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：每个代议制民主国家都必须明确选民选择代表的机制。美国最常见的机制——赢家通吃的单一议员选区——既助长了严重的党派性选区划分不公，又限制了"公平"的选区重划，导致立法机构无法实现比例代表制。我们研究了\textit{多议员选区（MMDs）}的设计方案，即每个选区通过非赢家通吃的投票规则选举多名代表。我们针对采用不同社会选择函数的多议员选区制度下的美国众议院进行了大规模实证分析，这些分析基于算法生成的选区地图，这些地图分别以党派利益或比例代表性为优化目标。实现这一目标需要将多种多赢家社会选择函数下的预测党派结果高效整合到优化地图集合的算法中。研究发现，采用可转移单票制的三议员选区制度下，秉持公平理念的独立委员会能够在每个州实现近似比例代表性的选举结果（四舍五入范围内），\textit{同时}谋求优势的党派势力将显著削弱其操纵选区划分的能力。此类选区还能保持地域连贯性。通过模拟分析，我们发现这些结论在跨党派投票情境下具有稳健性。在此过程中，我们推动了社会选择理论与计算选区划分学交叉领域的丰富研究议程。