Gerrymandering: The Efficiency Gap

Abstract

As gerrymandering has become a powerful tool to create and maintain power in an increasingly polarized U.S. political landscape, the courts have wrestled with ways to definitively identify gerrymandering. We describe the efficiency gap, a quantitative measure of gerrymandering, together with a prescription for how to use it, developed by lawyer Nicholas Stephanopoulos and political scientist Eric McGhee [2015] to identify political gerrymanders to the courts.

We introduce definitions relevant to the efficiency gap; we offer examples to help understand it further. Because the efficiency gap was developed for use in litigation, we include some legal history so as to understand the motivations for the definitions. Finally, we look at some mathematical properties of the efficiency gap.

Note: The information below was created with the assistance of AI.

1. Level of Mathematics
Target Level:
This paper is best suited for upper-level high school students (AP-level) or undergraduate students, particularly those studying:

Introductory discrete mathematics

Statistics

Political science with a quantitative focus

Operations research or mathematical modeling

Complexity Justification:

The math includes ratios, proportions, linear equations, and basic algebraic manipulation.

Advanced mathematical analysis includes efficiency metrics, graph interpretation, and sensitivity analysis.

The paper avoids heavy formalism, focusing instead on applied and intuitive approaches to mathematical reasoning in civic contexts.

2. Application Areas
Primary Applications:

Political Science and Public Policy: Identifying and evaluating gerrymandering practices in electoral districting.

Civic Education: Teaching mathematical literacy in the context of democracy.

Legal Analysis: Quantitative tools for evidence in judicial arguments about fair representation.

Secondary Applications:

Data Journalism and Civic Tech: Developing tools to visualize election fairness.

Educational Outreach: Engaging students in real-world issues using math.

3. Prerequisites
Students engaging with this paper should have knowledge of:

Basic algebra and fractions

Understanding of percentages and proportions

Concept of voting systems and electoral processes

Some exposure to:

Basic data analysis

Use of graphs to interpret relationships

Concept of "wasted votes" and fairness in representation

4. Subject Matter
Key Topics Covered:

Definition of the Efficiency Gap: A measure of wasted votes to assess potential gerrymandering.

Wasted Votes Concept: Votes for losing candidates or excess votes for winners.

Mathematical Formulation:

Basic formula:

Use of seat-vote plots: Notably, Figure 2 on page 11 plots statewide seat share against vote share for U.S. congressional elections (1976–2022), showcasing the efficiency gap's typical slope of ~2.

Legal context: Includes discussions of key Supreme Court cases like Baker v. Carr, Vieth v. Jubelirer, and Rucho v. Common Cause.

Strengths and Weaknesses:

Strengths: Simplicity, judicial applicability, alignment with empirical data.

Weaknesses: Sensitive to turnout variation, unopposed races, and large majorities.

Alternative Methods: Mentions ensemble sampling as a more computational but statistically rich approach.

5. Correlation to Mathematics Standards
NCTM High School Standards:

Reasoning and Proof: Justifying measures of fairness.

Representation: Using graphs (seats-votes plots) to represent mathematical relationships.

Connections: Applying math to social and political contexts.

Problem Solving: Framing gerrymandering as a quantifiable issue.

Common Core State Standards (CCSSM):

High School Statistics & Probability (S-ID):

Interpreting linear models and regression lines.

Understanding residuals and variability.

Algebra (A-CED, A-REI):

Creating equations to model relationships.

Analyzing and solving linear equations.

Postsecondary Curriculum (MAA, CRAFTY):

Emphasis on:

Mathematical modeling

Interdisciplinary applications

Quantitative reasoning