Misrepresentation in District-Based Elections

State delegations are often chosen through single-member district elections, creating a tension between respecting district majorities and reflecting the statewide electorate. First-past-the-post (FPTP) follows each district's majority but can yield a delegation seat share far from the party's statewide vote share. In contrast, proportional representation (PR), which makes a party's seat share correspond to its statewide vote share, requires departing from local majorities in some districts. We measure misrepresentation as a weighted sum of within-district misrepresentation, measured by the share of voters locally represented by their non-preferred party, and statewide misrepresentation, measured by the deviation of a party's seat share from its statewide vote share. The misrepresentation-minimizing rule is a cutoff rule determined by the relative weight of statewide misrepresentation. As this weight rises, the cutoff continuously shifts from FPTP's 50% to the PR cutoff that aligns the delegation's seat share with statewide vote shares. This shift makes gerrymandering harder, offering an alternative lever to limit gerrymandering. Using a majorization-based metric of geographic concentration, we show that concentrating support reduces misrepresentation only under the misrepresentation-minimizing rule. Within this class, FPTP and PR are uniquely characterized by the absence of cross-district spillovers and by gerrymandering-proofness, respectively. Using U.S. House elections, we infer the weights that rationalize outcomes, offering a novel metric for evaluating representativeness of district boundaries and electoral reform proposals.