Readings: Metrics

Compactness

Barnes, R., & Solomon, J. (2020). Gerrymandering and compactness: Implementation flexibility and abuse. Political Analysis, 1-19.

Definition: Compactness measures examine the shape of districts, specifically the "compactness of a district is a geometric quantity intended to capture how “contorted” or “oddly shaped” a district is" (p. 2). However, compactness measures are problematic and complicated in implementation due to confounding factors of "geography, topography, cartographic projections, and resolution" (p. 2). Compactness scores suffer from degrees of ambiguity and ability to be gamed.

Legal and Legislative History: Compactness measures appear in Reynolds v. Sims (1964), Gaffney v. Cummings (1973), Thornburg v. Gingles (1986), Shaw v. Reno (1993), Bush v. Vera (1996), Karcher v. Daggett (1983), and Cooper v. Harris (2017); 37 states have legal language requiring state legislative districts to be compact and 18 require congressional districts to be compact

The authors identified 9 factors that describe the complications behind the use of compactness measures.

(1) Choice of mathematical definition // 24 different measures in the literature; 3 most widely used listed below, each with a range of [0, 1] with higher values indicating greater compactness and lower values indicating potential gerrymandering:

• Polsby-Popper: Given as 4πA/P2 where A is the area of a district and P its perimeter, also known as the “isoperimetric ratio".

• Reock: The ratio of a district’s area to the area of its minimum bounding circle

• Convex Hull: The ratio of a district’s area to the area of its convex hull, the minimum convex shape that completely contains the district

(2) Contiguity // There's no federal requirement that districts must be contiguous and compactness measures assume contiguity. While there are ways to incorporate non-contiguity into compactness sores, each has a large effect. Polsby-Popper contain formula extensions to accommodate non-contiguity.

(3) Topological holes // Compactness scores make assumptions that districts are made up of a singular planar polygon; neglecting complications with islands (and their naturally shaped boundaries), and holes that are naturally occurring (e.g., lakes, ponds) or exist for historical reasons (e.g., town annexation)

(4) Boundaries of political superunits // Compactness scores do not account for the placement and position of political jurisdictional boundaries (e.g., town, county lines) and natural boundaries such as coastlines. Boundary data do not always align, authors recommend using data that co-align (e.g., U.S. Census data).

Scores can be updated to accommodate political boundaries. For Convex hull and Reock scores, the hull or minimum bounding circle can be intersected with a state polygon which results in "a better representation of what was possible and, therefore, a better indicator of whether gerrymandering took place" (p. 7).

(5) Map projection // The Earth is not flat and map projects should accommodate the curvature of a sphere to avoid distortion. This is usually not an issue (when "reasonable choices among localized (country-scale) map projections used in practice" (p. 10)), but states such as Alaska can pose a challenge (difference up to 20% between a projection of conterminous United States and an Alaska-specific projection). The use of global maps can also lead to distortions.

(6) Topography // Maps with topographical information and non-topographical information matter. Authors calculated in Polsby–Popper scores between the topographic and non- topographic data with results indicating less than 0.03 difference for all districts, with 75% of districts having deviations less than 0.005.

(7) Data resolution // Resolution can be defined as "density of points describing a boundary" (p. 11) and resolution matters for compactness scores. Lower resolution lead to simpler shapes often with shorter perimeters and choice of data resolution impacts compactness score calculations, particularily the Polsby-Popper score. Authors recommend avoiding low-resolution data, even when doing so sacrifices benefits in computational time.

(8) Floating-point calculations // 32-bit single-precision type contain 7 decimal places of precision while 64-bit double-precision type provide about 15 decimal places of precision. Authors recommend using 64-bit IEE754 compliant types as the “true” value, as 32-bit types can potentially give erroneous results.

(9) Whether alternative choices were possible in drawing a district’s boundaries

Efficiency Gap

Bernstein, M., & Dutchin, M. (2017). A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap. Notices of the American Mathematical Society, 1-6.

Summary:

Gerrymandering is manipulating shapes of U.S. congressional and legislative districts in order to obtain a preferred outcome. A proposal has been advanced to detect unconstitutional partisan gerrymandering with a simple formula called the efficiency gap (EG) which is a straight proportional comparison of votes to seats. EG can be computed based on voting data from a single election: if the result exceeds a certain threshold (0.08), then the districting plan has been found to have discriminatory partisan effect. The court affirms EG not as a conclusive indicator, but only as persuasive “corroborative evidence of an aggressive partisan gerrymander.” EG could be a useful component of a broader analysis. One of the most promising directions in the detection of gerrymandering is the use of supercomputing algorithms that can take multiple indicators into account simultaneously.

Specific Items of Note:

• Packing: when you seek to stuff your opponents into districts with very high percentages.
• Cracking: when you disperse your opponents into several districts in numbers too small to predominate.
• Both Packing and Cracking generate wasted votes by your opponents and thus reduce your opponent share of seats relative to their share of the vote cast.
• The efficiency gap (EG) is a simple numeric score proposed to detect and reject unfair congressional and legislative maps that are rigged to keep one party on top in a way that is unresponsive to voter preferences.
• Wasted votes, in the EG formulation, are any votes cast for the losing side or votes cast for the winning side in excess of the 50% needed to win.
• If (nearly) all the wasted votes belong to the winning side, it’s a “packed” district.
• If (nearly) all the wasted votes belong to the losing side, it’s a “competitive” district.
• And if there are several adjacent districts where most of the wasted votes are on the losing side, then it may be a “cracked” plan.
• EG is a signed measure of how much more vote share is wasted by supporters of party A than B.
• If EG is large and positive, the districting plan is deemed unfair to A.
• EG = 0 indicates a fair plan, as the two parties waste about an equal number of votes.
• Stephanopoulos-McGhee writes that this definition “captures, in a single tidy number, all of the packing and cracking decisions that go into a district plan.”
• Since in real life no districting plan will have efficiency gap of exactly 0, EG = 0.8 correspond to a historically robust threshold for unacceptable partisan gerrymandering.

General thoughts (Next Steps):

• If the Supreme Court were to enshrine the efficiency gap as a dispositive indicator of partisan gerrymandering, it would be sure to produce false positives as well as false negatives with respect to any common-sense understanding of political unfairness.
• The role of EG in legal landscape is much more complex. A lot of care, including further statistical testing and modeling, is required to use EG responsibly.
• On this view, the efficiency gap quantifies the level of bonus that should be enjoyed by the winner: a party with 60% of the vote should have 70% of the seats, a party with 70% of the vote should have 90% of the seats, etc.

Partisan Fairness

Deford, Daryl et al. (2021). “Implementing partisan symmetry: Problems and paradoxes.” Political Analysis.

The political science literature has proposed several measures of partisan symmetry for practical application in the evaluation of partisan gerrymandering cases, most recently by Katz, King, and Rosenblatt (2020).

This paper assesses the usefulness of these partisan symmetry metrics by applying some basic mathematical analysis to these concepts and examining their usefulness in the context of recent voting patterns in Utah, Texas, and North Carolina.

Combined, these analyses raise several significant concerns with the practical application of these metrics in the assessment of gerrymandering cases.

The underlying logic of most symmetry scores is the axiom that 50% of the votes should translate to 50% of the seats for a given political party.

The mean-median metric is vote-denominated: it produces a signed number that is often described as measuring how far short of half of the votes a party can fall while still securing half the seats.

A similar metric, partisan bias, is seat-denominated. Given the same input, it is said to measure how much more than half of the seats will be secured with half of the votes.

The ideal value of both scores is zero and both are signed scores, meaning they are supposed to identify which party has an advantage and by what amount.

Realistic implementations of these scores are also highly gameable. In all threes case studies examined in this paper (Utah, Texas, and North Carolina) it was possible to achieve neutral partisan symmetry scores (regardless of type) while still creating extreme partisan outcomes for at least one party.

The use of these scores can create paradoxes in which the score makes a sign error, and thus flags one party as being the beneficiary of partisan gerrymandering when in fact the electoral map ensures that it receives the fewest seats possible based on the political geography.

“We argue that there is at present no workable framework to make good on the idea of partisan symmetry” (p. 2).

“Proportionality is not the neutral tendency of redistricting and in some cases it may be literally impossible to secure (Duchin et al. 2019)” (p. 4).

“When there is an extremely skewed outcome (with a vote share for one party exceeding 75%), we will show that paradoxes always occur, just as a matter of arithmetic. But even for less skewed elections with a vote share between 62.5 and 75% for the leading party—which frequently occur in practice!—mundane realities of political geography can force these sign paradoxes” (p. 9)

“The Characterization Theorem shows that a putatively perfect symmetry score is nothing more and nothing less than a requirement that the vote shares in the districts be arranged symmetrically on the number line. Someone who wishes to assert that partisan symmetry is really about some principle—majority rule, responsiveness, equality of opportunity, etc—would have to explain why that principle is captured by the simple arithmetic of vote share spacing. With this framing, it is more difficult to argue that symmetry captures any essential ingredient of civic fairness” (p. 17).