JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS2020, VOL. 29, NO. 4, 715–728https://doi.org/10.1080/10618600.2020.1739532

Automated Redistricting Simulation Using Markov Chain Monte Carlo

Benjamin Fifielda , Michael Higginsb , Kosuke Imaia,c,d , and Alexander Tarre

aInstitute for Quantitative Social Science, Harvard University, Cambridge, MA; bDepartment of Statistics, Kansas State University, Manhattan, KS;cDepartment of Government, Harvard University, Cambridge, MA; dDepartment of Statistics, Harvard University, Cambridge, MA; eDepartment ofElectrical Engineering, Princeton University, Princeton, NJ

ABSTRACTLegislative redistricting is a critical element of representative democracy. A number of political scientistshave used simulation methods to sample redistricting plans under various constraints to assess their impacton partisanship and other aspects of representation. However, while many optimization algorithms havebeen proposed, surprisingly few simulation methods exist in the published scholarship. Furthermore, thestandard algorithm has no theoretical justification, scales poorly, and is unable to incorporate fundamentalconstraints required by redistricting processes in the real world. To fill this gap, we formulate redistrictingas a graph-cut problem and for the first time in the literature propose a new automated redistrictingsimulator based on Markov chain Monte Carlo. The proposed algorithm can incorporate contiguity andequal population constraints at the same time. We apply simulated and parallel tempering to improve themixing of the resulting Markov chain. Through a small-scale validation study, we show that the proposedalgorithm can approximate a target distribution more accurately than the standard algorithm. We also applythe proposed methodology to data from Pennsylvania to demonstrate the applicability of our algorithmto real-world redistricting problems. The open-source software package is available so that researchersand practitioners can implement the proposed methodology. Supplementary materials for this article areavailable online.

ARTICLE HISTORYReceived October 2018Revised December 2019

KEYWORDSGerrymandering; Graph cuts;Metropolis–Hastingsalgorithm; Paralleltempering; Simulatedtempering; Swendsen–Wangalgorithm

1. Introduction

Legislative redistricting is a critical element of representativedemocracy. Scholars have found that redistricting influencesturnout and representation (e.g., Ansolabehere, Snyder, andStewart 2000; McCarty, Poole, and Rosenthal 2009). Unfortu-nately, redistricting is also potentially subject to partisan gerry-mandering. After the controversial 2003 redistricting in Texas,for example, Republicans won 21 congressional seats in the2004 election (Democrats won 11) whereas they had only 15seats in 2002. Moreover, the United States Supreme Court hasrecently ruled in Rucho v. Common Cause that federal courtscannot address partisan gerrymandering, essentially leavingstate legislatures, state courts, and Congress to resolve the issue.Scholars have proposed numerous remedies for partisan gerry-mandering, including geographical compactness and partisansymmetry requirements (e.g., Grofman and King 2007; Fryerand Holden 2011).

The development of automated redistricting algorithms,which is the goal of this article, began in the 1960s. Vickrey(1961) argued that such an “automatic and impersonal proce-dure” can eliminate gerrymandering (p. 110). After Baker v. Carr(1962) where the Supreme Court ruled that federal courts mayreview the constitutionality of state legislative apportionment,citizens, policy makers, and scholars became interested in

CONTACT Kosuke Imai imai@harvard.edu Department of Government and Department of Statistics, Harvard University, Cambridge, MA 02138.An earlier version of this article was entitled, “A New Automated Redistricting Simulator Using Markov Chain Monte Carlo” (Fifield et al. 2014) and was presented at the2014 Annual Summer Meeting of the Society for Political Methodology (The University of Georgia).

Supplementary materials for this article are available online. Please go to www.tandfonline.com/r/JCGS.

redistricting. Weaver and Hess (1963) and Nagel (1965)were among the earliest attempts to develop automated redis-tricting algorithms. Since then, a large number of methods havebeen developed to find an optimal redistricting plan for a givenset of criteria (e.g., Bozkaya, Erkut, and Laporte 2003; Chou andLi 2006; Fryer and Holden 2011; Liu, Tam Cho, and Wang 2016;Tam Cho and Liu 2016). These optimization methods may serveas useful tools when drawing district boundaries (see Altman,MacDonald, and McDonald 2005, for an overview).

However, the main interest of scholars has been to charac-terize the population distribution of possible redistricting plansunder various criteria so that they can detect instances ofgerrymandering and better understand the causes and conse-quences of redistricting (e.g., Cirincione, Darling, and O’Rourke2000; McCarty, Poole, and Rosenthal 2009; Chen and Rod-den 2013). Because redistricting in any given state dependscritically on its geographical features including the residen-tial patterns of voters, it is essential to identify the distri-bution of possible redistricting maps under the basic condi-tions of contiguity and equal population. In a majority of theU.S. states, for example, state legislators control the redistrict-ing process and approve redistricting plans through standardstatutory means. Therefore, an important question is how toeffectively constrain these politicians through means such ascompactness requirements (e.g., Niemi et al. 1990), and prevent

© 2020 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America

716 B. FIFIELD ET AL.

the manipulation of redistricting for partisan ends. Simulationmethods allow scholars to answer these questions by approxi-mating distributions of possible electoral outcomes under vari-ous constraints.

Yet, until recently, surprisingly few simulation algorithmshave existed in the published scholarship. In fact, most of theseexisting studies use essentially the same Monte Carlo simulationalgorithm where a geographical unit is randomly selected as a“seed” for each district and then neighboring units are added tocontiguously grow this district until it reaches the prespecifiedpopulation threshold (e.g., Cirincione, Darling, and O’Rourke2000; Chen and Rodden 2013). Unfortunately, no theoreticaljustification is given for these simulation algorithms, and hencethey are unlikely to yield a representative sample of redistrictingplans for a target population. In addition, although a commonlyused algorithm of this type (Cirincione, Darling, and O’Rourke2000) is implemented by Altman and McDonald (2011) in theiropen-source software, the algorithm scales poorly.

To fulfill this methodological gap, in Section 2, we proposea new automated redistricting simulator using Markov chainMonte Carlo (MCMC). We first formulate the task of drawingdistrict boundaries as the problem of graph-cuts, that is, parti-tioning a graph into several connected subgraphs. We then adaptand modify the Swendsen–Wang algorithm to sample from arbi-trary distributions over a fixed number of contiguous districts(Swendsen and Wang 1987; Barbu and Zhu 2005). We then showhow this algorithm can be applied to sample redistricting plansthat incorporate the equal population constraint commonlyimposed by law. Finally, we apply simulated and parallel temper-ing to improve the mixing of the resulting Markov chain (Mari-nari and Parisi 1992; Geyer and Thompson 1995).1 We pro-vide an open-source R software package redist, which is freelyavailable on the Comprehensive R Archive Network (CRAN;https://CRAN.R-project.org/package=redist) so that researchersand practitioners can implement the proposed methodology(Fifield, Tarr, and Imai 2015).

Unlike the aforementioned standard simulation algorithms,the proposed algorithms are designed to yield a representativesample of redistricting plans under contiguity and equal popu-lation constraints. Since the first version of this article was madeavailable in 2014 (Fifield et al. 2014), MCMC algorithms havebecome a commonly used method for simulating redistrictingplans (e.g., Mattingly and Vaughn 2014; Chikina, Frieze, andPegden 2017; Herschlag, Ravier, and Mattingly 2017; DeFord,Duchin, and Solomon 2019). Our algorithms are similar tothe algorithm independently proposed at about the same timeby Mattingly and Vaughn (2014). While the latter algorithmsamples redistricting plans by swapping single precincts at eachiteration, our algorithms allow multiple, larger collections ofprecincts to be swapped, improving the mixing of the Markovchain. We also offer theoretical and empirical analyses of ouralgorithms in an effort to better understand their properties andperformance.

In Section 3, we conduct a small-scale validation study whereall possible redistricting plans under various constraints canbe enumerated in a reasonable amount of time. We show thatthe proposed algorithms successfully approximate the target

1The application of simulated tempering is presented in Appendix S4.

population distribution while the standard algorithm fails evenin this small-scale problem. We then apply our algorithms to alarge redistricting problem using data from Pennsylvania. Wefind that small changes to the adopted redistricting plan cannearly eliminate the partisan bias. Finally, Section 4 discussesremaining challenges and open problems for the developmentof automated redistricting simulation algorithms.

2. The Proposed Methodology

In this section, we describe the proposed methodology. Webegin by formulating redistricting as a graph-cut problem andpropose an MCMC algorithm to sample redistricting plans fromarbitrary distributions over the set of n contiguous districts.Next, we show how our algorithm can be used to sample plansthat have an equal population constraint, which is the most basicrequirement imposed in real-world redistricting problems. Wethen discuss how to apply parallel tempering to our MCMCalgorithm to improve the mixing of the Markov chain. Finally,a brief discussion comparing our algorithm with the existingredistricting algorithms is also given.

2.1. Redistricting as a Graph-Cut Problem

Consider a redistricting problem where a state consisting ofm geographical units (e.g., census blocks or voting precincts)must be divided into n contiguous districts. We formulate thisredistricting problem as that of graph-cuts, where a graph ispartitioned into a set of connected subgraphs (Altman 1997;Mehrotra, Johnson, and Nemhauser 1998). Figure 1 illustratesthe basic idea through two small examples used in our val-idation study (see Section 3.1). In each plot, a state is rep-resented by a graph where nodes are geographical units andedges between two nodes imply their contiguity. Thus, redis-tricting a state into n districts is equivalent to removingsome edges of a graph (light gray) and forming n connectedsubgraphs.

Formally, let G = (V , E) represent an undirected graphwhere V = {1, 2, . . . , m} is the set of nodes (i.e., geographicalunits of redistricting) to be partitioned, and E is the set ofundirected edges connecting neighboring nodes. This meansthat if two units i and j are contiguous, there is an edge betweentheir corresponding nodes on the graph, i ∼ j ∈ E. We partitionthe set of nodes V into n “districts” π = {V1, V2, . . . , Vn} whereeach district V� is a nonempty subset of V . Such a partition π

generates a subgraph G(π) = (V , E(π)) ⊆ G where an edgei ∼ j ∈ E(π) if and only if nodes i and j belong to the samedistrict of the partition V�, that is,

E(π) = {i ∼ j ∈ E : ∃ V� ∈ π s.t. i, j ∈ V�}. (1)

Because districts are formed by removing edges from E (i.e.,“cutting” them) to obtain E(π), redistricting can be thoughtof as a graph cut problem. Finally, since each resulting districtmust be contiguous, we require each district of the partition tobe connected. In this article, we call a partition comprised of nconnected districts valid and denote the set of all valid partitionsby �(G, n).

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25 Precincts, Two Districts

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50 Precincts, Three Districts

Figure 1. Redistricting as a graph-cut problem. A state is represented by a graph where nodes are geographical units and edges between two nodes imply their contiguity.Under this setting, redistricting is equivalent to removing or cutting some edges (light gray) to form connected subgraphs, which correspond to districts. Different districtsare represented by different colors. Two illustrative examples, used in our validation study in Section 3.1, are given here.

2.2. The Basic Algorithm for Sampling ContiguousDistricts Without Constraints

The goal of this article is to develop an algorithm that uni-formly samples valid redistricting plans with the equal pop-ulation constraint. We achieve this in several steps. We firstdescribe the basic algorithm that samples from an arbitrarydistribution f over the set of valid partitions, �(G, n) (Algo-rithm 1 in Sections 2.2 and 2.3). We then show how to mod-ify this basic algorithm to incorporate the equal populationconstraint, which is the most basic requirement imposed inreal-world redistricting problems (Algorithms 1.1 and 1.2 inSection 2.4). Finally, we apply parallel tempering to improvethe mixing of the proposed MCMC algorithm (Algorithm 2 inSection 2.5).

Our basic MCMC algorithm is designed to obtain adependent but representative sample from any distributionover valid redistricting plans. In particular, we modify theSWC-1 algorithm of Barbu and Zhu (2005), which uses aMetropolis–Hastings step (Metropolis et al. 1953; Hastings1970) to extend the Swendsen–Wang algorithm (Swendsenand Wang 1987) to arbitrary distributions over graph parti-tions. In contrast to our method, graph partitions sampledfrom the original SWC-1 algorithm are not restricted to becontiguous. Our basic algorithm begins with a valid par-tition π0 (e.g., an actual redistricting plan adopted by thestate) and transitions from a valid partition πt to anothervalid partition πt+1 at each iteration t + 1. We beginby describing the basic algorithm for sampling contiguousdistricts.

Figure 2 illustrates one iteration of Algorithm 1 using the50 precinct example with three districts given in the rightpanel of Figure 1. Our algorithm begins by randomly “turn-ing on” edges in E(πt) where each edge is turned on withprobability q. In the left upper plot of Figure 2, the edges thatare turned on are indicated with darker gray. Next, we iden-tify components that are connected through these “turned-on”edges and are on the boundaries of districts in πt . Each suchconnected component is indicated by a dotted polygon in theright upper plot. Third, among these, a subset of nonadjacent

connected components are randomly selected as shown in theleft lower plot (two in this case). These connected componentsare reassigned to adjacent districts to create a candidate par-tition after verifying that this reassignment does not shatterany district so that the number of districts remain unchanged.Finally, we accept or reject the candidate partition accordingto a probability that depends on, among other things, “turned-on” edges, and “turned-off ” edges from each of the selectedconnected components that are connected to adjacent districtsand are highlighted in the left lower plot. Section 2.3 providesa detailed explanation about how this acceptance probability iscomputed.

We now formally describe the basic algorithm.

Algorithm 1 (Sampling contiguous redistricting plans). Weinitialize the algorithm by obtaining a valid partition π0 ={V(0)

1 , V(0)2 , . . . , V(0)

n }, where each district V(0)� is contiguous,

that is, connected in the graph, and then repeat the followingsteps at each iteration t + 1,

Step 1 (“Turn on” edges): Starting from πt ={V(t)

1 , V(t)2 , . . . , V(t)

n } with graph G(πt) = (V , E(πt)), formthe edge set Eon(πt) ⊆ E(πt), where each edge e ∈ E(πt)is independently added to Eon(πt) with probability q. LetCP denote the set of connected components formed byEon(πt).

Step 2 (Gather connected components along boundaries):

(a) Identify all nodes which lie along a district boundary bycomparing E(πt) with E.

(b) Gather boundary connected components by perform-ing a breadth-first search over all boundary nodes. LetB(CP, πt) denote this set.

Step 3 (Select nonadjacent connected components alongboundaries):

(a) Generate R from a distribution with integer supportover the closed interval [1, |B(CP, πt)|].

(b) Initialize VCP = ∅. Repeat the following procedureuntil |VCP| = R:

718 B. FIFIELD ET AL.

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‘‘Turn on'' edges

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Gather connected components on boundaries

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Select components and propose swaps

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Accept or reject the proposal

Figure 2. The basic algorithm for sampling contiguous districts without constraints. The plots illustrate the proposed algorithm (Algorithm 1) using the 50 precinct datagiven in the right panel of Figure 1. First, in the left upper plot, each edge other than those which are cut in Figure 1 is “turned on” (dark gray) independently with certainprobability. Second, in the right upper plot, connected components on the boundaries are identified (dashed polygons). Third, in the left lower plot, a certain number ofnonadjacent connected components on boundaries are randomly selected (dashed polygons) and the acceptance ratio is calculated by counting certain edges (colorededges). Finally, in the right lower plot, the proposed swap is accepted according to the Metropolis–Hastings ratio.

(i) Randomly sample C from B(CP, πt) withoutreplacement, and add it to VCP.

(ii) If C is adjacent to another element of VCP, or if theremoval of VCP results in a noncontiguous district,remove C from VCP and B(CP, πt), and return toStep (i).

Step 4 (Propose swaps): Initialize π ′ = πt . For each C ∈VCP, perform the following procedure:

(a) Suppose C is currently assigned to district V(t)� . Propose

to assign C to a different, randomly selected neighbor-ing district in π ′.

(b) Suppose C is reassigned to district V(t)�′ . Update π ′ by

setting V(t)� = V(t)

� \ C, V(t)�′ = V(t)

�′ ∪ C, and π ′ ={V(t)

1 , V(t)2 , . . . , V(t)

� , . . . , V(t)�′ , . . . , V(t)

n}

.

Step 5 (Accept or reject the proposal): Generate u ∼ U(0, 1).Set

πt+1 ={

π ′ if u ≤ α(π ′, CP | πt),πt otherwise,

(2)

where the acceptance probability α(π ′, CP | πt) is definedin Theorem 1.

JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 719

Several remarks are in order. First, in Step 1, we choose q tobe relatively small such that it typically yields a large number ofsmall connected components on boundaries. In the validationstudies, we set q to 0.05, and for the Pennsylvania study, we setq to 0.04.

Second, in Step 3(a), we choose a distribution such that R |B(CP, π)| holds for most cases and the acceptance probability isin an ideal range of 20%–40%. In the applications of this article,for example, we use the zero-truncated Poisson distribution,rejecting and resampling values of R which fall outside theinterval [1, |B(CP, π)|]. We choose λ such that the chance ofrejection is small, with λ = 2 in our validation studies inSection 3.1 and λ = 10 in the Pennsylvania study presented inSection 3.2.

Alternatively, as in Barbu and Zhu (2005), we could fix Rto 1 and use a large value of q. Unfortunately, this strategy,though sensible in other settings, is not effective in redistrictingwith contiguous districts. The reason is that larger connectedcomponents typically include more units from the interior ofeach district. This in turn makes it more likely to shatter districtsand often dramatically lowers the acceptance probability, whichleads to slower convergence. Instead, our strategy is to improvethe mixing of the Markov chain through simultaneous swaps ofsmall connected components.

Third, Step 3(b) checks whether or not swapping connectedcomponents VCP leads to an invalid redistricting plan, that is, adistrict is no longer contiguous. This requirement does not existin the original algorithm of Barbu and Zhu (2005). Such “shat-tering” could occur, for example, if a district contains a narrowsliver that has width of a single precinct, and one of the precinctsin the middle of the sliver is selected to be swapped. Thischeck also enforces that VCP consists of nonadjacent connectedcomponents, and hence VCP is the unique set of components tobe reassigned in the move πt → π ′.

Finally, the acceptance probability in Step 5 is based on thefollowing Metropolis criterion,

α(π ′, CP | π) = min(

1,q(π , CP | π ′)q(π ′, CP | π)

· g(π ′)g(π)

), (3)

where q(π ′, CP | π) = q(π ′ | CP, π)q(CP | π) denotes theprobability that, starting from partition π , connected compo-nent set CP is formed in Step 1, and π ′ is formed through reas-signment of elements in CP in Steps 2–4. In addition, g(π) ∝f (π) is the unnormalized target distribution for sampling. Barbuand Zhu (2005) computed the acceptance ratio by marginalizingout CP in the proposal distributions, eliminating dependence onCP. However as we discuss in Section 2.3, the marginalizationof the proposal distributions is intractable in most redistrictingproblems. Thus, we cannot eliminate this dependence on CP,leading to slower convergence of the Markov chain.

Next, we present the acceptance probability used in Step 5 ofAlgorithm 1.

Theorem 1 (The acceptance probability of Algorithm 1). Letπ and π ′ be a pair of partitions which differ in the districtassignment of VCP, with |VCP| = R and R ≤ |B(CP, π ′)|, andlet g(π) ∝ f (π) be the unnormalized target distribution over�(G, n). Then, the acceptance probability for the move π → π ′

through one iteration of Algorithm 1 is

α(π ′, CP | π)

= min

⎛⎝1,

P(VCP | π ′, CP, R

)F(|B(CP, π)|) (

1 − q)∣∣C(π ′ ,VCP)

∣∣P (VCP | π , CP, R) F(|B(CP, π ′)|) (

1 − q)|C(π ,VCP)| · g(π ′)

g(π)

⎞⎠,

(4)

where

C(π , VCP) = {i ∼ j ∈ E(π) : ∃ C ∈ VCP s.t. i ∈ C, j /∈ C

}(5)

is a Swendsen–Wang cut, denoting the set of edges in E(π) thatneed to be cut to form VCP, F(·) is the cumulative distributionfunction for R whose support is a subset of the set of positiveintegers, and P(VCP | π , CP, R) is the probability of samplingVCP in Step 3(b) of Algorithm 1.

The proof is given in Appendix S1. Note that in the case ofsampling contiguous redistricting plans without constraints, thetarget distribution is uniform over all contiguous n-partitionsof the graph, �(G, n), and g(π) = g(π ′) cancels in the ratio ofEquation (4).

Finally, the following theorem shows that under a set ofassumptions, the unique stationary distribution of Algorithm 1is g(π), whose support is restricted to �(G, n).

Theorem 2 (Unique stationary distribution of Algorithm 1).Suppose that the following two conditions hold:

1. (Irreducibility) For all pairs π , π ′ ∈ �(G, n), partition π cantransition to partition π ′ through a finite number of iterationsof Algorithm 1.

2. (Aperiodicity) There exists a pair of partitions π , π ′ and set ofconnected components CP such that 0 < α(π ′, CP | π) < 1.

Then, the unique stationary distribution of the Markov chaingiven by Algorithm 1 is f (π).

The proof is given in Appendix S2. We remark that althoughit is difficult to provide a formal argument, our experiencesuggests that Conditions 1 and 2 may hold for real-world redis-tricting data unless additional constraints are imposed (seeSection 2.4 for a discussion of how we modify the algorithmto incorporate the equal population constraint). For the small-scale validation example we have examined, in which the com-plete enumeration of valid partitions is possible, we were ableto verify that the two conditions are satisfied. In addition, forall the empirical examples we have analyzed including the NewHampshire and Pennsylvania data, we have found at least oneset of initial and candidate partitions such that 0 < α(π ′, CP |CP) < 1.

2.3. Approximating the Acceptance Probability

The acceptance probability given in Equation (4) of Theorem 1differs from the ratio derived in Barbu and Zhu (2005) in the twoterms, F(|B(CP, π)|) and P(VCP, | π , CP, R), which correspondto the probability of sampling R and VCP in Step 3, respectively.Note that the computation of F(|B(CP, π)|) is straightforward.

720 B. FIFIELD ET AL.

As mentioned earlier, in our applications, we sample R from thezero-truncated Poisson distribution truncated from above by|B(CP, π)|. Hence, the required cumulative distribution func-tion can be easily computed. The key question is then how tocompute P(VCP, | π , CP, R).

In the original algorithm of Barbu and Zhu (2005), R isfixed to 1, and VCP is selected uniformly at random from CP,so P(VCP, | π ′, CP, R)/P(VCP, | π , CP, R) = 1. For redistrictingproblems, however, this cancellation does not occur, even whenR = 1, due to the possibility of shattering and the requirementthat components VCP lie along district boundaries. In addition,as mentioned earlier, for our applications, setting R = 1 andusing a large value of q lead to a high rate of rejection andpoor mixing. Thus, we let R vary, which further complicatescomputation of P(VCP, | π , CP, R) due to the nonadjacencyrestriction of VCP. Below, we propose an approximation ofthe acceptance probability when the exact computation of thisprobability is computationally infeasible.

Recall that in Step 3(b), we iteratively sample without replace-ment R connected components from the sampling set B(CP, π)

to form the set VCP. During this process, a sampled connectedcomponent whose addition to VCP would lead to either com-ponent adjacency or the creation of disconnected districts isrejected and removed from B(CP, π). Let B†

R(CP, π) representthe collection of nonadjacent connected component sets VCP ⊂B(CP, π), with |VCP| = R, whose removal from the graphpartition π does not result in disconnected districts. Ideally, therejection sampling strategy described above would correspondto uniform sampling from B†

R(CP, π), in which case,

P(VCP | π , CP, R) = 1|B†

R(CP, π)| . (6)

However, since components in B(CP, π) have a differing num-ber of neighbors, some elements of B†

R(CP, π) are more likely tobe sampled than others. Consequently, P(VCP | π , CP, R) has amore complicated form that depends on B†

R(CP, π).Additionally, identification of B†

R(CP, π) is computation-ally intensive even for the case R = 1 because it requireschecking whether each connected component C ∈ B(CP, π)

can shatter its district V�; this has worst-case complexityO (|V�| + |E(V�)|). The problem quickly becomes intractablewhen R > 1, even for small graphs, since we have to check bothconnectedness and nonadjacency for a combinatorial number ofsubsets of B(CP, π). We therefore approximate this probability.

Fortunately, in large-scale problems, so long as q and Rare small, the number of connected component sets VCP thatcan shatter a district is likely to be small relative to the totalnumber of boundary connected component sets. When this isthe case, at most VCP ∈ B†

R(CP, π) can be formed by samplingR nonadjacent components from B(CP, π). Additionally, smallq and R tend to lead to scenarios where R |B(CP, π)|. Inthis case, without-replacement sampling can be approximatedby with-replacement sampling since the probability of samplingthe same or a neighboring component in R draws is small.Therefore, we propose the following approximation,

P(VCP | π , CP, R) ≈ R!(

1|B(CP, π)|

)R. (7)

Substituting the approximation into the acceptance probabilitydefined in Equation (4), we obtain

α(π ′, CP | π)

≈ min

⎛⎝1,

( |B(CP, π)||B(CP, π ′)|

)R F(|B(CP, π)|) (1 − q

)|C(π ′ ,VCP)|F(|B(CP, π ′)|) (

1 − q)|C(π ,VCP)| · g(π ′)

g(π)

⎞⎠ .

(8)

This approximation is valid under the assumption that werarely reject samples drawn in Step 3(b) for adjacency or shat-tering issues. Indeed, in the Pennsylvania study, we find that only190 out of 30,000 iterations of the algorithm (or <0.1%) rejecteda proposed swap due to it either shattering an existing district orbeing adjacent to a swap proposed in the same iteration. Evenfor the small-scale validation study, at most 545 out of 10,000iterations of the algorithm rejected a proposed swap. Finally,we only need the approximation to work in the ratio P(VCP, |π , CP, R)/P(VCP, | π ′, CP, R) rather than the numerator anddenominator separately.

Nevertheless, we acknowledge that it is difficult to developa rigorous theoretical justification for the proposed approxi-mation. Therefore, in Appendix S8, swe conduct small-scalevalidation studies, in which a stronger approximation basedon B†

R(CP, π) can be computed, to examine both the accu-racy of the proposed approximation and the effects of ignoringadjacency when computing the selection probability P(VCP |CP, π , R). We find that the stronger approximation nearly per-fectly samples from the target population (see the first row ofFigure S7), suggesting the effects of component adjacency have anegligible impact on the ratio of selection probabilities. We alsofind that even in small maps, where our weaker approximation isexpected to perform poorly, the proposed approximation is rea-sonably accurate (see the second row of Figure S7) and becomesmore accurate as the size of redistricting problem increases (seeFigures S7–S9).

2.4. Incorporating the Equal Population Constraint

The basic algorithm we have developed so far samples from arbi-trary distributions over the set of redistricting maps with n con-tiguous districts. We can incorporate the constraints imposedin typical redistricting processes through our choice of thetarget unnormalized distribution g. Such constraints includeequal population, geographical compactness, and preservationof communities of interest. Here, we focus on the equal popula-tion constraint, which is the most basic factor to be consideredfor redistricting. While the proposed framework can accom-modate other constraints as shown in Section 3.2, we leave thechallenge of simultaneously incorporating multiple constraintsto future research.

Let pi denote the population size of node i where the popu-lation parity for the state is given by p̄ = ∑m

i=1 pi/n. Then, theequal population constraint is defined as,

max1≤�≤n

∣∣∣∣∑

i∈V�pi

p̄− 1

∣∣∣∣ ≤ δ, (9)

where δ determines the maximal deviation from the populationparity. For example, δ = 0.03 implies that the population of alldistricts must be within 3% of the population parity.

JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 721

How can we uniformly sample redistricting plans that meetthis population constraint using Algorithm 1? The major chal-lenge we must deal with is that our algorithm generates samplesfrom the set of contiguous partitions, �(G, n), while we wouldlike to generate samples only from the much smaller set of plansmeeting the constraint. One possible strategy for dealing withthis issue, which is often used in the literature, is to sample uni-formly from �(G, n) and discard any candidate partition thatviolates Equation (9). In Algorithm 1, for example, after Step 4,one could check whether the candidate partition π ′ satisfiesthe constraint, and if not, go back to Step 3. This procedure issummarized in the following algorithm, which we henceforthrefer to in the text as the basic algorithm with hard constraint.

Algorithm 1.1 (Sampling contiguous redistricting plans withhard constraint). We initialize the algorithm by obtaining avalid partition π0 = {V(0)

1 , V(0)2 , . . . , V(0)

n }, where each districtV(0)

� is contiguous, that is, connected in the graph, and thenrepeat the following steps at each iteration t + 1,

Step 1 (Run the basic algorithm): Starting from πt = {V(t)1 ,

V(t)2 , . . . , V(t)

n } with graph G(πt) = (V , E(πt)), run the firstfour steps of Algorithm 1 to obtain the proposed sample π ′.

Step 2 (Reject samples failing to meet constraint): For a spec-ified population constraint δ, reject π ′ and return to Step 3 ofAlgorithm 1 if

max1≤�≤n

∣∣∣∣∣∑

i∈V(t)�

pi

p̄− 1

∣∣∣∣∣ > δ,

otherwise continue to Step 3 of Algorithm 1.1.

Step 3 (Accept or reject the proposal): Generate u ∼ U(0, 1).Set

πt+1 ={

π ′ if u ≤ α(π ′, CP | πt),πt otherwise,

(10)

where the approximated acceptance probability α(π ′, CP |πt) is given by

α(π ′, CP | πt)

≈ min

⎛⎝1,

( |B(CP, πt)||B(CP, π ′)|

)R F(|B(CP, πt)|)(1 − q

)|C(π ′ ,VCP)|F(|B(CP, π ′)|) (

1 − q)|C(πt ,VCP)|

⎞⎠ .

There are several potential problems with this approach.First, the additional rejection sampling done in Step 2 furthercomplicates exact calculation of the acceptance probability. Forsimplicity, we use the same approximation given in Section 2.3,but this approximation may be poor especially for small valuesof the constraint δ. Second, there may be a vast majority of plansproposed through Steps 1–4 do not meet the equal populationconstraint. It may become impossible for the Markov chain toreach from one valid redistricting plan to another, which wouldviolate the irreducibility assumption in Theorem 2.

Alternatively, researchers could run Algorithm 1 to generatesamples from the uniform distribution and then simply discardany invalid sampled plans post hoc. Unfortunately, this naivestrategy, which overcomes the theoretical issues of the previous

algorithm, will also discard many sampled plans and hencegenerate vanishingly few valid plans in a reasonable amount oftime.

To overcome the issues with the preceding methods, we mod-ify Algorithm 1 by first choosing the stationary distribution ofour Markov chain such that redistricting plans likely to meet theequal population constraint are oversampled. The idea behindthis strategy is to use those invalid partitions that are likely tobe proposed by Algorithm 1 as a way for the Markov chain totransition between valid partitions. We use the following Gibbsdistribution as our target distribution,

fβ(π) = 1z(β)

exp

⎛⎝−β

∑V�∈π

∣∣∣∣∣∣∑i∈V�

pip̄

− 1

∣∣∣∣∣∣⎞⎠ , (11)

where β ≥ 0 is called the inverse temperature and z(β) is thenormalizing constant. Thus, fβ(π) has peaks around plans thatmeet the equal population constraint.

To sample from this distribution, we modify the accep-tance probability in Equation (8) by replacing g(π ′)/g(π) withgβ(π ′)/gβ(π), where gβ is the unnormalized form of the Gibbsdistribution, that is, gβ(π) ∝ fβ(π). Fortunately, once thisratio is computed for the initial state, we can compute the ratiofor subsequent states by only computing district populationchanges due to reassignment of each C ∈ VCP. Thus, the ratiocomputation has O(|VCP|) complexity, which has a negligibleimpact on the speed of the algorithm.

As the final step, we discard invalid sampled plans that donot satisfy Equation (9) and reweight the remaining sampledplans by 1/gβ(π) to approximate the uniform sampling from thepopulation of valid redistricting plans meeting the constraint.If we resample the sampled plans with replacement using thisimportance weight, then the procedure is equivalent to thesampling/importance resampling (SIR) algorithm (Rubin 1987).The steps described above are summarized in the followingalgorithm, which we refer to as the basic algorithm with softconstraint.

Algorithm 1.2 (Sampling contiguous redistricting plans with softconstraint). We initialize the algorithm by obtaining a validpartition π0 = {V(0)

1 , V(0)2 , . . . , V(0)

n }, where each district V(0)�

is contiguous, that is, connected in the graph, and then performthe following steps,

Step 1 (Run the basic algorithm with Gibbs distribution):Generate M samples using Algorithm 1 with edge cut prob-ability q, Gibbs parameter β , and approximated acceptanceprobability

α(π ′, CP | π)

≈ min

⎛⎝1,

( |B(CP, π)||B(CP, π ′)|

)R F(|B(CP, π)|) (1 − q

)∣∣C(π ′ ,VCP)∣∣

F(|B(CP, π ′)|) (1 − q

)|C(π ,VCP)| · gβ(π ′)gβ(π)

⎞⎠.

Step 2 (Reject samples failing to meet constraint): For a spec-ified population constraint δ, identify all samples generatedthrough Step 1 such that

max1≤�≤n

∣∣∣∣∑

i∈V�pi

p̄− 1

∣∣∣∣ > δ,

722 B. FIFIELD ET AL.

and discard them.

Step 3 (Resample using SIR): Draw S samples with replace-ment from the set resulting from Step 2 using samplingweights 1/gβ(π).

2.5. Improving the Convergence With Tempering

The fundamental challenge of incorporating the equal pop-ulation constraint through Algorithm 1.2 is that it becomesexceedingly difficult for the Markov chain to efficiently traversefrom one valid partition to another. On one hand, if the inversetemperature parameter β of fβ is set too high, the Markov chainmay never visit certain parts of the target distribution. On theother hand, if the β is set too low, many of the sampled plans willbe invalid. We consider simulated (Marinari and Parisi 1992)and parallel tempering (Geyer 1991) with Algorithm 1.2 to aidin efficient exploration of the sample space without compro-mising the desired stationary distribution. In the applications ofthis article, we opted to use parallel tempering due to its ease ofimplementation as well as its robustness to parameter selection.We discuss the details of parallel tempering below and referthe reader to Appendix S4 for more information on simulatedtempering (Algorithm S1).

In parallel tempering, r copies of an MCMC algorithm arerun at r different temperatures,2 and after a fixed number ofiterations we exchange the corresponding temperatures betweentwo randomly selected adjacent chains using the Metropoliscriterion. The idea is that chains at higher temperatures betterexplore the different parts of the distribution since the lowprobability states separating one set of valid partitions fromanother set of valid partitions now have a higher probability. Theswapping mechanism allows the chains at lower temperatures tojump between different parts of the target distribution more eas-ily without changing the stationary distribution of the Markovchain.

The nature of parallel tempering suggests that it should beimplemented in a parallel architecture to minimize computationtime. Altekar et al. (2004) described such an implementationusing parallel computing and MPI, which we use as the basis forimplementing our algorithm for uniformly sampling redistrict-ing plans meeting a specified population constraint. The algo-rithm, which we henceforth refer to as the parallel temperingalgorithm with soft constraint, is given below.

Algorithm 2 (Sampling contiguous plans with parallel tem-pering and soft constraint). Given r initial valid partitionsπ

(0)0 , π(1)

0 , . . . , π(r−1)0 , a sequence of r increasing inverse tem-

peratures β(0) > β(1) > · · · > β(r−1) = 0 with β(0) thetarget inverse temperature for inference, a swapping interval T,and a population constraint δ, the parallel tempering with softconstraint algorithm performs the following steps,

Step 1 (Generate samples): Generate MT samples by repeat-ing the following steps M times

2The temperature referenced here is different from the temperature in theexpression for fβ . For the sake of simplicity, we absorb the distributionparameter β into the sequence of tempering parameters β(0) , . . . , β(r−1) .

(a) (Run the basic algorithm with Gibbs distribution): Foreach chain i ∈ {0, 1, . . . , r − 1}, using the current parti-tion π

(i)t and the corresponding inverse temperature β(i),

obtain a valid partition π(i)t+T by running T iterations of

Algorithm 1 with the following approximate acceptanceprobability,

α(π ′, CP | π(i)t ) ≈ min

⎛⎝1,

(|B(CP, π(i)

t )||B(CP, π ′)|

)R

F(|B(CP, π(i)t )|) (

1 − q)|C(π ′,VCP)|

F(|B(CP, π ′)|) (1 − q

)∣∣∣C(π(i)t ,VCP)

∣∣∣ · gβ(j) (π ′))

gβ(j)(π(i)t )

⎞⎠ .

(12)

This step is executed concurrently for each chain.(b) (Propose a temperature exchange between two chains):

Randomly select two adjacent chains j and k andexchange information about the temperatures β(j), β(k)

and the unnormalized likelihoods of the current parti-tions gβ(j)

(π

(j)t+T

), gβ(k)

(π

(k)t+T

)using MPI

(c) (Accept or reject the temperature exchange): Exchangetemperatures (i.e., β(j) � β(k)) with probabilityγ

(β(j) � β(k)) where

γ(β(j) � β(k)

)=min

⎛⎝1,

gβ(j)

(π

(k)t+T

)gβ(k)

(π

(j)t+T

)gβ(j)

(π

(j)t+T

)gβ(k)

(π

(k)t+T

)⎞⎠.

(13)All previously generated samples are assumed to havebeen generated at the current temperature of the chain

Step 2 (Reject samples failing to meet constraint): Takingthe samples generated under temperature β(0), identify allsamples such that

max1≤�≤n

∣∣∣∣∑

i∈V�pi

p̄− 1

∣∣∣∣ > δ,

and discard them.

Step 3 (Resample using SIR): Draw S samples with replace-ment from the set resulting from Step 2 using samplingweights 1/gβ(0) (π).

The mixing performance of Algorithm 2 may be affectedby the choice of the temperature sequence β(i), the swappinginterval T, and the number of chains r. While much work hasbeen done on the choice of temperatures (e.g., Earl and Deem2005), no sequence has been shown to be best. However, ingeneral, sequences with geometric spacing have been shownto produce reasonable results (see, e.g., Kone and Kofke 2005;Atchadé, Roberts, and Rosenthal 2011). For this reason, we usethe sequence β(i) = (

β(0))1− i

r−1 , i ∈ {0, 1, . . . , r − 1} for ourimplementation, which we find works well in our applications.The swapping interval should be set high enough to allow forchains to adequately explore the local state space around themodes of the target distribution, but not so high that theseregions are oversampled. We use a value of T = 100 for ouralgorithm, which we found to work well in our applications. For

JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 723

the number of chains used, if r is too small, then swaps betweenchains are unlikely and mixing is still poor. If r is too large, toomany swaps are required before cooler chains have access tostates generated from the hot chains.

2.6. Comparison With Existing Algorithms

A number of researchers use Monte Carlo simulation algorithmsto sample possible redistricting plans under various criteria todetect instances of gerrymandering and understand the causesand consequences of redistricting (e.g., Engstrom and Wild-gen 1977; O’Loughlin 1982; Cirincione, Darling, and O’Rourke2000; McCarty, Poole, and Rosenthal 2009; Chen and Rod-den 2013). Most of these studies use a similar Monte Carlosimulation algorithm where a geographical unit is randomlyselected as a “seed” for each district and then neighboring unitsare added to contiguously grow this district until it reachesthe prespecified population threshold. A representative of suchalgorithms, proposed by Cirincione, Darling, and O’Rourke(2000) and implemented by Altman and McDonald (2011) intheir open-source BARD package, is given here.

Algorithm 3 (The standard redistricting simulator (Cirincione,Darling, and O’Rourke 2000)). For each district, we repeat thefollowing steps.

Step 1: From the set of unassigned units, randomly select theseed unit of the district.

Step 2: Identify all unassigned units adjacent to the district.

Step 3: Randomly select one of the adjacent units and add itto the district.

Step 4: Repeat Steps 2 and 3 until the district reaches thepredetermined population threshold.

Additional criteria can be incorporated by modifying Step 3to select certain units. For example, to improve the compactnessof the resulting districts, one may choose an adjacent unassignedunit that falls entirely within the minimum bounding rectangleof the emerging district. Alternatively, an adjacent unassignedunit that is the closest to emerging district can be selected (Chenand Rodden 2013).

Nevertheless, the major problem of these simulation algo-rithms is their adhoc nature. For example, as the documenta-tion of BARD package warns, the creation of earlier districtsmay make it impossible to yield contiguous districts. Thesealgorithms also rely on rejection sampling to incorporate con-straints, which is an inefficient strategy since many of the sam-ples created are discarded. More importantly, the algorithmscome with no theoretical result and are not even designed touniformly sample redistricting plans. In contrast, the proposedalgorithms described in Sections 2.2–2.5 are built upon the well-known theories and strategies about the MCMC methods. Fur-thermore, although our algorithm also uses rejection samplingto incorporate constraints, the rejected samples are still usedto assist in finding valid partitions, which is a more efficientstrategy. The disadvantage of our algorithms, however, is thatthey yield a dependent sample and hence their performancewill hinge upon the degree of mixing. Thus, we now turn to

the assessment of the empirical performance of the proposedalgorithms.

3. Empirical Studies

In this section, we assess the performance of the proposedMCMC algorithms (Algorithms 1, 1.1, 1.2, and 2) in two ways.First, we conduct a small-scale validation study where, due tothe size of the tested maps, all valid redistricting plans can beenumerated. We show that the proposed MCMC algorithmscan approximate the target distribution more accurately thanthe standard algorithm (Algorithm 3). The proposed basic algo-rithms (Algorithms 1.1 and 1.2) also scales much better thanthe standard algorithm. Second, we apply the proposed paralleltempering algorithm (Algorithm 2) to actual redistricting datafrom Pennsylvania. We demonstrate that small changes to theactual redistricting map can nearly eliminate partisan bias.

To conduct these analyses, we utilize precinct-level shapefiles and electoral returns data from the Harvard Election DataArchive to determine precinct adjacency and voting behavior.We supplement this data with basic demographic informationfrom the U.S. Census Bureau P.L. 94-171 summary files, whichare disseminated to the 50 states to obtain population parity indecennial redistricting.

3.1. A Small-Scale Validation Study

We conduct a validation study based on the 25 precinct set,which is shown as a graph in Figure 1 (see Fifield, Imai, et al.2019, for a more comprehensive validation study). We beginby considering the problem of partitioning this graph intothree contiguous districts. Even for this small-scale study, theenumeration of all valid partitions is a nontrivial problem (seeAppendix S3, which describes our enumeration algorithm). Ofthe roughly 325/6 ≈ 1.41 × 1011 possible partitions, only117,688 have three contiguous districts, and 3617 have districtpopulations within 20% of parity.

The results of the proposed algorithms are based on a singlechain of 10,000 draws while those of the standard algorithmare based on the same number of independent draws. For allsimulations, we set q to 0.05 and λ = 2, where λ is the meanof the truncated Poisson distribution. To implement paralleltempering (Algorithm 2), we specify a sequence of temperatures{β(i)}r−1

i=0 . For the population deviation of 20%, we chose a targettemperature of β(r−1) = 5.4, and for the population deviationof 10%, we chose a target temperature of β(r−1) = 9. In bothcases, we use β(0) = 0 and set r = 10.

For both analyses, we conducted an initial grid search overpossible values of β(r−1) and kept track of the acceptance prob-ability of the chain and the number of samples that were drawnwithin the target level of population parity. We selected a targettemperature such that a sufficient number of plans meet thepopulation constraint, and that the acceptance probability isin the recommended 20%–40% range (Geyer and Thompson1995). As described in Sections 2.4 and 2.5, we use a subset ofdraws taken under the target temperature, discard those that falloutside the population target, and then resample the remainingdraws using the importance weights 1/gβ(i) (π).

724 B. FIFIELD ET AL.

0.00 0.05 0.10 0.15 0.20 0.25 0.30

010

2030

4050

Republican Dissimilarity Index

Den

sity

No Equal Population Constraint

True DistributionAlgorithm 1Algorithm 3

0.00 0.05 0.10 0.15 0.20 0.25 0.30

010

2030

4050

Republican Dissimilarity Index

Den

sity

20% Equal Population Constraint

True DistributionAlgorithm 3

Algorithm 1.1Algorithm 1.2Algorithm 2

Our Algorithms

0.00 0.05 0.10 0.15 0.20 0.25 0.30

010

2030

4050

Republican Dissimilarity Index

Den

sity

10% Equal Population Constraint

True DistributionAlgorithm 3

Algorithm 1.1Algorithm 1.2Algorithm 2

Our Algorithms

Figure 3. A small-scale validation study with three districts using the Republican Dissimilarity Index. The underlying data are the 25 precinct set shown in the left plot ofFigure 1. We find that the proposed algorithm with a hard constraint (Algorithms 1 and 1.1; solid black lines), which discards invalid plans, approximates the true targetdistribution (gray histograms) well when no (left column) or a moderate (middle column) equal population constraint is imposed. However, the algorithm exhibits poorperformance when a stricter equal population constraint (right column) is imposed. The proposed algorithm with a soft constraint (Algorithm 1.2; purple dot-dashedline), which is based on Gibbs distribution, or parallel tempering (Algorithm 2; blue dot-dashed line) perform well even in this case. In contrast, the standard algorithm(Algorithm 3; red dashed lines) as implemented in the BARD package fails to approximate the target distribution in all cases and the performance deteriorates as theconstraint becomes stricter.

We investigate the degree to which each algorithm canapproximate the target distribution using the Republican dis-similarity index, which is a common measure of residentialsegregation,

D = 12

n∑�=1

t�T

· |p� − P|P(1 − P)

, (14)

where � indexes districts in a state, t� is the population of district�, p� is the share of district � that identifies as Republican, T is thetotal population in the state, and P is the share of Republicansacross all districts. The dissimilarity index represents the pro-portion of the minority group that would have to be moved tobe randomly distributed across districts within a state (Masseyand Denton 1988). We have examined other statistics and theresults are, as expected, similar to those presented here.

Figure 3 presents the results. The left column shows thatwhen the equal population constraint is not imposed, the pro-posed basic algorithm (Algorithm 1; black solid lines) approxi-mates the target distribution well while the standard algorithm(Algorithm 3; red dashed lines), as implemented in the BARDpackage (Altman and McDonald 2011), fails even in this easiestcase. In the middle and right columns, we impose the equalpopulation constraint where only up to 20% and 10% deviationfrom population parity is allowed, respectively. Not surprisingly,the standard algorithm (Algorithm 3; red dashed lines) againcompletely fails. The proposed algorithm with a hard con-straint (Algorithm 1.1; black solid line), which simply discardsinvalid plans as discussed in Section 2.4, also has difficulty inapproximating the target distribution. However, the proposedalgorithm with a soft constraint (Algorithm 1.2; purple dot-dashed line), which uses the Gibbs distribution of Equation (11)

as discussed in Section 2.4, performs well in both cases. Theproposed algorithm with parallel tempering (Algorithm 2; bluedot-dash lines) also performs reasonably well even when a strictequal population constraint is imposed.

3.2. Pennsylvania: Local Simulations

3.2.1. The SetupNext, we analyze the 2008 election data from Pennsylvania toexamine how small changes to the actual congressional map(Figure 4) would affect partisan bias. Pennsylvania representsa challenging application because its 2008 plan had a total of19 congressional districts and 9256 precincts. Our experiencesuggests that the proposed algorithm cannot reliably approx-imate the uniform distribution of all valid redistricting plansunder a strong equal population constraint for such a large state.Thus, we conduct “local simulations,” in which the goal is toobtain a representative sample of valid redistricting plans withina prespecified degree of deviation from the 2008 redistrict-ing plan. This represents a realistic application because manyredistricting cases do not drastically alter the existing plans.In Appendix S6, we conduct “global simulations,” in which thetarget distribution is uniform on all valid redistricting plansunder an equal population constraint, using the data from NewHampshire with only two districts and 327 precincts.

We use the proposed algorithm with parallel tempering(Algorithm 2), as described in Section 2.5. To conduct localsimulations, we replace Equation (11) with the following,

fβ(π) = 1z(β)

exp

⎛⎝− β

19∑

V�∈π

ψ(V�)

⎞⎠ , (15)

JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 725

Figure 4. Pennsylvania’s 2008 redistricting plan. Pennsylvania has 9256 precincts that were divided into 19 congressional districts in their 2008 redistricting plan.

where ψ(V�) represents the proportion of precincts in district� = 1, 2, . . . , 19 that are not shared with the corresponding dis-trict of the 2008 redistricting plan. We then wish to approximatethe population of valid redistricting plans which satisfies theconstraint

∑19�=1 ψ(V�)/19 ≤ δ for some δ. For example, if we

choose δ = 0.05, then we obtain a sample of valid redistrictingplans, 5% of whose precincts are switched from other districtsof the 2008 plan.

Following the recommendations given in the literature(Kone and Kofke 2005; Atchadé, Roberts, and Rosenthal 2011),we chose β(i) to be geometrically spaced, that is, β(i) =(β(0)

)1− ir−1 , i ∈ {0, 1, . . . , r−1}. After a preliminary grid search

over possible values of the target temperature and of R, we setthe target temperature to be 2500, q = 0.04, and λ = 10 (whereλ is the mean of the truncated Poisson distribution), which ledto a good mixing behavior, reasonable acceptance probabilities,and provided a sufficient number of valid plans for subsequentanalyses. Since we are only exploring the local neighborhood ofthe 2008 plan, we initialize the Markov chain three times fromthe 2008 plan, but using different random seeds so that they startto explore different parts of the target distribution. We run eachof the 10 temperature chains for 120,000 simulations for eachinitialization and thin the chain by 12. We then base inferenceoff of those 10,000 simulations for each of the three initializedMPI chains.

3.2.2. Empirical FindingsWe examine how the partisan bias changes as the simulated planmoves farther from the 2008 plan. Following the literature (seeGrofman and King 2007), we use deviation from partisan sym-metry as a measure of partisan bias. This measure evaluates thepartisan implications of the counterfactual election outcomesunder hypothetical uniform swings of various degrees acrossdistricts. Specifically, we consider all vote swings where thestatewide 2008 two-party vote shares are between 40% and 60%.These vote shares are then aggregated according to each redis-tricting plan, yielding the counterfactual two-party vote shares

Figure 5. Local simulations for the 2008 Pennsylvania redistricting plan. The plotshows that the average partisan bias of the simulated plans decreases nearly to zeroas more precincts are switched out of the original redistricting plan. The solid linerepresents the partisan bias of the original plan, while the red dashed line indicatesan unbiased redistricting plan, in which the seats gained from a vote swing of anyvalue is equal to the number of seats lost in response to a vote swing of equalmagnitude in the opposite direction. The solid red circle represents the minimal-bias plan when less than 3% of all precincts are swapped.

at the district level and the number of hypothetical Democratand Republican winners of the election. The resulting informa-tion is summarized as the seats-votes curve (Tufte 1973), whichis a nondecreasing step-function f (x) evaluated within the rangeof vote shares x from 40% to 60%.

Formally, let f ∗(x) be a nondecreasing step-function that issymmetric around the 50%–50% two-party vote share. Then,the partisan bias is defined as the standardized differencebetween the area below f (x) and the area below f ∗(x),

726 B. FIFIELD ET AL.

partisan bias = 1η

∫ 0.5+η

0.5−η

(f (x) − f ∗(x)

)dx

= 1η

∫ 0.5+η

0.5−η

f (x)dx − 1, (16)

where the area under f ∗(x) is always equal to a uniform voteswing η. In our case, 0 ≤ η ≤ 0.1. This partisan bias measure isscaled so that −1 indicates a maximally Republican-biased plan(where no vote swing of any magnitude favoring Democratscan ever flip a seat away from Republicans), while 1 indicatesa maximally Democrat-biased plan (where no vote swing of anymagnitude favoring Republicans can ever flip a seat away fromDemocrats). A bias measure of 0 represents an unbiased plan,

where uniform vote swings result in equal seat gains or lossesfor each party.

Figure 5 presents the analysis where the solid black lineindicates the partisan bias and electoral competitiveness of the2008 plan, and the red dashed line represents an unbiased plan.The 2008 plan is biased in favor of the Democrats, with astandardized partisan bias measure of approximately 0.035. Wefind that the partisan bias of this plan can be nearly eliminatedby swapping approximately 3% of all precincts in Pennsylvania.

We further explore this finding by closely examining the min-imally biased plan found by swapping fewer than 3% of precinctsin the local simulations (represented by the red solid circle inFigure 5). This specific plan was obtained by swapping 277

Figure 6. Partisan patterns of the swapped and unswapped precincts in Pennsylvania local simulations. The left column plots the distribution of the Democraticcongressional vote-share for the precincts left untouched (black solid line), swapped in (red dot-dashed line), and swapped out (blue dashed line) of a congressional districtto obtain a nearly unbiased plan. The right column shows the geography of the swapped and unswapped precincts in Congressional District 1 (top row) and CongressionalDistrict 3 (bottom row). In both congressional districts, the algorithm swaps out more partisan precincts for more moderate precincts to achieve more symmetric responsesto voter swings.

JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 727

precincts from the original 2008 map into new districts. Figure 6plots the distributions of Democratic voteshare in swapped andunswapped precincts in two congressional districts. District 1(top row) encompasses Central and South Philadelphia, andsupported President Obama with nearly 90% of its vote. As aresult, Democratic control of the seat is heavily insulated frompartisan swings. The minimally biased plan substitutes out 48precincts with an average Democratic vote share of 90.5% for34 new precincts from the surrounding suburbs with an averageDemocratic vote share of 84%. While still heavily Democratic,these changes make the new district more vulnerable to voterswings in the direction of the Republicans.

The bottom row of Figure 6 conducts the same analysis forDistrict 3, located in the northwest corner of Pennsylvania. In2008, this was the only district in Pennsylvania to flip parties—in the case of this district, from Republicans to Democrats.In a Democrat-biased redistricting plan, Republican-held seatsshould be more vulnerable to swings. Indeed, Democrats flipped21 seats in the House of Representatives in 2008. To achieve amore unbiased statewide plan, the proposed algorithm swaps inmore conservative precincts (average Democratic vote share of38.2%) for moderate precincts (average Democratic vote shareof 43.5%), thereby making it more resilient to such swings.While the change in total Republican votes (an unbiased planwould have gained approximately 1000 additional Republicanvotes in District 3) would not have been enough to keep theseat from switching parties, the proposed change would haveincreased the bar necessary to flip the seat in Democratic-leaning uniform swing scenarios.

4. Concluding Remarks

Over the last half century, a number of automated redistrictingalgorithms have been proposed in the methodological literature.Most of these algorithms have been designed to find an opti-mal redistricting plan given a certain set of criteria. However,many researchers have been interested in characterizing thedistribution of redistricting plans under various constraints.Unfortunately, few such simulation algorithms exist and eventhe commonly used ones do not have a theoretical justification.

In this article, we propose an automated redistricting simu-lator using Markov chain Monte Carlo. Unlike the existing stan-dard algorithm, the proposed algorithms have the theoreticalproperty to approximate a target distribution. In a small-scalevalidation study, we show that the proposed algorithms workwell whereas the standard algorithm fails to obtain a represen-tative sample. Even in more realistic settings where the compu-tational challenge is greater, our analyses show the promisingperformance of the proposed algorithms. Nevertheless, it is stillunclear whether these algorithms scale to even larger states thanthose considered here.

Future research should consider further improvements ofthe proposed simulation algorithms, including a strategy toincorporate multiple constraints and a method to generate over-dispersed starting values. It is also important to conduct morevalidation studies so that we can better understand the condi-tions under which the proposed methodologies fail. An initialsuch validation study shows some promising results regarding

the empirical performance of the MCMC algorithm proposedin this article (see Fifield, Imai, et al. 2019).

Supplementary Materials

In the supplementary materials, we provide proofs of the theorems pre-sented in the article, as well as additional empirical examples.

Acknowledgments

We thank Jowei Chen, Jacob Montgomery, and seminar participants atChicago Booth, Dartmouth, Duke, Microsoft Research, and SAMSI foruseful comments and suggestions. We thank James Lo, Jonathan Olmsted,and Radhika Saksena for their advice on computation. The replicationarchive for this article is available in Fifield, Higgins, et al. (2019). The open-source R package redist for implementing the proposed methodologyis available in Fifield, Tarr, and Imai (2015). Replication materials can befound in Dataverse at https://doi.org/10.7910/DVN/VCIW2I.

ORCID

Benjamin Fifield http://orcid.org/0000-0002-2247-0201Michael Higgins http://orcid.org/0000-0003-4969-2733Kosuke Imai http://orcid.org/0000-0002-2748-1022

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