Measuring the Compactness of Political Districting Plans

52 Pages Posted: 5 Oct 2007 Last revised: 9 Oct 2022

See all articles by Roland G. Fryer

Roland G. Fryer

Harvard University - Department of Economics; National Bureau of Economic Research (NBER); American Bar Foundation; University of Chicago

Richard Holden

Massachusetts Institute of Technology (MIT); National Bureau of Economic Research (NBER)

Date Written: October 2007

Abstract

The United States Supreme Court has long recognized compactness as an important principle in assessing the constitutionality of political districting plans. We propose a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable -- which we coin the relative proximity index. We prove that any compactness measure which satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, requiring us to solve for each state's maximal compactness; an NP-hard problem. Using two properties of maximally compact districts, we prove they are power diagrams and develop an algorithm based on these insights. The correlation between our index and the commonly-used measures of dispersion and perimeter is -.22 and -.06, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when their average vote increases is significantly greater under maximally compact districting plans, relative to the existing plans.

Suggested Citation

Fryer, Roland G. and Holden, Richard, Measuring the Compactness of Political Districting Plans (October 2007). NBER Working Paper No. w13456, Available at SSRN: https://ssrn.com/abstract=1019460

Roland G. Fryer (Contact Author)

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Richard Holden

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