ABSTRACT

In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and these scores are used to compare two or more districts or plans. In this paper, we prove mathematically that any map projection from the sphere to the plane reverses the ordering of the scores of some pair of regions for all three of these scores. We evaluate these results empirically on United States congressional districts and demonstrate that this order-reversal does occur in practice with respect to commonly used projections. Furthermore, the Reock score ordering in particular appears to be quite sensitive to the choice of map projection.

摘要

在政治重新划分中，地区的紧凑性被用作其公平性的定量代理。几个公认的，但相互竞争的地理紧凑性概念通常用于评估区域的形状，包括Polsby-Popper得分，凸包得分和Reock得分，这些得分用于比较两个或多个地区或计划。在本文中，我们通过数学方法证明了任何地图投影从球体到平面，从这三个分数中的某些对区域的分数颠倒顺序。我们在美国国会地区对这些结果进行经验评估，并证明相对于常用的预测，这种顺序逆转的确确实存在。此外，Reock分数排序似乎对地图投影的选择非常敏感。