The design of algorithms for political redistricting generally takes one of two approaches: optimize an objective such as compactness or, drawing on fair division, construct a protocol whose outcomes guarantee partisan fairness. We aim to have the best of both worlds by optimizing an objective subject to a binary fairness constraint. As the fairness constraint we adopt the geometric target, which requires the number of seats won by each party to be at least the average (rounded down) of its outcomes under the worst and best partitions of the state. To study the feasibility of this approach, we introduce a new model of redistricting that closely mirrors the classic model of cake-cutting. This model has two innovative features. First, in any part of the state there is an underlying 'density' of voters with political leanings toward any given party, making it impossible to finely separate voters for different parties into different districts. This captures a realistic constraint that previously existing theoretical models of redistricting tend to ignore. Second, parties may disagree on the distribution of voters - whether by genuine disagreement or attempted strategic behavior. In the absence of a 'ground truth' distribution, a redistricting algorithm must therefore aim to simultaneously be fair to each party with respect to its own reported data. Our main theoretical result is that, surprisingly, the geometric target is always feasible with respect to arbitrarily diverging data sets on how voters are distributed. Any standard for fairness is only useful if it can be readily satisfied in practice. Our empirical results, which use real election data and maps of six US states, demonstrate that the geometric target is always feasible, and that imposing it as a fairness constraint comes at almost no cost to three well-studied optimization objectives.

翻译：政治选区重划算法的设计通常采取两种途径之一：优化紧凑性等目标，或借鉴公平分割理论构建能保证党派公平性的协议。我们旨在通过优化服从二元公平约束的目标来实现两全其美。采用几何目标作为公平约束，它要求每个政党赢得的席位数至少等于该州最差和最佳划分结果的平均值（向下取整）。为研究该方法的可行性，我们引入了一个与经典蛋糕切割模型高度相似的新选区重划模型。该模型具有两个创新特征：首先，州内任何区域都存在选民对特定政党的政治倾向“密度”，使得无法将不同政党的选民精细分离到不同选区，这反映了现有选区重划理论模型往往忽略的现实约束；其次，各政党可能对选民分布存在分歧——无论是源自真实认知差异还是策略性行为。在缺乏“真实分布”的情况下，选区重划算法必须同时公平对待各政党基于其自身报告数据提出的诉求。我们的主要理论结果出人意料地表明：即使面对任意分歧的选民分布数据集，几何目标始终是可行的。任何公平标准只有在实践中易于满足时才有实际价值。使用六个美国州的真实选举数据和地图进行的实证研究表明，几何目标始终可行，且将其作为公平约束对三个经过充分研究的优化目标几乎不产生成本。