Combatting Gerrymandering with Ranked Choice Voting: An experimental analysis of Multi-member Districts in the United States

TL;DR

This paper investigates how multi-member districts (MMDs) and multi-winner voting rules can mitigate partisan gerrymandering in the U.S. House. It develops a scalable empirical framework that jointly optimizes district maps and social choice functions (notably STV and Thiele rules like PAV and Thiele Squared) and evaluates outcomes using vote-share proportionality, competitiveness, and intra-party diversity. The key finding is that 2–3 member districts with non-WTA rules can achieve proportional representation up to rounding and substantially limit gerrymandering, while preserving geographic cohesion and remaining robust to cross-party voting and data noise. The results suggest a practical design space, including the Fair Representation Act’s 3–5 member structure, that balances flexibility across states with strong safeguards against extreme partisan manipulation. The work thereby advances computational social choice on gerrymandering and provides actionable insights for reform proposals and independent redistricting commissions.

Abstract

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.

Figures (16)

• Figure 1: The Republican seat share over all states as the number of districts is varied in each state. The horizontal line denotes the vote share fraction, i.e., the proportionality value. Each point is composed of every state, with rounding and weighting by number of seats in the state, and vertical lines corresponding to when $N/K$, the average number of seats per district, is an integer. For example, the vertical line at $0.5$ corresponds to two-member districts in states with an even number of seats and all two-member districts except one one-member or three-member district in states with an odd number of seats. The right-most point is with SMDs, and the left-most point is if each state has one large MMD. "Median" refers to the median value found across random maps from the SHP algorithm, and "Most fair in each state" to the maps with the smallest proportionality gap. Overall, MMDs are effective at preventing the worst gerrymanders, especially with non-Winner takes all rules. Note that national seat share with Median and Fair maps only look invariant to district size because gaps cancel out between states (some favor Democrats, others Republicans). \ref{['fig:propbystate']} shows that seat share is not invariant at the per-state level: even when optimizing for fairness, we require MMDs to reduce the per-state proportionality gap.
• Figure 2: How the partisan lean and proportionality gap vary at the state level with voting method and the number of districts. (a). The absolute value of the state-wise proportionality gap in the "Most Fair" map in each state. Even if a redistricting agent wanted to close the proportionality gap, it could not do so with SMDs. (b) The distribution of the "gerrymandering range" (difference in Republican seat share between extreme Republican-optimized and Democrat-optimized maps), as a measure of how much redistricting matters. Each violin plot shows the distribution over states. While the range varies substantially at the state level for SMDs, it shrinks across states as district size increases. (c). The per-state version of \ref{['fig:propdifferentmethods']}, showing the full distribution of maps and the extreme gerrymanders for four states. While there are substantive state-level gaps with SMDs even in the most proportional maps, the gaps become negligible with even just two-member districts and STV. Note that in \ref{['fig:propdifferentmethods']}, these gaps cancel out nationally, as some state-wise gaps favor Democrats and others favor Republicans. Qualitatively similar results to (a) are shown for the Median and gerrymandered maps in Appendix \ref{['fig:propmethods_median']}, and plots for all states as in (b) are in Appendix \ref{['fig:boxall']}.
• Figure 3: The average vote shift needed in each map (averaged across districts in a map and across states) to shift the number of seats won by each party by at least one. The larger the vote shift needed in a district, the less competitive is the district. The four plots show the various maps selected. For example, (a) shows the vote shift needed when the median (in terms of party seat share) maps are selected. With PAV/ STV, competitiveness generally increases with district size.
• Figure 4: Comparison of Republican seat share distribution of the Fair Representation Act compliant ensembles using STV/PAV voting rule to single-member district baseline for all states with 10 or more seats (see Appendix \ref{['fig:hr4000_by_rule']} for additional voting rules).
• Figure 5: Gerrymandering range (difference in Republican seat share between Republican-optimized and Democrat-optimized maps) for different voter ranking noise levels, to simulate cross-party voting. Here, the voter idiosyncratic noise for each candidate is drawn from a Gumbel distribution with the noise level corresponding to $\beta$ (with higher noise corresponding to higher variance). Results with Normal distribution noise are in \ref{['fig:noisegerrymanderinggap_normal']} and are qualitatively similar. As noise levels increase, the range of seat shares feasible using gerrymandering decreases, and the effect is on top of that of using bigger districts with STV. Intuitively, higher levels of voter noise correspond to gerrymandering being harder, since individual voters differ more from what is predicted.

Theorems & Definitions (5)

• Proposition 1: Seat shares under STV
• Lemma C.1
• proof
• Proposition 1: Seat shares under STV
• proof