Can Gerrymandering be Measured? Here Come the Mathematicians

By Brian Hollenbeck and Michael Smith of Emporia State University

Just weeks ago, the U.S. Supreme Court acted to sharply limit the role of the courts with regards to partisan gerrymandering. In Rucho v. Common Cause, the Court majority upheld the Davis v. Bandemer case of 1976, reaffirming that partisan gerrymandering is a “political question” and refusing to intervene. In Rucho, the Court found that “None of the proposed ‘tests’ for evaluating partisan gerrymandering claims meets the need for a limited and precise standard that judicially discernible and manageable.” They also noted that racial gerrymandering may be held to a different standard, because “race-based decision making…is ‘inherently suspect’ [as per] Miller v. Johnson [1995].”

Are they right? In recent years, mathematicians and mathematically-trained political scientists have begun to weigh in on the gerrymandering question. While the struggle to identify and analyze instances of potential partisan gerrymandering is more than 200 years old, new insights and computer models move it into new territory. Was a state’s congressional district map intentionally drawn to favor one political party?

There are four main criteria one can check to determine if a district map should be flagged for potential partisan gerrymandering:

  1. Does a district contain significantly more or fewer voters than another?
  2. Does the shape of a district appear to be unnatural and thus indicate manipulation?
  3. Does the distribution of voters among the districts negatively affect one party more than another in an election?
  4. Does the outcome of a potential election for a particular district map drastically differ from the expected outcome of a non-partisan map?

Measuring Compactness

The first criterion is known as “one-person, one-vote” and is simple to check. This criterion requires each district contain approximately the same number of voters. In a hypothetical community of 100 people, to be divided into 4 equally-populated districts, there are 1.6 x 1057 possible configurations!

The second criterion stems from the original case of gerrymandering, where the bizarre shape of a state senate election district in Massachusetts provoked a now-famous political cartoon mocking its likeness to a salamander. States have tried to combat this by requiring the shapes of districts to be “compact.” Intuitively, this means the district should not zigzag unnecessarily around the state. But extra constraints such as county lines, rivers, mountains, and population centers necessitate the need for exceptions. Thus, deviation from perfection is to be expected for most districts in most states. To quantify the magnitude of this deviation, mathematicians have created several definitions for compactness.

One perimeter-based definition is known as Polsby-Popper, introduced in 1991. The Polsby-Popper score uses the ratio of the district’s area to the square of its perimeter. This method is advantageous because it is simple to understand and penalizes any shape that meanders a lot. However, this means any district with long borders due to rivers or other physical obstacles will also be penalized.

A second definition makes use of the convex hull of a district. The convex hull can be thought of as the shape a rubber band would make if it were wrapped around the boundary of the district. The score is calculated by finding the ratio of the district’s area and the area of its convex hull. This score can sometimes be easier to calculate than a perimeter-based score since the hull “smooths” convoluted edges. However, this feature could minimize the impact of gerrymandering on a district’s score. Convex hull scores often reach similar overall results as perimeter scores, when comparing districts for compactness.

A third definition of compactness, known as Reock, compares the ratio of the district’s area with the area of a circle that circumscribes the district. This is both simple to calculate and understand. However, the Reock score can be misleading since a district with a large distance in one dimension will automatically require a large circle to contain, thus scoring low for compactness. This is true even if there are natural formations such as a coastline, which may offer a nonpartisan explanation for why the boundary meanders.

In short, there is no one, best standard to use in measuring compactness. Real-world geographical boundaries often complicate matters too much to reach a final conclusion.

Measuring Partisan Bias

These attempts to measure gerrymandering via the district’s shape have led us to a muddle. Perhaps it is time for a different approach, one which focuses on the outcome of an election based on voter distribution, rather than the shape of a district. In this case, we are trying to identify maps drawn in which voters from one party have been spread out among several districts (known as cracking), or grouped together in a few districts (known as packing).

The efficiency gap was introduced by Stephanopoulos and McGhee in 2015 and is calculated by finding the number of wasted votes for each party. A wasted vote is any vote that did not contribute to a party winning its district. Any votes above the minimum needed for a party to win the district are considered unnecessary and therefore “wasted.” Likewise, all votes cast by the losing party in a district are also wasted. The efficiency gap is calculated by finding the difference between wasted votes for the two parties and expressing this difference as a percentage of the total number of voters in a state.

One cannot assume that a high compactness score will always correspond to a low efficiency gap. Alexeev and Mixon have concluded in some situations, “a small efficiency gap is only possible with bizarrely-shaped districts.” In fact, they proved that every districting system will be flagged by at least one of our first three criteria.

Furthermore, convoluted attempts to undermine the minority party can have unintended consequences. The Court’s majority opinion in Rucho noted, “Democrats also challenged the Pennsylvania congressional districting plan at issue in Vieth. Two years after that challenge failed, they gained four seats in the delegation, going from a 12-7 minority to an 11-8 majority. At the next election, they flipped another Republican seat.”

Best Outcome among Many Possibilities

Criterion #4 requires simulation to find the most common outcomes for thousands of random maps. A map could be deemed “gerrymandered” if its election outcome does not fall into one of the expected distributions of seats. This is what the dissenting opinion proposed in Rucho: “Suppose now we have 1,000 maps, each with a partisan outcome attached to it. We can line up those maps on a continuum – the most favorable to Republicans on one end, the most favorable to Democrats on the other … And we can see where the State’s actual plan falls on the spectrum – at or near the median or way out on one of the tails?”

So, that is exactly what we did. Here at Emporia State, we randomly chose 100,000 possible maps for a hypothetical district of 100 people, divided into four districts. In this district, one party has a 52% majority, the other 48% supports a second party. For the sake of simplicity, these maps did not require the districts to be contiguous. While such districts might not be practical in reality, it does guarantee the most non-partisan maps possible since “urban electoral districts are often dominated by one political party-can itself lead to inherently packed districts” (Rucho). This simulation shows that for a state of 100 voters, about 54% of non-partisan maps will lead to the majority party winning two seats. Another 40% will yield three seats to the majority, while 5% will give the majority one seat.

However, results change dramatically when the parameters for a state are tweaked. As the table below shows, the expected distribution of seats quickly changes if the advantage of the majority party increases.

Number of votes out of 100 for Party X (the majority party)

Seats won by X 50 52 55 60 65 70
1 17% 5% 0% 0% 0% 0%
2 66% 54% 24% 2% 0% 0%
3 17% 40% 63% 43% 14% 3%
4 0% 1% 13% 54% 86% 97%

These trends become more pronounced as the population of a state increases. As the next table indicates, even a slim 52% majority will eventually guarantee Party X wins all four seats if the population is large enough. This fact was recognized by the majority opinion in Rucho: “[i]f all or most of the districts are competitive … even a narrow statewide preference for either party would produce an overwhelming majority for the winning party in the state legislature.”

Distribution of random map outcomes for various populations when Party X has 52% of the vote

Seats won by X 100 voters 1000 voters 10000 voters
1 5% 0% 0%
2 54% 19% 0%
3 40% 64% 5%
4 1% 17% 95%

A more sophisticated simulation will generate different results. The fact that states generally do not have all their districts vote in favor of a single party indicates that contiguousness of districts affects the outcome. In other words, party affiliation is not randomly distributed across a state. Thus, the minority party is likely to have enough votes concentrated in one region of a state to win at least one district. Simulations that take into account contiguousness, county lines, or other state-specific restrictions will be less random and more likely to benefit the minority party.

North Carolina 2016 House Districts Map

Now let’s try a real-world example. Consider the 13 congressional districts of North Carolina. In the 2016 election, 49.8% of voters selected the Republican nominee for President while 46.2% chose the Democratic nominee. Despite this slim difference, ten of 13 districts voted Republican. Using the given percentages from 2016, suppose we assign each of North Carolina’s 2,706 precincts a voter preference – Republican, Democrat, or neither. We next randomly distribute those precincts into 13 districts of approximately the same size. We repeat this experiment 1000 times.

The next table shows the results of this simulation, assuming any tied districts went equally to Republicans and Democrats. Notice about 40% of these maps will result in Republicans winning at least 10 seats. On the other hand, a less random simulation, conducted by an expert witness that takes into account North Carolina districting criteria, had zero maps out of 3000 give Republicans a 10-3 advantage or better (Rucho). In other words, the state’s districting criteria actually lead to a smaller Republican advantage than would be predicted by a random simulation.

Simulation of percentage of North Carolina districts won by Republicans

# of districts won by Republicans 7 or less 8 9 10 11 or more
% of maps 5% 18% 36% 30% 10%

Conclusion

Instead of viewing gerrymandering as a tool to pad the majority, it may make more sense to view it as a tool that may be used to increase minority representation. Furthermore, as political scientists have noted for years, multimember districts with proportional representation—while not required by the Constitution or Court rulings—remains by far the more effective method to ensure fair representation for minorities. However, this method is rarely used in U.S. Congressional or state legislative elections.

About the Authors: Bran Hollenbeck is a Professor of Mathematics at Emporia State University and Michael A. Smith is a Professor of Political Science at Emporia State University where he teaches classes on state and local politics, campaigns and elections, political philosophy, legislative politics, and nonprofit management. Read more from Smith on his blog and follow him on Twitter.


The views and opinions expressed on this blog are solely those of the original authors and other contributors. These views and opinions do not necessarily represent those of the Midwest Political Science Association, MPSA staff, and/or other site contributors.