  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 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, and advantage-seeking partisans would
have their power to gerrymander significantly curtailed. Simultaneously, such
districts would preserve geographic cohesion, an arguably important aspect of
representative democracies. In the process, we advance a rich research agenda
at the intersection of social choice and computational gerrymandering.
