Misrepresentation in District-Based Elections

TL;DR

This paper formalizes the tradeoff between honoring district-level majorities and achieving statewide proportionality by introducing two distortion measures, Dist and Agg, and a total misrepresentation objective $\Phi(\mathbf x; w_A) = Dist(\mathbf x) + w_A Agg(\mathbf x)$. It shows that, for any fixed weight $w_A$, the misrepresentation-minimizing rule is a uniform cutoff on district vote shares, with the cutoff moving smoothly from the FPTP threshold $1/2$ toward a proportional threshold as $w_A$ increases, thereby generating a one-parameter family of rules that interpolate between local and statewide aims. The paper develops a full conditional- and global-optimization analysis: the optimal district allocation is to award A its top-$S$ districts, $\Phi(S; w_A) = Dist(S) + w_A |a - S|$, and the forward difference $\Delta^+(S; w_A)$ governs the optimal seat total; the endpoints $w_A=0$ and $w_A=\infty$ recover FPTP and PR, respectively. It further provides majorization-based monotonicity results, characterizes the Pareto frontier of tradeoffs, and introduces a gerrymandering-cost metric showing that increasing $w_A$ makes it harder to gain seats via vote-shuffling. Finally, the authors calibrate the framework to U.S. House elections, inferring implied weights from observed outcomes to evaluate how district maps and reform proposals balance local versus statewide representation, thereby offering a practical instrument for comparing state boundary designs and reforms.

Abstract

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.

Figures (8)

• Figure 1: Optimal cutoff rule $t(w_A)$ as the weight on aggregate misrepresentation $w_A$ varies. Left: Proportionality rounds down. Right: Proportionality rounds up.
• Figure 2: Optimal seat total $S^*(w_A)$ for the profile in the right panel of Figure \ref{['fig:mainthm']}.
• Figure 3: Feasible misrepresentation pairs $(\mathop{\mathrm{Dist}}\nolimits,\mathop{\mathrm{Agg}}\nolimits)$ for the profile in the right panel of Figure \ref{['fig:mainthm']}. Red points are the top-$S$ allocations. Green squares trace the misrepresentation-minimizing allocation as $w_A$ varies. Blue points are all other feasible allocations.
• Figure 4: Aggregate weights needed to move away from FPTP and make PR optimal are averaged over the last three election cycles. Any weight displayed as $0.6$ is $\geq 0.6$ and capped at $0.6$ to aid in display of the much lower weights. Weights for each of these years are provided in Appendix \ref{['app:empirics_data_appendix']}.
• Figure 5: The average of the first three seats switching weights over election cycles is shown for selected states. The missing points indicate years where FPTP returns a seat count that corresponds to proportional representation.

Theorems & Definitions (23)

• Lemma 1
• Lemma 2
• Corollary 1
• Definition 1
• Definition 2
• Theorem 1
• Corollary 2
• Proposition 1
• Lemma 3
• Proposition 2