I’ve just publicly posted a paper on how (and how not to) measure area compactness for purposes of legislative redistricting. Some of the paper is highly mathematical (the proof that an entire class of compactness measures ranks any two redistricting plans exactly the same). However, most of the paper is very understandable, especially the counterexamples showing why most of the proposed measures of compactness can be used for gerrymandering because they do not reliably measure compactness.
My new paper on measuring compactness is publicly posted here:
A Single Compactness Measure for Legislative Redistricting
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1857944
Abstract:
Legislative districts in the 50 states are being redrawn following the completion of the 2010 United States census. Thirty-five states require that districts are compact, which is believed to make gerrymandering – designing legislative districts so as to advantage one political party – more difficult. There are now more than a dozen proposed competing numerical measures of the relative compactness of legislative districts. This article demonstrates that nine of the proposed measures of compactness do not reliably measure compactness. Pictorial counterexamples show that these nine proposed measures of area compactness assign the exact same value to shapes having visually distinct compactness levels. Next, this paper mathematically proves that all area-to-perimeter or area to square-of-perimeter measures (or their reciprocals or square roots) rank the compactness of any two sets of redistricting plans in the exact same order. Thus, these reliable proposed measures of district compactness are equivalent to the simplest such measure defined as the ratio of area to the square of the perimeter. The paper concludes with a discussion of the index of compactness we believe to be conceptually and computationally best because it has a maximum value of one (1) when the area is as compact as a circle, a minimum value approaching zero when the area’s perimeter is very large compared to its area, and provides a direct comparison of any two districts’ compactness regardless of district area size. This compactness measure is 4π times the ratio of the area of the district to the square-of-its-perimeter, known in mathematics as the isoperimetric quotient.
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Kathy Dopp, kathy (dot) dopp (at) gmail.com
http://electionmathematics.org
Town of Colonie, NY 12304
“One of the best ways to keep any conversation civil is to support the discussion with true facts.”