Random sampling of graph partitions under constraints has become a popular tool for evaluating legislative redistricting plans. Analysts detect partisan gerrymandering by comparing a proposed redistricting plan with an ensemble of sampled alternative plans. For successful application, sampling methods must scale to maps with a moderate or large number of districts, incorporate realistic legal constraints, and accurately and efficiently sample from a selected target distribution. Unfortunately, most existing methods struggle in at least one of these areas. We present a new Sequential Monte Carlo (SMC) algorithm that generates a sample of redistricting plans converging to a realistic target distribution. Because it draws many plans in parallel, the SMC algorithm can efficiently explore the relevant space of redistricting plans better than the existing Markov chain Monte Carlo (MCMC) algorithms that generate plans sequentially. Our algorithm can simultaneously incorporate several constraints commonly imposed in real-world redistricting problems, including equal population, compactness, and preservation of administrative boundaries. We validate the accuracy of the proposed algorithm by using a small map where all redistricting plans can be enumerated. We then apply the SMC algorithm to evaluate the partisan implications of several maps submitted by relevant parties in a recent high-profile redistricting case in the state of Pennsylvania. We find that the proposed algorithm converges faster and with fewer samples than a comparable MCMC algorithm. Open-source software is available for implementing the proposed methodology.
翻译:分析员通过将拟议的重新划分计划与一系列抽样替代计划进行比较,发现党派偏差。为了成功应用,抽样方法必须标出中小或大区地图,纳入现实的法律限制,并准确和高效地从选定的目标分布中抽取样本。不幸的是,大多数现有方法至少在其中一个地区挣扎。我们提出了一个新的顺序计算法,产生一个重新划分计划样本,以便与现实的目标分布相融合。由于它同时绘制了许多计划,因此SMC算法可以有效地探索重新划分计划的相关空间,比现有的马可夫链蒙特卡洛(MCMC)算法更好,按顺序制定计划。我们的算法可以同时纳入现实世界重新划分问题中通常存在的若干限制,包括人口平等、紧凑性以及行政边界的维护。我们用一个小地图来验证拟议的算法的准确性,所有重新划分计划都可以在现实的目标分布上进行对比。我们随后将SMC算法用于评估重新划分计划的有关空间空间,比现有的马尔科洛·卡洛(MC)计算法的相对性比例要好得多。我们提议采用SMC的公开算法来评估一些相关的州级比较性比例。我们建议采用比较性方法,以便比较比较比较地将一些州级算法的比较地分析方法,以便比较性地在比较地分析方法在提交的一些比较性分析方法中找到比较地比较性地分析方法,我们用比较一些比较一些比较了比比较了比较性的方法在比较性的方法在比较性的方法在比较性的方法在比较性方法中找到比较性方法,用比较性方法,比较性的方法在比较性方法在比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性地比较性的方法在比较性地比较性地比较性地算法的比较性地比较性地比较性地比较性地比较性地算法系的比较性地算法系。