Before revision:  For this reason, quantifying the shape of districts, in particular their compactness, is a key task in politics and civil rights. A growing body of literature suggests and analyzes compactness measures mathematically, but little consideration has been given to how these scores should be calculated in practice. Here, we consider the effects of a number of decisionsthat must be made in interpreting and implementing a set of popular compactness scores. We show that the choices made in quantifying compactness may themselves become political tools, with seemingly innocuous decisions leading to disparate scores. We show that when the full range of implementation flexibility is used, it can be abused to make clearly gerrymandered districts appear quantitatively reasonable. This complicates using compactness as a legislative or judicial standard to counteract unfair redistricting practices.
After revision:  A number of measurements have been suggested as ways to detect and prevent such behavior. These measures give concrete axes along which districts and districting plans can be compared. However, measurement values are affected by both noise and the compounding effects of seemingly innocuous implementation decisions. Such issues will arise for any measure. As a case study demonstrating the effect, we show that commonly-used measures of geometric compactness for district boundaries are affected by several factors irrelevant to fairness or compliance with civil rights law. We further show that an adversary could manipulate measurements to affect the assessment of a given plan. This instability complicates using these measurements as legislative or judicial standard to counteract unfair redistricting practices.