Measuring the Compactness of Political Districting Plans

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. AbstractWe 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 Ϫ.37 and Ϫ.29, respectively. We conclude by estimating seatvote 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.

Can Compactness Constrain the Gerrymander?

“…At the outset we noted that convexity is a necessary but not a sufficient criterion for an optimal design under a wide range of compactness measures. Thus one might argue that the results of Proposition 1 may be avoided for designs that require more than convexity (such as the dispersion requirement in Fryer and Holden (2007)). This is correct.…”

“…Moreover by most measures, compactness is maximized by convex shapes even if convexity is not an explicit part of the measures. For measures that focus on minimizing dispersion within districts, for example (see Fryer and Holden (2007) and related measures in Niemi, Grofman, Carlucci, and Hofeller (1990)), convexity is a necessary but not a sufficient condition for compactness.…”

Using Computer Simulations to Measure the Effect of Gerrymandering on Electoral Competition in the U.S. Congress

“…Since it is difficult to estimate the nongerrymandered counterfactual using comparable elections, researchers have recently turned to simulating it using computer‐automated districting algorithms (Chen and Cottrell ; Cirincione, Darling, and O’Rourke ; Fifield et al ; Fryer Jr. and Holden ; Krasno et al. ; Magleby and Mosesson ) .…”

“…Cirincione, Darling, and O’Rourke (), for example, look only at redistricting in South Carolina; Chen () and Krasno et al. () focus on the state legislature in Wisconsin; Fryer Jr. and Holden () limit their analysis to congressional districts in only four states, and Fifield et al. () limit their analysis to districts in only two states.…”

“…For the purposes of this study, I leverage a particular districting algorithm developed by Fryer Jr. and Holden () that employs a weighted Voronoi diagram to partition the population of a given state into equally populated districts just as other algorithms do. The benefit of this algorithm, however, is that it can be used to search for districts that are at least as compact as the current set of districts.…”