  Political districts may be drawn to favor one group or political party over
another, or gerrymandered. 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 standards to counteract unfair redistricting practices.
This paper accompanies the release of packages in C++, Python, and R that
correctly, efficiently, and reproducibly calculate a variety of compactness
scores.
