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.

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