Every representative democracy must specify a mechanism under which voters choose their representatives. The most common mechanism in the United States -- Winner takes all single-member districts -- both enables substantial partisan gerrymandering and constrains `fair' redistricting, preventing proportional representation in legislatures. We study the design of \textit{multi-member districts (MMDs)}, in which each district elects multiple representatives, potentially through a non-Winner takes all voting rule. We carry out large-scale empirical analyses for the U.S. House of Representatives under MMDs with different social choice functions, under algorithmically generated maps optimized for either partisan benefit or proportionality. Doing so requires efficiently incorporating predicted partisan outcomes -- under various multi-winner social choice functions -- into an algorithm that optimizes over an ensemble of maps. We find that with three-member districts using Single Transferable Vote, fairness-minded independent commissions would be able to achieve proportional outcomes in every state up to rounding, \textit{and} advantage-seeking partisans would have their power to gerrymander significantly curtailed. Simultaneously, such districts would preserve geographic cohesion. Through simulation, we find that the insights are robust to cross-party voting. In the process, we advance a rich research agenda at the intersection of social choice and computational gerrymandering.

翻译：每个代议制民主政体都必须明确选民选举代表的机制。美国最常见的机制——胜者全得制的单一席位选区——既助长了严重的党派选区划分不公，也限制了“公平”的选区重划，阻碍了立法机构的比例代表性。我们研究了《多席位选区（MMD）》的设计，其中每个选区选举多名代表，且可能采用非胜者全得制的投票规则。我们针对美国众议院，在不同社会选择函数下，对MMD进行了大规模实证分析，并采用了算法生成的、基于党派利益或比例代表性优化的选区地图。这需要将不同多赢家社会选择函数下的预测党派结果高效地整合到一种在选区集合上进行优化的算法中。我们发现，在使用单一可转移投票的三席位选区时，追求公平的独立委员会能够在每个州（四舍五入后）实现比例代表结果，同时，谋求私利的党派通过选区划分不公获取权力的能力将受到显著限制。与此同时，此类选区还能保持地理上的连贯性。通过模拟，我们发现这些洞察对跨党派投票具有稳健性。在此过程中，我们推进了一项融合社会选择理论与计算化选区划分不公研究的丰富研究议程。