Measuring the Compactness of Political Districting Plans by Roland G. Fryer and Richard Holden

We develop 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. Any compactness measure that 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, which requires us to solve for each state’s maximal compactness—a problem that is nondeterministic polynomial-time hard (NP hard). The correlations between our index and the commonly used measures of dispersion and perimeter are −.37 and −.29, 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 its average vote increases is significantly greater under maximally compact districting plans relative to the existing plans