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Combatting Gerrymandering with Social Choice: the Design of Multi-member Districts

Every representative democracy must specify a mechanism under which voters choose their representatives. The most common mechanism in the United States – winner-take-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 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 open up a rich research agenda at the intersection of social choice and computational redistricting.


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1 Introduction

This bill requires (1) that ranked choice voting …be used for all elections for Members of the House of Representatives, (2) that states entitled to six or more Representatives establish districts such that three to five Representatives are elected from each district, and (3) that states entitled to fewer than six Representatives elect all Representatives on an at-large basis—Fair Representation Act, H.R. 4000, 2019.111

The Fair Representation Act, first introduced in 2017 and reintroduced in 2019 and 2021,222 would mandate the use of multi-member districts (MMDs) to elect members to the United States House of Representatives, i.e., having fewer, larger districts each with multiple representatives. The bill is supported by good governance organizations such as FairVote [43]; the American Academy of Arts and Sciences in 2020 released a report advocating states to use multi-member districts – however, “on the condition that they adopt a non-winner-take-all election model” [3]. Despite the popular focus on single-member district (SMD) elections, such MMDs have a long history in the United States, especially at the state and local level. In 1962, 41 state legislatures had MMDs, often with winner-take-all models [65]; even today, 10 state legislatures elect representatives for at least one chamber in such a manner – Arizona, for example, has two-member districts, where each voter is given two votes and the top two vote-getters in each district are elected [14, 15], and there is an ongoing debate in Maryland on whether to keep their (winner-takes-all) MMDs. City councils, state parties, and other organizations often adopt more sophisticated techniques, using variations on Ranked Choice Voting (RCV) to elect multiple winners from each of several districts [44].

This paper considers the design of such multi-member districts and, in the process, opens a rich research agenda at the intersection of two well-studied, but hitherto distinct, aspects of the design of representative democracies: gerrymandering, and social choice for multiple winners. The gerrymandering literature assumes winner-takes-all in SMDs and, given a fixed set of voters, studies how to divide the voters into districts such that the set of winners across districts satisfies desirable properties (e.g., for a gerrymandering party, maximizing the number of winners belonging to their party; for a neutral rule-maker, devising rules to ensure ‘proportionality,’ such that the fraction of winners of each party matches the fraction of voters). The social choice literature, on the other hand, considers a single district and studies voting rules (functions from voter rankings to a set of winners) such that the set of winners in that district satisfies similar properties, including proportionality. A map consisting of multiple MMDs (e.g., 30 two-member districts in Arizona) requires both partitioning voters into districts and devising how each district collects and aggregates votes. The challenge cannot be decomposed: drawing districts depends on the social choice function, and the same social choice function has different effects depending on district size and composition.

We study this joint challenge, carrying out large-scale analyses for the U.S. House of Representatives under MMDs with different social choice functions, under maps that could be drawn by either partisan gerrymanderers or independent commissions. We show that indeed the choices of (how many and which) districts to draw and which voting method to use cannot be made independently. For example, while MMDs with winner-take-all elections are commonly believed to be discriminatory against minorities when compared to SMDs [64, 36, 23], we find that MMDs with more proportional voting methods support political minorities. We further find that ‘interior’ solutions in this joint optimization – with neither SMDs nor one large MMD – often best balance the multiple objectives promoted by good governance groups. In more detail, our contributions and findings are as follows.

First, in Section 2, we provide a scalable methodology to algorithmically study partisan gerrymandering and fair redistricting under MMDs. We derive an efficient way to calculate partisan outcomes under Single Transferable Voting (a common variant RCV for multiple winners), enabling the use of an algorithm by Gurnee and Shmoys [49] to calculate near-optimal multi-member maps.

Second, in Section 3, we study proportionality as a function of the number of districts and the voting rule used. Our results indicate that two-member or three-member districts with proportional voting rules are often sufficient to prevent the worst gerrymanders by an adversary and to enable independent commissions to find a proportional map – even small MMDs are enough to eliminate “natural” gerrymandering, in which geography and the distribution of citizens leads to a natural advantage for one party, and purposeful ones. However, poor design constraints (in either the number of districts or the voting rule used) enable more extreme gerrymanders than is possible with SMDs. These effects occur because MMDs with appropriate social choice functions protect concentrated (political) minorities by making packing and cracking more difficult, and support diffuse minorities by joining them in the same large multi-member district, with a lower threshold to win a seat. We further analyze H.R. 4000 and show that it achieves an effective balance in the MMD design space.

Third, in Section 4, we turn to intra-party diversity, measuring two competing claims: that small SMDs protect diverse, geographically-correlated interests within a party (e.g., urban, Black Democrats versus suburban, white Democrats), and conversely that RCV with large MMDs enable ideological differences within a party (e.g., Progressives vs moderate Democrats) [44, 66].

We find, under reasonable assumptions on how voters rank candidates within a party, that two- or three-member districts preserve geographic compactness, and thus the notion that legislature members represent a geographically cohesive set of voters.

More generally, outside the context of multi-member districts for legislatures in the United States, we believe that this work opens up a substantive new research area in the design of representative democracies at the intersection of gerrymandering and social choice, across both the computer and political sciences. Section 5 outlines a theoretical, empirical, and practical agenda for this area.

Related work

This work combines two rich literatures, showing how joint optimization over how voters are split into districts (gerrymandering) and how they vote within a district (social choice) yields outcomes that neither could achieve separately. It further contributes to the literature studying multi-winner districts in the United States. Given each field’s vast literature, we focus on the most relevant work.

Gerrymandering is the practice of using district boundaries to engineer electoral outcomes by “packing” and “cracking” voters within and between districts [52, 42, 61]. Past computational research has almost exclusively studied gerrymandering and redistricting more broadly in the context of a fixed number of single-member districts with winner-take-all elections.

Optimal political districting is well known to be an NP-hard computational problem [67, 56, 25]. Therefore, to study redistricting, researchers have developed ensemble methods [34, 59, 26, 11, 49] that generate huge quantities of legal district plans to explore the exponentially large space of feasible maps. Such techniques are commonly used to detect gerrymandering [28, 38, 40, 51] or to study the impact of different redistricting rules on the distribution of outcomes [33, 35]

. However, even under neutral ensembles, the natural geographic segregation among partisan voters can create skewed political outcomes

[19, 26].

There are many proposed definitions of (un)fairness metrics in redistricting such as the efficiency gap [73], the mean-median gap [81], partisan-symmetry [82], competitiveness [35], and most simply, proportionality. Such metrics can be used as the objective function of an optimization algorithm to generate district maps with (un)fair outcomes [75, 49, 54]. However, because of partisan segregation, in some states creating a fair plan with SMDs is actually impossible [39], motivating the use of alternate election procedures.

Social choice on the other hand, fixes the voters and number of winners in a given election and studies how various voting rules (how preferences are elicited and aggregated) affect the outcome [20, 10, 37]. Here, we focus on the literature on multi-winner elections.

The theoretical social choice literature considers properties of voting rules, often for arbitrary distributions of voters [24, 57, 12, 41, 29, 13, 46, 27]. Most relevant is Skowron [71], which studies the proportionality guarantees of various multi-winner voting rules, including Thiele rules; they find Proportional Approval Voting (a type of Thiele rule) to be the most proportional, though other Thiele methods may outperform it on other metrics. The optimality of PAV and the parametrizable nature of the class motivates us to focus on Thiele rules in this work.

On the applied side, a strand of the literature studies the effects of various voting reforms, especially on minority votes. McDaniel [60] find that RCV increased racially polarized voting in comparison to run-off elections. McGhee and Shor [62] find mixed evidence for whether top-two primaries have reduced polarization, and Rogowski and Langella [68] find no evidence that open vs closed primaries affect candidate ideology; Grose [48] finds the opposite on both points. Spencer et al. [72] advocate for ranked choice voting as a VRA remedy, supporting its legality.

Multi-member districts in the United States. As chronicled by Klain [55] in 1955, there is a rich history of multi-member districts in the United States in state legislatures and city councils. In 1962, for example, 30 states used MMDs to elect state senators, and 41 states used them to elect state representatives, mostly in two- to four-member districts with winner-takes-all procedures [65]. Historically, at-large city council members have often also been elected with equivalent procedures. On the other hand, since 1967 federal law has banned the use of MMDs to elect members to the U.S. House of Representatives, a status quo H.R. 4000 would repeal.

The political science community has studied these districts, often focusing on their effect on race [70, 31, 16], though some have questioned the optimal district size [50]. In particular, while the empirical evidence is mixed due to the observational nature of analysis, the predominant view is that winner-take-all MMDs harmed racial minorities, especially in Southern states [64, 36, 23]. Similarly, winner-take-all at-large city council elections are thought to dilute the votes of minorities [74, 22, 80, 77, 32]. As theoretical models [47] elucidate, such districts enable the majority – even without substantially distorting district lines as might be necessary with SMDs – to ensure that a minority even with 49% of the vote elects none of its preferred candidates. Partially due to such evidence, by 1982 the majority of states had eliminated the use of MMDs for their state legislatures, and several cities have eliminated their at-large council seats under court mandate. Most recently, Lempert [58] analyzes the legality of RCV and multi-member districts, and advocates for it as a court-ordered remedy to illegal gerrymanders.

Most related is more recent work that has favorably compared MMDs using rank choice voting to SMDs for city councils. Benade et al. [18] compare outcomes using RCV versus single-member districts in four empirical case studies, finding that it yields better representations for demographic minorities. From the same research group, Buck et al. [21] and Angulu et al. [4] propose using RCV with several MMDs instead of SMDs for the Lowell, MA and Chicago, IL city councils, respectively. Their work primarily considers non-partisan representation, such as race and other demographic characteristics – these factors are more salient in municipal contexts, where such MMDs with RCV are more common today. Our Section 4 explores similar questions.

To this literature, our work adds a systematic analysis of the partisan effects of such districts: providing a scalable methodology (Section 2), characterizing the effects on proportional representation under both adversarial gerrymandering and ‘fair’ redistricting, and providing design recommendations for legislation (Section 3).

2 Model and methods

Section 2.1 formalizes the joint redistricting and social choice challenge, Section 2.2 introduces the social choice functions we analyze and shows how to efficiently compute per-party seat shares under them, and Section 2.3 presents our empirical method.

2.1 Problem definition

The task is to elect a legislature composed of seats. There is a population of voters , where each voter lives in an atomic block and belongs to a party . The atomic blocks each have a corresponding population and are organized into an adjacency graph where two blocks share an edge when they are geographically adjacent to each other.

Multi-member redistricting. The blocks are partitioned into geographically contiguous districts, where each district is allocated seats in the legislature such that , and . Each district also must be population balanced in accordance with , such that the population ratio of district to the whole state is bounded between for population tolerance . The resulting set of districts is referred to as a map or a plan. Maps can be drawn to fulfill various objective functions. For example, a party can gerrymander a map to maximize the seats won by their party, or an independent commission can attempt to satisfy one or more notions of fairness.

Social choice function. Each district runs a separate election to fill its seats, with each voter voting in the district to which its block belongs. Each election is run according to a social choice function that determines what information each voter provides (in this work, either a set of approved candidates or a ranking over candidates) and how votes are aggregated.

Together, the choice of map (composed of districts with seats each) and social choice function define an election procedure. Given data on voters and assumptions on how they select or rank candidates (as detailed in Section 2.2), the procedure determines the set of winners.

Evaluation metrics. An election procedure can be evaluated based on the set of winners it produces. To measure outcomes across parties, we primarily consider proportionality: how does the fraction of winners belonging to each party (seat share) compare to the fraction of voters (vote share); the proportionality gap is the difference . The larger the gap, the more that the procedure favors one party over another. In Section 4, we further consider intra-party measures, i.e., how the election procedure influences within-party differences.

Research questions. In this work, we ask: how does the election procedure determine the induced outcomes, and what is the joint influence of the social choice function and the map? Note that the various components of the election procedure may be determined by different actors with competing interests. The quoted section from H.R. 4000, for example, would mandate that and a RCV-based social choice function be used. However, barring a separate mandate concerning independent commissions, in each state a partisan state legislature may still seek to draw maps most favorable to their party.333Of course, to the extent that these laws are enforced in court, maps have to be in accordance with the Voting Rights Act and any state-specific laws concerning partisan balance. Thus, we in particular are interested in the following design question: how do various constraints on the election procedure design space (restrictions on , , or ) affect the range of outcomes possible, under maps drawn by either partisan actors or independent commissions?

2.2 Social choice functions, voter assumptions, and calculating seat shares

We study the above research questions empirically, analyzing outcomes under counterfactual election procedure designs. However, doing so requires converting historical voting data to expected electoral outcomes under various social choice functions and with hypothetical candidates. Further, outcomes must be able to be efficiently computed, as our redistricting optimization algorithm (introduced in Section 2.3) calculates as a sub-routine the seat share under a given district and social choice function. We now lay the groundwork for our empirical method, by introducing our social choice functions and assumptions and showing how to efficiently compute per-party seat shares in a district.

Social choice functions considered. We consider two well-studied classes of social choice functions from the literature: Thiele rules and Single Transferable Vote (STV).

A Thiele rule is characterized by a function . Each voter provides a set of the candidates they like most. The set of winners are determined as follows. Consider a potential set of winners . The amount of points that voter contributes to is , and the set with the most points across voters (after tie-breaking) is selected. The (non-increasing) function establishes that votes have diminishing returns; the th approved-of candidate in the set is worth . For example, winner-takes-all approval voting is a Thiele rule with : each set receives a number of points corresponding to the number of candidates in the set that the voter likes, and so the candidates with the most votes win. We further consider Proportional Approval Voting (PAV), a Thiele rule with ; and Thiele Squared, with . PAV is well-studied in the theoretical social choice literature and comes with optimality guarantees (in terms of proportionality) [71]. We use Thiele Squared as a benchmark for the extent to which a social choice function can induce a balanced (but not necessary proportional) legislature.

In Single Transferable Vote (STV), a type of rank choice voting for multiple winners, each voter instead submits a ranking over candidates. Consider a district with seats and voters; the set of winners is created iteratively, as follows. The number of first place votes for each candidate is counted. Any candidate with a number of votes at least the “Droop” threshold is selected as a winner, and their surplus votes (number of votes minus )444“Surplus votes” is typically defined as subtracting , not . However, this alternative definition better handles ties, ensuring that exactly candidates are elected and each candidate selected has at least votes. are transferred to each voters’ next preferences. Details can differ on how such votes are transferred; in our simulations in Section 4, we use the Scottish (or “fractional”) STV rule, in which each voter keeps a fractional number of votes equal to their share of the winning candidate’s surplus votes, but our results in Section 3 do not depend on this rule. If no candidate has enough votes, then the candidate with the least number of votes (with tie-breaking) is eliminated, and their votes are transferred. The process continues until candidates have been selected, or the number of remaining candidates equals the number of remaining seats. STV is commonly used in practice in multi-winner elections.

In both cases, for consistency, we assume that ties are broken in favor of candidates from party , but randomly within each party.

Voter assumptions. Next, we require assumptions on how voters rank or approve candidates. For each Thiele rule, we assume that in each district there are exactly candidates from each party, and each voter simply approves all the candidates in their party. For STV, we assume that in each district there are at least candidates from each party, and each voter ranks all candidates of their party over each candidate of the other party, though intra-party rankings may vary.

These assumptions reflect that party membership is a strong predictor of the party of the candidate for which a voter votes. Such an assumption, using historical party vote shares to predict vote shares in future elections, is used throughout the redistricting literature for SMDs; our additional assumption is that this behavior extends to partisan voting in multi-member elections.

Calculating seat share given vote shares. Now, suppose – for a given district with seats and voters – we know the fraction of voters that belong to each party (vote share). How do we calculate the number of winners from each party for a given social choice function?

Given our assumptions, it is straightforward to efficiently do so for Thiele rules, as candidates from a given party receive the same votes and so we do not have to consider individual candidates, only the number of winners from each party directly. Then, we have that the party R seat share is:

The outer encodes tie-breaking in favor of party ,555Ties occur when for some . and the inner comes directly from the rule definition. The party D seat share is .

On the other hand, in STV, candidates of the same party may have different numbers of votes (in any round), as intra-party voter rankings may be arbitrary. One may thus, naively, believe that calculating party seat shares would require further assumptions on voter behavior (on voters’ intra-party rankings) and carrying out the iterative procedure defined above, as the elimination and transfer scheme introduces substantial path dependence across candidates and rounds; in general with arbitrary voter rankings, such calculations are required. Doing so, with many voters and candidates, would be prohibitively computationally expensive as a subroutine in a redistricting optimization algorithm. The following proposition establishes that under our assumptions, party seat shares can be efficiently calculated just from the vote shares, without dependence on voter rankings.

Proposition 1 (Seat shares under STV).

Suppose – for a given district with seats and voters – that a fraction of voters belong to each party , and that there are at least candidates per party. Assume that each party’s voters rank all same-party candidates above each other-party candidates, and ties are broken in party D’s favor.

Then, the number of winners belonging to party using STV is , where is the unique integer such that

Further, STV is equivalent to proportional approval voting (PAV) in terms of partisan seat share,

At a high level, the proof establishes that no candidate receives meaningful (in terms of party vote share) votes from voters of the other party, and so in terms of party vote share, we can proceed with running two separate STV processes. Then, the sequential round path dependence does not matter, as the overall number of first place votes for candidates of each party is invariant, and so the final partisan outcome depends just on initial vote shares. We note that the proof does not depend on our assumption of fractional STV; other transfer rules (such as random voter transfer) that preserve the number of surplus votes would induce identical seat shares.

Importantly for this work, creftypecap 1 enables calculating party seat shares in a district that uses STV, without needing to (either computationally or data-wise) consider voters’ individual rankings – substantially simplifying the task for a (human or algorithmic) map drawer. As explained next in Section 2.3, we leverage this result to draw gerrymandered and fair maps for STV.

Perhaps surprisingly, the result also establishes that PAV and STV are equivalent in terms of party seat share, under the assumption that voters belong to one of two parties and rank all members of their party above members of the other party. This equivalence is not true in general [45], and to our knowledge is novel for our setting. As multi-winner elections in which candidates belong to parties are common, the insight may be of independent interest in developing proportionality guarantees for STV (while the result does not directly extend to more than two parties, there may be more general, natural conditions under which analogous results hold). In this work, we use the equivalence to transfer intuition from PAV to STV.

2.3 Empirical method

We study our research questions empirically, in the context of the United States House of Representatives. At a high level, we proceed as follows, separately for each state. Given historical voting data, we algorithmically generate maps for each state, social choice function , and district size : the most gerrymandered map for each party (the map that, given the voter data and the social choice rule, would produce the most winners from that party), the map with the smallest proportionality gap, and a neutral ensemble of random maps. For each map, we then calculate our metrics of interest for the relevant social choice rule. We detail steps of this process next. In Section 4, we simulate full STV elections to study intra-party effects and detail the relevant methodological differences there.

Data. To calculate partisan seat shares, we require vote shares in each atomic block that will be used to compose districts. For each multi-district state, we use the geography matched precinct-level statewide election returns from multiple election data repositories [63, 30, 78, 79, 5, 9, 8, 7, 6] for elections since 2008 aggregated by Gurnee and Shmoys [49]

. We filter these results to just Republican and Democratic candidates and average the two-party vote share across elections at both the block-level and the state-level. For the atomic geographic blocks, we use census tracts, with population counts provided by the 5-year American Community Survey estimates.

666All political and geographic data can be found in the Box repository As a general caveat, given the infrequency of statewide elections, these averages are noisy and sensitive to the idiosyncrasies of each race, candidate, and political environment, as well as true underlying shifts over time.

For the intra-party analysis in Section 4, we further require individual-level characteristics that can be used to simulate voter rankings. There, we use a voter file provided to us by a private election analytics company, with individual level voter data: in each block, an anonymized list of voters, along with their (potentially modeled) demographics (including ethnicity, gender, census block of home address); each voter is further assigned a party (either modeled or ground truth in states with party registration) and is scored on several ideological dimensions through a mixture of methods, including survey modeling, ecological inference, and individual level voting history.777Calculating accurate scores is a challenging task, but we note that the data originates from a prominent company whose scores are used by many state and national parties, campaigns, and outside groups. We calibrate the voter file to match the calculated party vote shares.

Generating maps. To generate maps, we use the stochastic hierarchical partitioning (SHP) algorithm by Gurnee and Shmoys [49], which recursively samples subdivisions of a state to create a large random888These districts are drawn without partisan information and favor compact districts; however, the exact distribution is difficult to characterize given the nested hierarchical dependencies. ensemble of population balanced and contiguous districts. These subdivisions are organized into a tree of nested regions that implicitly encodes an exponential number of distinct maps. Each district (leaf node) is scored on induced party seat share for each social choice function,999This step for STV is enabled by creftypecap 1. Due to the equivalence, we use the same maps for PAV and STV.

and the tree can be efficiently traversed with a dynamic program to select the most gerrymandered map, or the tree leaves can be gathered into a secondary integer program to select the districts which create the most fair plan (for arbitrary definitions of fairness). To ensure robustness, we estimate the variance of vote-share to calculate the expected seat-share per district and optimize for this expectation, rather than simply optimizing for the number of strict majority seats. This prevents, for example, the algorithm from drawing districts in which the historical vote share was 50.1% for one party and declaring them safe seats for that party.

We adopt this approach over the standard recombination Markov chain algorithm

[34] to be able to optimize and ensemble concurrently because we want to model both partisan and nonpartisan mapmakers. In particular, given vote shares in each block, as explained above, it is efficient to calculate the partisan composition of the winners in a given district for our social choice functions; thus, the algorithm can efficiently produce the extreme gerrymanders and fair maps under any such function, as the same district calculation can be used for many maps. The same outcome under recombination methods would require sampling many maps and then selecting the most extreme outcomes (which have been shown to be less extreme than SHP optimized maps [49]).

The hierarchical nature of the algorithm further naturally accommodates creating multi-member districts of various sizes, as internal nodes in the sample tree are themselves valid multi-member districts. For example, to simulate H.R. 4000, instead of running 29 chains for the 29 different combinations of three-, four-, and five-member districts to fill California’s 53 seats, we can simply generate one tree which bounds the leaf capacity to be between three and five.

Our experiments. Putting things together, for each generated map, we can simply calculate our metrics of interest. Note that while the map optimization step adds randomness to vote shares, we calculate the metrics without this imposed randomness.

We carry out the process for each social choice function and each state with 2 or more seats in the House, fixing as the number of seats allocated to that state after the 2010 census. The number of districts is varied from (a large MMD with seats) to (only SMDs). For each , the set of district sizes is selected such that all districts differ in size by at most one seat.101010Formally, let , and . Then there are districts of size and districts of size . For example, when and , . When , .

Overall, generating districts and calculating the metrics utilized over 40 CPU-weeks of compute, not counting the full STV elections simulated in Section 4.

3 Proportionality with MMDs

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 vertical lines correspond to when , the average number of seats per district, is an integer (up to rounding). For example, the vertical line at

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.

(a) Avg per-state gap in “Most Fair” maps
(b) Partisan lean distribution under STV and PAV
Figure 2: How the partisan lean and proportionality gap vary at the state level with voting method and the number of districts. (a). The state-wise proportionality gap in the “Most Fair” map in each state. Even if an independent commission wanted to close the proportionality gap, it could not do so with SMDs. (b). The per-state version of Figure 1, showing the full distribution of maps and the extreme gerrymanders. 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 Figure 1, 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 Figure 8, and plots for all states as in (b) are in Appendix Figure 6.

We begin our analysis by analyzing how MMDs affect partisan proportionality, both in how MMDs empower independent commissions and in how they constrain partisan gerrymanderers. We first illustrate the effects of each social choice function and district size, for both extreme gerrymanders and fair redistricting. We conclude this section by applying our insights to provide design recommendations for legislation such as H.R. 4000. Recall that creftypecap 1 proves that, in terms of proportionality, STV is equivalent to PAV. Throughout this section, then, we use the two terms interchangeably.

Figure 1 contains our main result; for each social choice function, it shows how the overall (across states) partisan seat share varies with the number of districts. Figure 2 zooms in on the state level.

Preventing exreme gerrymanders. MMDs are effective at preventing the most extreme gerrymanders, especially with non-winner-takes-all rules. Moving from SMDs to two-member districts provides about half the benefit in terms of reducing the proportionality gap for the most extreme gerrymanders, with further benefits to moving to larger MMDs. In the limit, with one large MMD in each state and using STV (equivalently, PAV), the proportionality gap in each state is negligible.

What explains these results? With SMDs, a R gerrymander would draw a map such that as many districts as possible have a majority of Rs (up to a tolerance for robustness). It does so by cracking, creating many districts in which the D vote share is just below (thus electing an candidate), and, as necessary, packing, creating a few districts in which the D vote share is as close to 1 as possible. Such a map would maximize D wasted votes. Now, consider PAV, i.e., a Thiele rule with , and two-member districts. Any district in which the D vote share is between and would elect one member from each party. Cracking thus requires creating districts in which the D vote share is less than . Packing means targeting the D vote share to be just below (e.g., a D vote share of to elect one member for each party), or close to but then giving up both seats in the district. In each case, gerrymandering requires more precision and wastes fewer votes for the opposing party.111111Under our voting assumptions, for every PAV election with seats, of the votes are wasted in total, and so the potential wasted votes for either party also goes to with district size.

Simultaneously, fewer districts to draw reduces the degrees of freedom available. The corresponding bands become narrower as district size increases, and similar arguments apply to other Thiele rules with strictly decreasing


Note, however, that the trend toward proportionality is not monotonic; a mixture of districts of different sizes enables gerrymanders even more extreme than do SMDs, and the pattern repeats between every integer division of the number of seats. Consider two- and one- member districts; the gerrymandering party R could then pack or crack party D in the one-member districts, and use the two-member districts to elect one candidate from each party with R vote share just above . Similarly, an urban gerrymandering party D could waste fewer votes winning one-member urban districts, while still getting above of the vote in two-member rural districts.

Enabling proportional redistricting. The above analysis considers the most extreme gerrymanders. However, commissions that draw ‘fair’ maps are increasingly common; we now show that MMDs enable the drawing of proportional maps that would not be possible under SMDs. Consider Figure 1(a), which shows the average absolute proportionality gap in each state for the map that minimizes this gap. We find that there is a substantive state-level gap that virtually disappears – up to rounding with a fixed number of districts – even with two-member districts and STV.

Figure 1(b) further demonstrates the SMD ‘Massachusetts problem,’ as elucidated by Duchin et al. [39], and shows that MMDs fix it;121212While Duchin et al. [39] find no SMD in MA with a majority of Republicans, we find it possible to draw such a district. The difference is data. They choose to only include presidential elections, where MA leans heavily Democratic. We choose to also include statewide senate and gubernatorial races where Republicans have had more electoral success, and so our averaging concludes that Republicans compose of the state. All redistricting approaches are sensitive to such data choices; results remain qualitatively the same, though details for any given election may differ. because of how evenly Republicans are distributed (diffused) across the state, there is no way to draw SMDs such that a proportional amount of Representatives are Republican. Intuitively, suppose party R has vote share . With SMDs, to achieve proportionality a commission would need to draw districts such that party R is in the majority in of the districts, which may not be possible or may require atypical, contorted maps.

A single MMD with PAV provably, approximately achieves proportionality [71], solving the Massachusetts problem up to rounding. Our results indicate that even multiple two-member districts enable fair maps and solve the problem in practice. Intuitively, in the above example, a commission would just need to draw a map with enough districts with R vote share above . Note that, by the pigeon-hole principle, with overall party vote share , for any map there will always be at least one district in which its vote share is at least ; so, as the threshold to win 1 seat decreases with district size, an ever-smaller minority party is guaranteed at least one seat.

Finally, we note that, as illustrated by the average proportionality gaps of the “Median” maps in Appendix Figure 8, even small multi-member districts with PAV or STV eliminate the issue of ‘natural’ gerrymanders, in which even ‘neutral’ maps – drawn by a redistricting algorithm that ignores partisan vote shares – favor one party, due to the natural geographic distribution of voters. Such natural gerrymanders have played a central role in discussion of a justiciable gerrymandering standard; our results indicate that with even small MMDs, an independent commission would not need to draw maps that substantially differ from neutral maps in order to achieve proportionality.

Figure 3: Comparison of Republican seat share distribution of H.R. 4000 compliant ensembles using STV/PAV voting rule to single-member district baseline for all states with 10 or more seats (see Appendix Figure 9 for additional voting rules).

H.R. 4000. In addition to testing a parameter sweep over the number of districts, we test the potential outcomes of redistricting under H.R. 4000, a proposal mandating that the number of seats per district, . Figure 3 suggests that H.R. 4000. would be successful: it enables proportionality almost everywhere and – despite offering additional degrees of freedom in the size of districts – it does not offer significant additional capacity to gerrymander over a baseline of using just three- or just five-member districts, in contrast to the case of mixing single- and two-member districts. This finding is important because coherent communities vary significantly in size and are therefore best represented by variable sized districts. For instance, in the case of Texas, the Houston metro area may be best served by a five-member district whereas, San Antonio, a city with roughly half the population, may best be served by a three-member district. Under a fixed number of seats per district, either Houston would be split, or San Antonio would be grouped together with disjoint communities.

(a) Median map
(b) Most fair map
(c) Most Republican map
(d) Most Democratic map
Figure 4: 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 the various maps selected. For example, (a) shows the vote shift needed when the median (in terms of party seat share) map is selected for each setting.

Competitiveness. Up to now, we have primarily considered proportionality – how the partisan seat share reflects the underlying vote share in each state. Here, we further consider competitiveness – the vote shift that one would need in a given district in order to change the partisan seat share. For example, in a Winner takes all election, a Republican vote share of 0.6 in a district would lead to a required vote shift of 0.1. In a PAV election with two winners, the same district would have a required vote shift of , as a Republican vote of (with appropriate tie-breaking) would lead to two elected Republicans as opposed to one.

While competitiveness is a controversial as a goal for redistricting (as an objective when drawing maps) [35], it is considered an important dimension along which to evaluate a map. Uncompetitive districts, for example, may depress participation or may contribute to polarization.

One potential concern with multi-member districts is that they may lead to uncompetitive districts if the number of members in each district is small. With two-member districts, for example, most districts will have one member from each party – and one of the parties would need at least of the vote to win both seats in the district. Our results, however, indicate that this concern is unfounded. Figure 4 illustrates the average vote shift needed to shift the seat share; regardless of which map is considered, with PAV or STV competitiveness goes up with district size. However, it is the case that most two-member districts would be split 1D-1R, even in what would have previously been considered politically safe regions. Similarly, all even-sized districts have the potentially undesirable property that a majority of votes does not always yield a strict majority of seats.

Design recommendations. Perhaps the most obvious takeaway is that using a winner-take-all procedure with MMDs, as was popular in the twentieth century and still in effect in some state legislatures, enables rampant gerrymandering at nearly all district sizes and should be avoided (in fact, while proportionality gaps with one large winner-takes-all MMD in each state cancel out at the national level in Figure 1, as shown in Figure 1(a) they increase to close to the theoretical maximum with larger MMDs). Second, a simple national rule mandating all districts elect three members (as possible depending on the number of seats) with STV or PAV would work well across most political conditions. In particular, fairness-minded independent commissions would be able to achieve proportional outcomes in every state up to rounding, and advantage-minded partisans would have their power to gerrymander significantly curtailed.

While two-member and three-member districts are effective on average, individual states can further promote proportional outcomes by tuning MMD constraints based on their population, partisan lean, and political geography (see Appendix Figure 6 for full state-by-state results). In particular, we observe that small and highly partisan states both benefit from larger districts. Smaller states have fewer opportunities for wasted votes to cancel out and so can more easily become disproportionate. Highly partisan states often suffer from the ‘Massachusetts’s problem’ and so minority parties have trouble breaking the requisite threshold. Similarly, states with significant geographic self-segregation could protect the political power of both concentrated and diffuse voters with larger districts that inherently waste fewer surplus and losing votes.

Finally, in all cases policymakers should be mindful of the trade-offs associated with the choice of valid configurations of district sizes. In particular, allowing a mix of district sizes enables more flexibility in representing coherent communities, however these extra degrees of freedom can offer opportunities to gerrymander above what is possible with just SMDs, when mixing SMDs with MMDs. We believe H.R. 4000 strikes an ideal balance in this regard.

4 Intra-party diversity

So far, we have studied how multi-member districts affect the balance of power between parties. However, one of the primary justifications of Rank Choice Voting (either for SMDs or MMDs) is that it enables minor parties or ideologies to gain seats – it blunts the game-theoretic logic that tends Winner takes all democracies toward two parties. In this section, we analyze such claims and show the effects of MMDs on intra-party diversity. Our results are mixed, indicating that there may be costs to increasing district size. Section 4.1 contains method details and Section 4.2 contains results.

4.1 Methods and assumptions

In this section, we exclusively consider STV, as it allows members within a party to prioritize different candidates, without risking their party overall representation by not approving a same-party member (as could happen with approval voting based methods like Thiele rules).131313Studying such methods would also require analyzing primaries, to select exactly candidates from each party. However, doing so requires overcoming two challenges: first, we need construct plausible intra-party voter rankings; second, we need to simulate STV elections given a map, as creftypecap 1 does not apply to intra-party representations without a hierarchical ranking structure.

Constructing intra-party rankings. We require further assumptions on how voters differentially rank candidates within their party, to study how MMDs may change the characteristics of winners within a party. The key challenge is to develop a model for how a voter – given their characteristics – will vote given a menu of (hypothetical) candidates. Up to now, we have only assumed that voters approve all candidates of their party or rank them all above members of the other party.

We do so as follows, using the voter file described in Section 2.3. Recall that we have individual-level voter data in each block, along with demographic information and ideological scores; in particular we use the voter’s geographic location and a univariate partisan score indicating the strength of their party affiliation (most Republican to most Democratic). We generate many candidates for each party in each district, with varying demographic information and partisan scores. Finally, we simply assume that voters rank these candidates within each party in order of the distance between their characteristics (either partisan scores or geographic location, in different simulations).141414This approach is conceptually related to that of Becker et al. [17], who use Ecological Inference (EI) to study racial groups’ vote choices in primaries, to study how to draw SMDs such that a minority group’s preferred candidate wins both a primary and the general election – with the insight that racial composition does not solely determine whether a district is effective for minorities, as within-party vote choice may not only depend on race. Our approach replaces EI with our calibrated, individual level voter file; i.e., our results that assume that voters order candidates using their individual-level partisan scores allow for the possibility that voters with different demographic characteristics may nevertheless vote similarly. For MMD analysis, both methods require extrapolating the scores to rankings, and thus requires assumptions on how voters will vote for hypothetical candidates.

We note, however, that while there has been much work on spatial voter models aiming to characterize such behavior based on the “distance” between the voter and each candidate [2, 76, 69, 53], it is a hard challenge – it is not clear that voters behave according to such ideological spatial positioning, beyond the relative consistency of party membership. Thus, our results should not be interpreted as what would necessarily happen with multi-member districts, but what intra-party coalitions could or could not be formed with MMDs given voter interest, and how these coalitions differ from those possible under SMDs.

Running STV elections. Given the candidates, sampled voters, and the voters’ rankings over the candidates, we simply run fractional STV for each district in each given map. We do so for the random, neutral maps calculated above, as we wish to study intra-party effects as distinct from partisan ones. Running these STV elections utilized over 60 CPU-weeks of compute, on top of the map generation discussed in Section 2.3. Note that this run-time is for given maps, underscoring the methodological necessity of creftypecap 1 to study partisan seat shares without needing to simulate STV during the redistricting optimization process.

(a) Diversity of winning candidate set
(b) Diversity of coalitions supporting each winner
Figure 5: How intra-party diversity measures vary with the number of districts. (a) The diversity of the winning candidate set. (b) The diversity of the voters who supported a given winning candidate, averaged by party. These results establish that, with STV, a more diverse set of winners can be elected, i.e., minority viewpoints within a party are supported. Simultaneously, each winning candidate draws support from a more diverse coalition of voters.

4.2 Results

Figure 5 contains our intra-party results, when we assume that within a party voters rank according to partisan score. First, Figure 4(a)

illustrates how MMDs affect the diversity of the winning set within each party. For each map, we determine the set of winners for each party and then calculate the standard deviation of their partisan scores. For geographic diversity, we calculate the average Euclidean distance of each winner from the centroid of the winners’ locations. We find that with STV and large districts, a more diverse set of winners emerge, within each party. The intuition for partisan score is simple and is similar to that regarding diffusion of minority party voters across a state (the Massachusetts problem). When voters rank according to partisan score, then similar voters across the state can pool their votes to ensure that a favored candidate wins. Using SMDs, these voters may be split across different districts, such that in each a different intra-party coalition chooses the winning candidate. Surprisingly, larger districts simultaneously increase the geographic diversity of winners, even when voters rank according to partisan score.

Second, Figure 4(b) shows that with MMDs each winning candidate draws support from a more diverse coalition of voters. For each winner, we determine the voters who contributed votes in the STV round in which the candidate was elected, weighted by how many votes they provided (since we do fractional STV). For partisan score, we calculate the (weighted) standard deviation of the partisan scores of the voter coalition. For geographic diversity, we calculate the (weighted) average voter euclidean distance from the district centroid. Finally, we average across winners within each party. Especially in terms of geographic diversity, this result establishes that MMDs may come at a cost, in terms of the geographic representation aspect of our representative democracy: insofar as it is valuable for a representative to be accountable to a cohesive set of voters (such as a geographically cohesive population for which they provide constituent services or areas for which they acquire funding), large MMDs weaken such ties; voter coalitions move from about 25 kilometers from the coalition center on average to almost 160 kilometers. Two- or three-member districts, however, come at a far small cost.

Appendix Figure 11 contains the same plots when voters intra-party rank is according to geography. Perhaps as expected, under this assumption MMDs do not increase the diversity of the winners according to partisan scores – if voters do not band together based on partisan scores, then their mutually preferred candidates on this dimension do not win. However, the findings regarding coalition diversity remains virtually identical: even when voters rank within a party based on geographical distance to a candidate, our findings suggest that each winner represents a far more geographically dispersed coalition of voters than with SMDs. We note that while this finding may seem paradoxical, it can occur when party members are not evenly distributed throughout the state – whereas members in a (smaller) city may with SMDs be big enough to elect a preferred member, in MMDs they may be over-ruled by a larger group of same-party members in another city.

These findings suggests caution in choosing too big a district size for MMDs, especially as our results in Section 3 suggest that most of the proportionality benefits can be achieved with two- or three-member districts—and results in this section suggest that such districts come at a smaller cost in terms of geographic cohesiveness.

5 Discussion

We introduce the joint gerrymandering and social choice problem, showing the promise of multi-member districts with non-winner-takes-all rules in ensuring partisan proportionality, in terms of both enabling independent commissions and constraining partisan gerrymanders. We show that H.R. 4000 achieves an ideal balance between flexibility of representation while ensuring proportionality.

There is much left to do in terms of understanding the effects of multi-member districts. Future work should adopt our methodology to study partisan gerrymandering state and local legislatures where democratic innovations are more common, especially in the states that already use (winner-takes-all) MMDs. Such legislatures stand to benefit even more from MMDs than congressional districts given that they cannot rely on interstate cancellation of partisan bias as is currently the case in the House of Representatives. While we expect the high level insights to hold, details may differ when zooming in with more seats in a smaller region due to, e.g., effects of geography. Also crucial is studying how intra-party effects such as those studied in Section 4 (and in the work of, e.g., Angulu et al. [4] and Buck et al. [21]) interact with partisan gerrymandering (or gerrymandering on other characteristics). Similarly, the 2021 version of H.R. 4000 has a provision151515Sec. 205, here: that MMDs should not be used in states where doing so would violate the Voting Rights Act; it would be crucial to analyze under what circumstances this could happen and how one would measure violations.

Finally, while this work is empirical, it opens an intersection of gerrymandering and social choice with many theoretical questions that we expect to be of interest to the computational social choice community. For example, while much is known about proportionality in the case of one large MMD (see, e.g., Skowron [71] and references therein), to our knowledge no work has considered it in the case of multiple MMDs. Our results suggest that under party-based voting, the proportionality gap decreases (non-monotonically) with district size; showing this effect under more general assumptions, including how it interacts with the intra-party effects highlighted in Section 4, is of interest.


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Appendix A Proofs

Lemma A.1.


and let be the unique integer such that

Then, , for all .


We have that is such that:

The first condition requires that choosing the th candidate from party R is strictly (due to tie-breaking against R) more valuable than choosing the ()th candidate from party D.

The second condition requires that choosing the ()th candidate from party R is no more valuable than choosing the ()th candidate from party D.

Rewriting the first condition, we have

Similarly, rewriting the second condition yields

See 1


For expositional simplicity, we assume that , i.e., the number of votes is evenly divisible by one plus the number of winners, though the same arguments extend. Then, let be the Droop quota.

To understand the intuition, note that the given seat shares would immediately hold if each party had a coordinator who could optimally decide how voters of that party rank candidates within the party. In that case, the coordinator would ensure that as many candidates have possible have first place votes equal to the Droop quota, with no first place votes going to candidates who will be eliminated. The resulting seat shares follow, given tie-breaking against party R (and, if needed, the surplus vote transfer procedure for the additional single vote needed in the Droop quota).

However, this argument is not sufficient because, in principle, candidates could receive meaningful votes from voters of the other party. Further, without such a coordinator, sub-optimal arrangement of votes could potentially lead to elimination of candidates who would be elected with such a coordinator. (The former reason is fundamental, and is why the formula does not hold in general without parties or even for more than 2 parties without further assumptions; the latter possibility is exactly the issue that STV surplus vote transfers are designed to avoid, and eliminating it is book-keeping). The proof centers around eliminating these two possibilities.

The key step in the proof is noting that under the assumptions, that votes can only transferred from a candidate of one party to a candidate of the other party (either after a candidate is eliminated or selected as a winner) if candidates from the sender party have been exhausted, as the sender party voters rank all other party members after all their own party members.

Thus, for any set of voter rankings under the assumptions, the per-party seat share remains the same under re-arrangements of how each voter ranks members of the other party (at the point that such rearrangements matter for who is elected, only one party candidates are left, and so the partisan seat share does not change). Thus, without loss of generality, for the rest of the proof we assume that all voters within a party share the same ordering for candidates of the other party.

Recall that the Droop quota is designed such that no more than total candidates can ever meet the Droop quota across rounds. Doing so would require at least total votes (as votes necessary to reach the quota minus 1 are never transferred and so not double counted, and the th candidate would require votes), and

The above facts establish that we can carry out the initial rounds of STV separately for each party, until in each party there are either no candidates remaining or all remaining candidates for that party have votes exactly equal to : no votes are transferred across parties up to this stage; as long as we only elect candidates with votes at or above the Droop Quota, we do not mistakenly elect any candidate separately that we would not have together; and if this stage is ever reached, the identities of the elected or eliminated candidates does not matter, since up to now there is a conservation of votes by party and so the per-party counts remain the same.

Now, consider each party , and suppose in current round that candidates have been elected for the party. Further suppose that there are candidates left for the party, and they have number of first place votes among them. If and there exists a candidate with vote share not exactly , then by the pigeonhole principle, there exists at least one candidate of the party such that the votes for that candidate meets the Droop quota and is selected. Then, that candidate is declared a winner, and its surplus votes are transferred to other candidates of the same party. Then, iterate with , , and . Otherwise, eliminate the candidate in the party with the least number of votes, and iterate with and until or the remaining candidates all have a number of votes equal to .

Suppose at the end of these separate processes, candidates from each party have been elected, and there are candidates tied with votes each. We know that , as at least this many votes were required to elect candidates and have the remaining tied at .


Other candidates eliminated

Furthermore, for each party , and so for each party , we have

Next, note that the inequality is strict for party , given the tie-breaking favoring party by assumption: either and so (the last candidate selected had votes at or above ), or there is a tie involving a candidate of party who does not get selected, .

The proof is finished by applying Lemma A.1.

Appendix B Map Generation

All details of the the Stochastic Hierarchical Partitioning (SHP) algorithm can be found in the original paper and associated appendices [49]. We further use the same geographic and electoral data (see Appendix Table 2). However, for sake of reproducibility, we discuss the relevant algorithmic parameters and differences from the original paper. Pending reorganization for clarity, we plan to make our github public.

Multi-member. The main algorithmic difference from the original SHP algorithm is that we adapted ours to generate MMDs. To do this, instead of parameterizing a sample tree node by a region and total number of seats , we needed to also specify the number of districts that node contains. This is required because at an intermediate node, the number of districts is not immediately derivable from the total number of seats in that node because of ambiguity in the number of versus sized districts. Therefore, the number of seats is used just to balance population, and the number of districts is used for all other tree operations (sampling valid splits, maintaining balance, etc.).

Ensembles. For each pair (state, ), with and , we sampled the root node times and each internal node times. These constants were chosen to balance computational cost and optimization quality. We used random-iterative center selection with Voronoi-weighted capacity matching to sample region centers and sizes. All districts are population balanced within a tolerance. Each of these ensembles were then scored, optimized, and subsampled to create a final distribution over partisan outcomes.

Appendix C Supplementary plots

Figure 6: Figure 1(b), repeated for each state with at least 2 Representatives. Y-Axis is Republican seat share.
Figure 7: Proportionality gap in favor of each party in most advantageous map
Figure 8: Figure 1(a), repeated for Median, Most Republican, and Most Democratic maps.
Figure 9: Seat share distribution of H.R. 4000 compliant ensemble by voting rule.
Figure 10: Comparison of Republican seat share distribution of H.R. 4000 compliant ensembles with baseline for all states with 10 or more seats using STV/PAV voting rule (see Appendix Figure 9 for more rules). H.R. 4000 mandates . The baseline is to use only three-member and only five-member districts (with four-member districts for overflow as necessary).
(a) Diversity of winning candidate set
(b) Diversity of coalitions supporting each winner
Figure 11: Same as Figure 5, except voters now rank within party based on geographic distance.

Figure 6 shows the complete state-by-state results for STV based elections where the high level patterns from Figure 1 can be seen in more granular details. For example the bump structure at uniform district sizes is very clear for Michigan (MI), Flordia (FL), Wisconsin (WI), California (CA), North Carolina (NC), and Pennsylvania (PA), among others. In CA, MA, NY, OK, and TN single-member proportional plans are not even possible due to the diffusion of minority party voters. In all states, the maximum gerrymandering capability peaks at about 1.5 seats per district and starts to decay rapidly with larger districts.

In Figure 7 we see that, under full control of the redistricting process, Democrats could actually more effectively gerrymander the House of Representatives than Republicans, a finding counter to the conventional wisdom that geography favors Republican gerrymandering [26, 19]. The main caveat, is that this analysis is still using only "natural" districts – those which aren’t wildly contorted – and therefore does not include the most surgical gerrymanders. This is a standard challenge in redistricting algorithms, which are effective for large-scale analysis but can often be out-performed (with respect to any metric) in any specific setting by experts. Regardless, as shown in Figure 6, it is true that Democrats can gerrymander their large states (CA, NY, and IL) more effectively than Republicans can gerrymander theirs (TX and FL), and that when ignoring the VRA, Democrats can crack large cities into many Democratically leaning wedge-shaped districts. Furthermore, the gap in advantage is largest at two- and three-member districts because these thresholds enable Democrats to more efficiently place their voters than Republicans. For STV in particular, Democrats can break the or threshold needed for a sweep in many urban districts, while also still clearing the or threshold needed to gain one seat in rural districts, resulting in fewer wasted votes across most types of districts. While critics might point to this as a deal-breaker, it is important to recognize that this is only true in the limit of Democratic control, and such a scenario is deeply unrealistic.

In Figure 8, we show how the median, max Republican, and max Democratic absolute proportionality gap changes as a function of the voting rule and ratio of districts to seats. The median gap drops by about a factor of three between single-member and two-member districts and continues to slowly decay with larger districts. The median gap is a relevant metric because it proxies how easy it is to create a proportional map. Similarly, if the median map is fair, then there exists many fair maps, and it becomes easier to optimize for other desirable criteria like proportional racial representation, maintaining political subdivisions, and compactness. Also in Figure 8, we see that STV and Thiele squared track each other very closely, with the exception of Democratic gerrymanders of two- and three-member districts. This follows similar logic as the overall Democratic advantage analysis above, except that because Thiele squared requires more votes for a sweep, Democrats can no longer rely on full control of districts with just above of votes, and so end up wasting many votes in more heavily democratic areas by just missing the sweep threshold.

Figure 12: Example fair multi-member plans for Texas.
Figure 13: Example fair multi-member plans for Illinois.