The Zeroeth Amendment: Apportionment and thePerpetuation of Partisan Bias

Daniel B. MaglebyBinghamton University, SUNYdmagleby@binghamton.edu

Gregory RobinsonBinghamton University, SUNYgrobinso@binghamton.edu

July 4, 2019SOMETHING

PRELIMINARY AND INCOMPLETE DRAFT

Abstract

The topic of district size was so important in the early American republic that evenbefore it passed what would become the first amendment, the 1st Congress approved anamendment to limit the size of constituencies in the House of Representatives. In con-trast to the amendment providing freedom of speech, assembly and religion, this earlieramendment dictating a ratio of representation failed to receive the requisite supportamong the states. Absent this constitutional provision, the number of representativesand thus district population is governed by statute, the most recent of which caps thesize of the House of Representatives at 435. With over 300 million people living in theUnited States, each representative is responsible for more than 700,000 constituents.

The framers were rightly concerned about how one person could provide adequaterepresentation to so many people; however, the level of apportionment used by theUnited States has other ill-effects. In particular, we show that low levels of appor-tionment (a high ratio of constituents to representatives) exacerbates bias in electoraloutcomes. We show that increasing the population while holding constant the numberof representatives can lead to more bias in electoral outcomes. We test this claim usingMonte Carlo simulations of hypothetical districts under varying levels of apportion-ment. Using a computer algorithm we produce tens of thousands of alternative mapsof congressional districts under existing levels of apportionment and under levels ofapportionment in which there are more members of the House. Using this large setof hypothetical districts, we find that as the number of representatives increases, theexpected level of bias decreases.

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There is absolutely no reason, philosophy, or common sense in arbitrarily fixing

the membership of the House at 435 or at any other number.

-Rep. Ralph Lozier (D-MO), 1928

In general it may be remarked on this subject, that no political problem is less

susceptible of a precise solution than that which relates to the number most con-

venient for a representative legislature.

-Federalist 55

1 Introduction

In August of 1789, the United States House of Representatives considered and approved

seventeen proposed amendments to the United States Constitution. The United States

Senate considered the articles of amendment and concurred in twelve of them. A month

later, the House and Senate agreed to a conference report largely in keeping with the Senate’s

proposal. Unchanged from the House’s proposal was the wording of Article the First:

After the first enumeration, required by the first Article of the Constitution,

there shall be one Representative for every thirty thousand, until the number

shall amount to one hundred, after which the proportion shall be so regulated

by Congress, that there shall be not less than one hundred Representatives, nor

less than one Representative for every forty thousand persons, until the num-

ber of Representatives shall amount to two hundred, after which the proportion

shall be so regulated by Congress, that there shall not be less than two hun-

dred Representatives, nor less than one Representative for every fifty thousand

persons.

2

The initial amendment proposed by Congress was never ratified, but it came first because

it addressed one of the most contentious issues in debate leading up to the ratification

of the Constitution, the size of congressional districts. The amendment’s inclusion was a

strategic concession to the anti-federalist who were skeptical of the powers vested in the

federal government by the new Constitution. A few years earlier, district size elicited the

only substantive contribution to the deliberations of the Constitutional Convention from

its presiding officer, George Washington,1 who encouraged delegates to the Convention to

support a proposal changing representation in the House from one member for every forty

thousand inhabitants to one for every thirty thousand in the final draft of the Convention’s

document.2

The amendment’s purpose was to provide for a certain level of apportionment, for our

purposes, the ratio of representatives to those they represent. From our contemporary

perspective, this “Zeroeth Amendment” is striking, because it would produce a House of

Representatives with between 6,200 and 10,300 members, according to the 2010 census.

Apportionment at this level would almost certainly have led to differences in congressional

organization, congressional parties, and the policy making process from what we observe in

the modern Congress. While these effects are worth considering, we set those aside in order

to explore how changes in apportionment might affect how votes cast in congressional (or

other legislative) elections are translated into seats. In particular, we will show how the

level of apportionment conditions the bias by which votes cast for a party’s candidates in

elections are translated into seats controlled by that party in a legislature. We call the affect

of constituency size on the seat to vote translation, apportionment bias.

1Historians disagree about Washington’s sincerity. It is possible that Washington intended to put pressureon delegates, perhaps life-long friend George Mason in particular, to stop dragging their feet and supportthe draft Constitution. Mason eventually became a prominent opponent of constitutional ratification and aleader of the anti-federalists.

2Madison’s Notes on the Debates in the Federal Convention, September 17, 1787

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The implications of constituency size are many. Almond and Verba (2015) suggest that

the institution of single-member districts is one of the factors that made Americans more

satisfied with their government because of the feeling of close connection between constituent

and representative. This understanding is something that cannot exist in any straightfor-

ward manner in proportional representation systems with party voting and large district

magnitudes or even a single national voting district. The notion of constituency size is also

implicit in most comparisons of elections and member behavior between House and Senate.

The idea that the Senate is a place of (relative) moderation in part because senators have

larger, more diverse constituencies is received wisdom in the study of American politics.

To take one substantive example, scholars have suggested that constituency size can affect

legislator behavior with respect to issues of trade policy as constituencies comprise a greater

proportion of the national population. Thus interests are thought to be less particularistic

and more representative of the national interest (Rogowski 1987; Ehrlich 2009; Karol 2007;

Hearn 2018).

Our argument is not necessarily intuitive. We suggest that dividing states into fewer,

larger districts allows for the sort of bias that can advantage one party over another and

possibly exacerbate polarization. This outcome arises even if districts are drawn in a po-

litically neutral manner. The reason our argument may be unintuitive is because sampling

theory would seem to suggest that larger samples are more representative of the underlying

population. But this effect only holds when samples are independent. The legislative dis-

tricting process results in dependent samples in at least two ways. First, districts are sampled

without replacement. When each observation that is drawn for one sample is unavailable

for other subsequent samples the sampling distribution ceases to be “well-behaved.” As the

sample size comprises a greater share of the population from which it is drawn, sample means

can diverge from the population. Second, the probability that neighbors are assigned to the

same district is not uniformly distributed. Due to requirements that districts be contiguous,

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neighbors are more likely to be assigned to the same district than to different districts. Since

neighbors and neighborhoods are not randomly assigned, it follows that the collection of

people assigned to districts are likewise not randomly assigned to districts.

The second part of our argument introduces a bit more of a real world complication

in the form of residential sorting by political affiliation. Finite population sampling can

only introduce so much bias. But if voters sort themselves into relatively more politically

homogeneous neighborhoods (Bishop 2009; Nall 2015; Rodden 2019) the bias can be quite

significant. We lay out this argument in the rest of the paper.

An example puts these issues into perspective. Consider New York’s 27 congressional

districts, 63 state senate districts, and 150 state assembly districts. These districts divide

a large and highly sorted population with Democrats concentrated in New York City and

other urban areas like Buffalo, Rochester, Syracuse and Albany, and Republicans more evenly

distributed across suburban areas outside of New York City and in rural upstate regions.

In the 2016 election, Democrats carried 18 of 27 (66.6%) congressional districts. Democrats

had similar success in the State Assembly where they controlled 102 to 150 (67.3%) seats

after the 2016 election. Given that nearly 60% of New York residents cast their votes for the

Democratic candidate for president in the 2016, Democratic dominance of legislative elections

may not be surpising. On the other hand, in the same election, Democrats only managed

to carry 31 of 63 (49.2%) seats in the State Senate.3 By New York state law, the same

set of people are responsible for drawing the state assembly, state senate, and congressional

districts, so it is reasonable to expect that the process of drawing 27, 63, and 150 districts

was the same in every respect except the number of divisions in which mapmakers divided

New York’s population. What then can explain the puzzling pattern we observe in New

York? Gerrymandering is a common scapegoat for distortions in representation, and a

3In fact, two more candidates ran as Democrats, but caucused with Republicans once they arrived inAlbany giving the GOP control of the state senate. Even including those state senators among the Democrats,they only managed to carry 52.4% of the vote.

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gerrymandered map of senate districts is at least partly to blame for the patterns we observe

in New York. Likewise, we know high levels of partisan sorting like we see in New York

– with Democrats concentrated in urban areas and Republicans spread out over suburban

and rural areas – leads to distortions in partisan representation (see e.g. Chen and Rodden

2013; Chen and Cottrell 2014; Magleby and Mosesson 2018). In addition to these sources of

bias, we will show that the expected amount of bias in an electoral map in New York and

places like it is conditioned on the size of legislative districts. In short, jurisdictions exhibit

patterns of apportionment bias conditional on the number of districts into which they are

divided.

The paper proceeds as follows. In the next section we lay out an approach to identifying

apportionment and partisan sorting as a source of bias in the districting process. After that,

we discuss findings from a simulation that considers the outcome of 48,000 hypothetical

redistricting scenarios of a hypothetical jurisdiction in which different groups are sorted to

varying degrees. Next, we use data from a real world example to show how bias emerges

and attenuates as jurisdictions are divided into increasing numbers of districts. We end the

paper by offering conclusions and discussing the implications of our findings for partisan

polarization in the Congress and other legislatures.

2 Legislative District Size in the Ratification Debate

At the same time that James Madison, Alexander Hamilton, and John Jay were writing in

favor of constitutional ratification under the pseudonym Publius, a New York judge named

Robert Yates took to the pages of the New York Journal to write in opposition to the

new constitution under the pseudonym Brutus. Brutus expressed a number of objections to

the constitution, and the process by which it was drafted and proposed, over the course of

sixteen essays addressed “To the Citizens of the State of New York.” These essays form an

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important part of what have come to be called The Anti-Federalist Papers. Two of those

essays, “Brutus III” and “Brutus IV,” focus on a critique of the proposed new legislative

branch, and of its lower chamber in particular.

While Brutus highlights (in obvious contrast to the Southern Anti-Federalists) the injus-

tice of counting slaves as even three-fifths of an “inhabitant” for the purposes of enumeration,

he spends the bulk of both of these essays on the problem of the proposed size of what was

to become the House of Representatives. For Brutus, the small size of the House would, in

part, make the legislative process more susceptible to corruption by outside interests or the

executive branch and would ensure that the membership of the House would be exclusively

selected from among the wealthiest and most connected citizens, yielding only, as he puts it,

“the shadow of representation” (Brutus IV).

It is this last point that begins to outline the problem of size, or the problem of num-

bers, in the nascent House of Representatives. For Brutus, so small an assembly fails as a

representative body because, as we will put it in social scientific terms, it cannot provide a

representative sample of the breadth of interests in American society. As he writes, “[S]o

small a number could not resemble the people, or possess their sentiments and dispositions.”

Brutus writes further, as Publius also contemplates in Federalist 35, that the various pro-

fessions, classes, and orders of people in the economic and social scene cannot be fairly

represented by, for example, the six members of the House that New York would elect:

[W]e require a larger representation in proportion to our numbers, than Great

Britain, because this country is much more extensive, and differs more in its

productions, interests, manners, and habits.

Brutus was concerned with the possible effects of corrupting influences in a small legisla-

ture, and he was likewise concerned that it would be difficult to represent a diverse society in

with a small set of representatives. In addition, the small size of the House, in and of itself,

7

is problematic for Brutus: “No free people on earth, who have elected persons to legislate

for them, ever reposed that confidence in so small a number.” A small House can be a prob-

lem in-and-of-itself because it undermines the popular character of “the People’s House” by

weakening the relationship between the representatives and the represented. Indeed, Riker’s

study of the rhetorical battle over constitutional ratification suggests that the small size of

the proposed House of Representatives was one of the most frequent points raised by Anti-

Federalists in their campaign to stop ratification (1991b; 1991a). A legislature with too few

districts can also increase the incidence of apportionment paradoxes (Neubauer and Gartner

2011).4

3 Identifying Apportionment as a Source of Bias

Attribution of electoral distortions to apportionment is complicated by the fact that distor-

tions may arise for any number of reasons. Scholarship on this question has pointed to two

important considerations: gerrymandering and patterns of geographic sorting. Gerryman-

ders exist to distort electoral outcomes for “sinister” purposes (Garner 1999). Geographic

sorting introduces distortions because it tends to generate situations where voters of a par-

ticular type are over-concentrated in a minority of districts. Our theoretical claim is straight

forward: the way that votes are translated into seats is affected by the number (size) of

districts that divide a population; however, “real world” redistricting scenarios introduce

the possibility of bias arising from other sources. Thus, we begin our analysis by consider-

ing an imagined world free of gerrymandering and other sources of bias. Using a computer

algorithm, we divide a hypothetical world into maps with different numbers of districts and

show that apportionment conditions the translation of votes into seats but only when dis-

tricts divide sorted populations. The exercise occupies space on the boundary of an empirical

4Beyond that, and closer to our analysis going forward, a too-small House can exacerbate other potentialproblems like partisan electoral bias and polarization, especially when residential sorting by party is present.

8

and a theoretical demonstration. It is not an agent-based model, although it shares Monte

Carlo methods with many ABMs. We view our analysis as closely related to stress-testing in

finance: what happens when we push aspects of our American electoral system to its limits?

In this section, we focus our analysis on a hypothetical jurisdiction. For the purposes of

our analysis, we will presume that society is divided between two groups – greens and blues.

To the extent that sorting occurs, blues tend to congregate near one another in a single,

more densely populated area. An analogy to Republicans and Democrats in the United

States is intentional. Scholars and more casual observers of politics have long noted that

Democrats have come to dominate more densely populated urban areas. On the other hand,

Republicans tend to reside in less densely populated areas outside of cities. For more than

100 years, analysts have observed that, all else equal, the relative geographic concentration

of one group or party will disadvantage the concentrated group (see Edgeworth 1898; for

a more recent exploration of this phenomenon see Chen and Rodden 2013). Given that

this pattern of sorting has mostly occurred organically – without any exogenous influence

aimed at making it occur – some analysts have taken to calling the resulting patterns of

bias “natural” (see Krasno et al. 2018; Chen and Cottrell 2014). Here, we have engineered

scenarios where the blue group is increasingly more concentrated than the green group.

A computer simulation allows for causal identification of the relationship between appor-

tionment and electoral bias and allows us to condition our claims on the degree to which the

underlying population is sorted. Because of advances in automated districting algorithms,

we can claim to have plausibly randomized the districting process (see Chen and Rodden

2013; Tam Cho and Liu 2016; Magleby and Mosesson 2018; Chen and Cottrell 2014; Cirin-

cione, Darling and O’Rourke 2000). We use a computer algorithm to draw a large number

of unique, constitutionally valid maps of districts based on our engineered distributions of

voters. Crucially, we set (and can vary) the number of districts the computer algorithm

draws. Since the process for drawing districts is the same for every map, any differences in

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bias that we observe must be attributable to either the number of districts or the distribution

of voter characteristics we impose. We create a set of underlying distributions of voters in

which they are less and more sorted. For each distribution of voters, we randomly divide its

hypothetical geography into different numbers of districts.

3.1 Patterns of Spatial Sorting

To generate hypothetical distributions of voters, we populate a series of 200×200 grids with

two types of voters, green and blue. In every grid, 25% of the positions are occupied and

75% are empty. A position may be occupied by one and only one voter. Thus there are

10, 000 voters distributed across the grid. Half of the voters belong to the green group, and

half belong to the blue group. In all, we evaluate 6 grids in which the two groups are more

and less sorted. We represent these grids in Figure 1.

[Figure 1 about here.]

In each of the grids, green voters are assigned a position according to a uniform probability

distribution. We maintain the positions of the green voters in every iteration of the grid we

consider. In panel A. of Figure 1, all the blue voters are likewise distributed according to a

uniform probability function. Panels B. through F. represent a series of grids in which the

blue group is more and more clustered at the center of the grid. In these grids, half the blues

maintain their position determined by a uniform probability distribution. The other half are

assigned a position according to a distribution that makes it more likely that a voter falls

closer to the middle of the grid than further away.

More specifically, we generate each grid independently with all of the greens and half of

the blues being assigned a position according to a uniform probability distribution. For the

second set of blues the probability that a blue is assigned to position i is 0.0 if the position

10

A. B.

C. D.

E. F.

Figure 1: The underlying distributions of voters take the form of grids A. through F. In gridA., k = 1 so voters of the green and blue groups are distributed uniformly through the space.Each successive grid shows a more sorted populated with blues concentrating towards themiddle of the graph. In Grid F., k = 10, so voters are extremely sorted with most of theblue voters clustered in the center of the hypothetical map.

11

is already occupied, or the following function if the position is open.

di1∑

j∈N\M dj) (1)

where N = {1, 2, . . . , 40, 000} is the set of all positions in the grid, and M is the locations

in the grid occupied already occupied. We calculate di as follows.

di = 1−(

ci

20, 00012

)k(2)

In equation 2, ci is the distance of the furthest point in unit i of the grid. Thus di

represents the normalized distance of each unit from the outside of the grid where di is close

to 1 for units close to the center of the grid, and 0 for units furthest from the center of the

grid. Observe that as k increases, equation 1 places relatively more density on points in the

middle of the grid. In other words, using equation 1 we may randomly assign members of

the blue group to more and less sorted positions.

The distributions in Figure 1 represent different hypothetical residential patterns of voters

in a political jurisdiction. In each successive panel all the green voters and half of the blue

voters are distributed according to a uniform probability distribution. The other half are

distributed according to equation 1 where, in each successive panel, blues are more likely to

fall in a cluster towards the center of the grid. In Panel A. of Figure 1, members of neither

group are more concentrated than the other. By contrast, half of the members of the blue

group are clustered in the center of the grid in Panel F.

For our purposes, the nature of these grids allows us to make inferences about the influ-

ence of increased sorting in a population of voters. Observe that we restrict the proportion

of voters belonging to either group to 0.50 of the total population, so neither group exhibits

an advantage ex-ante. All that differs between the grids we analyze is the relatively likeli-

12

hood that one group, the blues in these data, is relatively clustered. Without information

positioning these hypothetical voters in space, we would expect that neither party would

receive any advantage in the aggregation of votes into representation.

3.2 A Neutral Redistricting Algorithm

In order to make claims about the effect of apportionment on representation, we must hold

constant the method by which districts are drawn. In the previous section we describe

our method for generating hypothetical patterns of voters. In this section, we turn to the

method of drawing districts. For every level of apportionment we consider, we apply the same

process for drawing districts. More specifically, we use an algorithm proposed by Magleby

and Mosesson for drawing neutral districts (2018). As this algorithm features prominently

in our analysis, we provide an intuitive description of the algorithm here. We give a more

technical description of the algorithm in Appendix A.

In the maps of districts we draw using the algorithm we are concerned with two consti-

tutional constraints on district delineation. First, districts must be contiguous. That is, it

must be possible to move from any part of a district to any other part of a district without

leaving the district. Second, the set of districts generated by the delineation process must

also contain equal population.5 The Magleby and Mosesson algorithm accounts for both of

these constraints in the way it delineates maps of districts (2018, 4).

To implement the Magleby and Mosesson algorithm, we conceive of the map upon which

the resulting set of districts is based as a connected graph. In this application, a connected

graph represents a set of geographic units that are connected to other geographic units if

they share a border. Using the grid we presented in the previous section as an example,

5Technically, the constitution requires that congressional districts be as nearly perfectly balanced aspossible. In practice, the court has allowed for very slight deviations from that standard. For state legislativedistricts and other districting scenarios, the Court has shown tolerance for much looser constraints onpopulation parity between districts.

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we may represent the grid as a graph if we presume that each unit of the grid is a vertex,

vi, of the graph G. An edge, e(i,j), extends from one vertex, vi, to another vertex, vj, if

the geographic units represented by the vertex share a boundary. Thus, we may formally

represent the grids presented in the previous section as a graph G(V,E) where V is the set

of all vertices (geographic units) and E is the set of all edges between vertices that share

boundaries.

We presume that vertices are connected (share an edge), if they represent units of geog-

raphy that share more than a single point of their boundaries.6 Thus, a unit of the grids we

analyze here shares a boundary with up to four adjacent units – one above, one to the left,

one to the right, and one below. Units on the outside of the grid will share boundaries with

fewer units.

In addition, we presume that vertices are weighted according to the population that

resides in the geographic unit represented in a vertex. Thus the weight of vi is wi ≥ 0. In

the grids we analyze here, weights are either 0, if we assign no voter to that unit or 1 if a

voter is assigned to that unit. Observe that the identity of the voter (green or blue) does

not factor into the calculation of the weight. Therefore we may say that the algorithm is

neutral in its treatment of either group.

The Magleby and Mosesson approach conceives of the districting process as a weighted

graph partitioning problem. It seeks to divide G(V,E) into a set of subgraphs

{G1(V1, E1), . . . , Gk(Vk, Ek)}6This type of connectedness is called rook contiguity because units are only considered to be adjacent if

a rook chess piece could move into that space. Rook contiguity contrasts with queen contiguity where unitsare adjacent if they share a single point, again mimicking the movement of a chess piece. Both forms ofcontiguity are constitutionally acceptable; however, we find rook contiguity to more closely follow intuitionsof how contiguity works in electoral settings.

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where each subgraph corresponds to an electoral district 1, . . . , k. Any two vertices in each

of these subgraphs must be connected by a chain of vertices and edges also contained in

the subgraph. In other words, each subgraph must be contiguous. Likewise, the weight of

each subgraph must be close to equal – each must contain an equal number of voters. The

algorithm accomplishes this task in four steps.

1. First, the algorithm coarsens the graph. It collects connected vertices into many small

sets. Every small set of vertices is a contiguous subgraph of G(V,E). Each of the re-

sulting subgraphs is generated by randomly combining adjacent vertices. If any vertex

of the subgraph shares an edge with a vertex in another subgraph, those subgraphs are

considered to be connected. After forming an initial set of connected subgraphs, the

algorithm proceeds to join connected subgraphs randomly until a number of subgraphs

smaller than the number of vertices but larger than the number of desired districts is

achieved. The result of the coarsening step is a vastly simplified graph of subgraphs.

2. The algorithm partitions the graph. In this step, the algorithm divides the set of

subgraphs that resulted from the coarsening step into district subgraphs. These district

subgraphs are contiguous, but they are not necessarily balanced in terms of weight

(population).

3. The algorithm uncoarsens the graph. The algorithm reverses the coarsening step. Each

vertex contained in a district subgraph is assigned to that district, thus the rough

division of the coarsened graph is projected into the original graph derived made up

of vertices that correspond to geographic units.

4. The algorithm refines the graph. Recall that in Step 2, the algorithm partitions the

coarse graph into the appropriate number of districts, but the populations of the result-

ing districts are not exactly equal. In the final step, the algorithm uses an algorithm

15

developed by Kernighan and Lin (1970) to switch vertices between district subgraphs

in order to bring the the weight of the district subgraphs into balance. The Kernighan

and Lin algorithm stops when it achieves the desired level of balance. For our purposes,

we consider graphs to be balanced if the maximum deviation from perfectly balanced

is 1% or two voters, whichever is greater.

The algorithm repeats these steps until it achieves the desired number of valid maps. Each

map included in our analysis contains contiguous districts that are also balanced. Observe

that balance varies from districting scenario to districting scenario. For example, in a map

that divides a population of 10,000 into ten districts, we would consider a map with a district

containing 1,009 people to be balanced. By contrast, a map that divides a population of

10,000 into 100 districts that contained a district with a population of 109 would be out of

balance. Thus, we restrict maps with more districts to maximum deviation of two voters

from ideal balance.

Magleby and Mosesson (2018) assert that their algorithm draws maps without “indication

of bias” (2018, 147); however, others have expressed skepticism that any algorithm can draw

maps in an unbiased way Altman et al. 2015; Altman and McDonald 2011; Altman, Gill

and McDonald 2004. We may set aside this debate for the purposes of our analysis because

what matters is that the method of districting remains constant as we increase the number

of districts we afford a jurisdiction. The algorithm we choose to deploy may not exhibit bias,

but even if it did, our findings will be dispositive if the amount of bias present varies with

the number of representatives elected within that geographic space.

We use the algorithm to divide the population distributions represented in Figure 1

into maps of 2, 5, 11, 25, 50, 100, 200, and 400 districts. For each of these districting

scenarios, we generate 10,000 unique maps using the Magleby and Mosesson algorithm. In

all, we analyze 80,000 unique maps that divide the hypothetical grids of voters into various

numbers of districts. Figure 1 provides a representation of the districts that the algorithm

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produces when the population is moderately sorted. Observe that the districts are relatively

compact; the shape of the districts is not bizarre. Since the population is sorted in a way

that concentrates population towards the middle of the grid, districts formed in the center

of the grid tend to be smaller than districts towards the outsides of the grid.

[Figure 2 about here.]

4 Findings

We conduct two analyses of the maps generated by the algorithm. First, we evaluate the

effect of apportionment on districts carried conditional on the underlying distribution of

voters in the population. Second, we analyze the relationship between relative vote weight

and apportionment conditional on the underlying distribution of voters. We expect both

analyses to reveal the similar patterns. Namely, as the population becomes more sorted, the

outcomes will be biased against the relatively concentrated groups. Additionally, we expect

that patterns of bias will increase and then attenuate as apportionment increases.

4.1 Districts Carried

We summarize our findings related to the number of districts carried by either party in Figure

3. We say that a group carries a district if it makes up more than half of the population

residing in that district. The top panel in Figure 3 summarizes our findings in the least

sorted version of the underlying electoral map. The bottom panel summarizes our findings

vis-à-vis the most sorted version of the underlying distribution of voters. Each iteration of

the algorithm is represented as a light blue horizontal bar. When the same outcome occurs

more than once, we overlay the appropriate number of bars. The result is that relatively

common outcomes show up as darker bars and relatively uncommon outcomes show up as

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Figure 2: Nine examples represent possible outcomes from the 80,000 districting simulations.Here we included the first iteration of the algorithm when applied to an underlying distri-bution of voters that are moderately sorted with a single cluster towards the middle of thegrid. Map A. through I. correspond to districting plans of 2, 5, 11, 25, 50, 100, 200, 400,and 1,000 districts.

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lighter bars. Since each group casts exactly half of the votes, an unbiased translation of votes

into seats would generate a set of districts divided evenly between the two groups. Thus, in

each panel we include a gray line indicating 0.50. To the extent that the distribution deviates

below (above) that line, it is biased against (in favor of) the blue party. The patterns evident

in Figure 3 are consistent with our expectations. The relatively concentrated group, the blue

party, carries fewer seats as that group becomes more concentrated. Likewise, we find that

the effect of the increase in concentration attenuates as apportionment increases.7

[Figure 3 about here.]

7We summarize the data upon which we base Figure 3 in Appendix B.

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Number of Districts

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0.25

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