and legal lectures have explored compactness for decades and offer dozens of ways to define it. Yet despite these traditions and this panoply of definitions, it is clear that compactness is not enforced:

Non-compact congressional districts from the 114th Congress: NC-12, OH-9, FL-5, NY-10, TX-35.
The five least compact congressional districts from the 114th Congress.

Why not? The variety of definitions is actually part of the problem. With just a few shapes derived from the district's boundaries (illustrated below), one can construct innumerable ratios to quantify compactness. Among the many possibilities presented in the paper, we could measure:

  1. The ratio of a district's area with respect to that of a circle of equal perimeter (Isoperimeter Quotient, IPQ).
  2. The ratio of the area of the largest inscribed circle to that of the district.
  3. The average distance on the surface to its perimeter, as compared to a circle.
  4. The ratio of a district's population to the population contained within its convex hull.

Building blocks for compactness measures: equal perimeter circle, equal area circle,
              convex hull, smallest circumscribing circle, largest inscribed circle, and 
              distances to the perimeter or centroid, or between people.
With a few "derived" shapes, one can construct many different definitions
of compactness. Shown is Pennsylvania's 7th Congressional District.

There is no consensus as to which of these definitions is best, and there is a concern that the choice among definitions is simply politics in disguise. Since Democrats in America are concentrated in urban areas, Democrats and Republicans might prefer definitions that split cities in different ways.

But are the definitions really different? I propose two complementary tests:

  1. Compare the political outcomes of districts constructed to optimize each of the definitions. Do some measures treat the two parties differently?
  2. Use plans enacted over the last three decades to evaluate how consistently the various definitions rank different plans. Do the measures give similar answers as to the question "is the district compact?"

District optimization is in a class of problems called "NP-hard" in computer science. This is just a formal way of saying that it is computationally unfeasible to identify the single best solution. So instead of deriving the optimal map, we can hope at best to generate "populations" of good maps. For each map, I simply run the algorithm with a different starting configuration, called a seed. For each seed, the algorithm manipulates the district boundaries to (a) ensure equal populations among the districts, and (b) improve their compactness (according to one measure), while preserving the contiguity of the individual districts. The algorithm ends with one good-quality statewide plan. (To visualize one run, check out this gif.) With many maps, we can get a sense of "good maps" for that algorithm

Many others have done this before me. See for instance Chen and Rodden or Cho and Liu. I depart from past work by systematically implementing several algorithms and compactness "scores." I find that the various definitions and algorithms do in fact treat cities differently. Below I display a sample of maps based on different algorithms but the same starting configuration (seed). Note for instance, the varying divisions of Pittsburgh -- halfway down, along the western edge of the state:

A collection of simulated maps are presented for Pennsylvania.
              The varying compactness definitions do yield different geometries.
A collection of simulated maps are presented for Pennsylvania.
The varying compactness definitions do yield different geometries.

Nevertheless, differences in shape do not necessarily translate into different practical outcomes. Re-aggregating the votes from real (presidential) elections, we can simulate these outcomes on the generated maps. How does each party perform, under the various mapping strategies? Below, I plot the seat shares for Pennsylvania Democrats in four elections. There are several important observations:

Plotted are seats accruing to Pennsylvania Democrats in four successive elections,
              using maps generated according to a variety of compactness measures and algorithms.
              The simulated distributions are compared to the expectations for enacted plans.
Seat shares for Pennsylvania Democrats are plotted for maps generated with eighteen different compactness procedures. Each distribution represents a collection of statewide districting plans, with real votes aggregated within the borders of those plans to determine district winners, and a statewide seat share for Democrats. To the left of each distribution is its average (mean): the number of seats expected to accrue to Democrats. Democrats won a majority of the presidential votes in each of the elections shown: 50.6 to 46.4% in 2000, 50.9 to 48.4% in 2004, 54.5 to 44.1% in 2008, and 52.0 to 46.6% in 2012. But they would expect to capture a majority of seats only in 2008, when Barack Obama won by more than 10%.
  1. The various definitions of compactness yield remarkably consistent seat shares to the two parties. It's true as some on the Supreme Court have suggested, that adopting a compactness standard would affect the outcomes. But it also appears that the choice among definitions isn't "playing politics." The Supreme Court decisions in the Reapportionment Revolution (Baker, &c.) also affected outcomes. Indeed, it is the very nature of any voting law to affect outcomes!
  2. In the 2000, 2004, and 2012 elections, Pennsylvania Democrats captured a minority of seats despite support from a majority of votes. (The results were 50.6 to 46.4% in 2000, 50.9 to 48.4% in 2004, 54.5 to 44.1% in 2008, and 52.0 to 46.6% in 2012.) This is Chen and Rodden's observation on unintentional gerrymandering. The Pennsylvania Democrats are at a structural disadvantage, independent of any machinations by the Legislature.
  3. This said, comparing the results simulated under enacted plans to the distributions from simulation (black v. red), the scale of intentional gerrymandering for the 2010 Census is apparent. The map that Pennsylvania Republicans have adopted has given them an additional seat advantage, expected to endure over the entire decade. Similar conclusions hold in Maryland and North Carolina, where Democratic and Republican legislatures have adopted plans that give them significant advantages with respect to simulations.

But we can ask another question. The maps created algorithmically are not representative of enacted plans. The fact that maps optimized for different definitions of compactness have similar political effects does not imply that those definitions would score real plans consistently. So, from a purely spatial perspective, is it meaningful to ask whether a district is compact? I argue that it is. From a technical standpoint, 70% of the variance among plans from the last three districting cycles (1990, 2000, 2010) is explained by a single variable. In other words, the various definitions are largely consistent with a single concept.

Alternatively, one may simply examine the correlations between plans' scores or ranks, below. The color shows the predictive power of one variable for another. Yellow means that the two scores agree 100% of the time; bright green denotes ~70% agreement. With the notable exception of the (dubiously-useful) Axis Ratio, the various definitions of compactness yield consistent scores and rankings.

Correlations are shown among compactness measures.
Score and rank correlations are presented among compactness measures, for the population of maps enacted for the last three Censuses. The first and second components are from a Principle Component Analysis; the first component soaks up 70% of the variance among measures.

When considering districting reform, it's important to understand how it is likely to affect minority representation. In evaluating this, there is a significant complication. Unlike the partisan questions above, where a plurality of the votes is required to win an election, there is no logical threshold for which a share of minority voters will result in a minority representative. Adopting simplified strategies to derive this threshold, I predict a drop in minority representation as high as 10% from adopting a purely compactness-based approach.

That said, there are four big caveats. First, one of the methods shows virtually no change in minority representation. Second, if the threshold for electing a minority falls (as it has done over recent elections), compactness-based approaches are likely to yield higher minority representation. Third, procedures for generating collections of compact districts described in the paper build in a work-around: it's possible to simply preferentially select the statewide plans with higher expected minority constituencies. Finally, given the downfall of the VRA preclearance formula in the aftermath of Shelby County v. Holder, algorithmic strategies provide a new potential pathway for "preventative" protections of voting rights at the Federal level.

Check out the paper for details on the algorithm, data, and a review of the legal history of apportionment and districting. Or explore your own state with the map!

Copyright (c) 2017 James Saxon