Representation in District-Based Elections

Abstract

District-based elections must reconcile two objectives: representing each district’s electorate and matching the statewide delegation to overall political support. We study the design of seat allocation rules when these objectives conflict. The optimal rule is characterized by an endogenous cutoff that compares the district-level and statewide representational effects of awarding a party an additional seat. In a linear benchmark, the optimal rules form a family ranging from first-past-the-post to proportional representation as the weight on statewide representation increases. Along this path, it becomes increasingly difficult for an already overrepresented party to gain further seats through gerrymandering. The characterization yields a measure of first-past-the-post robustness to statewide representation concerns and provides a unified framework for comparing it with established electoral diagnostics. Applying these measures to U.S. House elections from 2002 to 2024, we find that maps drawn by single-party legislatures generally perform worse across our measures than those produced by other institutions, and Republican-legislature maps fare worse on average than Democratic-legislature maps. The efficiency gap, a widely used wasted-vote measure of partisan gerrymandering, is more strongly associated with struck or litigated maps than the other measures, yet it considerably amplifies the Republican–Democratic asymmetry.