Optimizing Opportunity: An Algorithmic Approach to Redistricting for Fairer School Funding

Introduction

Public school district borders are some of the most impactful boundaries in the United States, yet we rarely stop to think about what they define. Not only are they responsible for determining where children are educated, but also for outlining taxing jurisdictions. Because a significant portion of school district funding is drawn from local property taxes, these boundaries directly impact the resources and opportunities that are available to students. Even students living in neighboring communities can have very different levels of funding and educational experiences. While nearly all states’ school funding formulas have provisions that account for local tax capacity, and federal dollars are largely distributed through programming targeted to high-poverty districts, these sources are often not enough to bridge the gaps created by disparities in local tax revenue.

Local funds play an outsize role in creating and reinforcing inter-district funding disparities, because districts in almost every state are permitted to raise and keep local property taxes in excess of their state-calculated funding targets (Kenyon and Munteanu 2021). As a result, high-wealth districts can easily increase their budgets beyond what lower-wealth districts can match. Nationwide, property taxes reportedly comprised 65% of local revenues in 2021, but can be much higher in some states (Common Core of Data (CCD) - Local Education Agency (School District) Finance Survey (F-33) Data n.d.). At a fixed tax rate, properties with higher assessed values will yield more revenue. Given the connection between property values and neighborhood affordability, the students that lose out due to lower district budgets tend to be those from low-income backgrounds.

The problem also has a troubling racial dimension. This funding system is layered on top of generations of policies and government practices that have created and entrenched racial and economic inequality in housing (Kuhn, Schularick, and Steins 2018). At different phases of America’s past and present, this has included redlining and racially discriminatory mortgage lending; court enforcement of racially restrictive covenants; government-funded construction of segregated housing developments; exclusionary zoning policies; and unfair property assessment, among other forms of discrimination. These factors have shaped both the racial composition of neighborhoods and the property values in the taxing jurisdictions from which school districts raise local dollars.

The result is a map of highly segregated residential communities that demonstrate stark economic divides (S. Reardon and Weathers 2019). Left unmitigated, the legacies of historical discrimination have the potential to intersect to create, shape, and enforce new patterns of segregation (S. F. Reardon and Owens 2014). This problem is further reified by the ways in which school district borders function: both as geographic areas that are home to district students, and as the taxing jurisdictions that yield their local funding. Because these boundaries determine the students served by a district and the local funding available for its schools, they function to separate students from resources—and from each other.

Furthermore, students from low-income backgrounds, and with other needs and challenges have demonstrably higher funding needs than students from high-income families (Jackson, Johnson, and Persico 2015). While state aid can and should be used as a tool to provide additional support to students with higher needs, this goal is undermined when these funds are eaten up in pursuit of achieving basic funding parity with high wealth districts. This unfortunate reality burdens state education budgets with compensating for existing inequity, rather than achieving equity. Funding equity—meeting greater needs with greater, rather than equal, resources—is the only acceptable outcome to provide students with the resources they need to succeed. Given the high degree of alignment between segregated school districts, patterns of residential segregation in the communities they serve, and funding divides, one option is to consider drawing better school district borders.

This project explores two related questions:

1. When school district boundaries are redrawn to encompass a more equitable distribution of property wealth, how does the demographic composition of districts change, and to what extent can both racial and economic segregation be reduced?

2. How are educational outcomes for students impacted by changes to funding levels and possible reductions in racial and economic segregation as a result of redistricting?

Drawing inspiration from the pioneering work on congressional redistricting conducted by The Algorithm-Assisted Redistricting Methodology (ALARM) project at Harvard University, this project adapts machine learning methods to the context of school district boundaries. The challenge of identifying optimal boundary configurations can be viewed as a combinatorial optimization problem that seeks optimal solutions through use of Markov Chain Monte Carlo (MCMC). In the context of this research question, the algorithm prioritizes low variance of aggregate property wealth among districts, while aiming to reduce levels of both racial and economic segregation while requiring districts to be contiguous and compact.

To address the second research question, in the discussion section, a post-hoc evaluation mechanism using a difference in differences (DiD) research design framework is outlined. While it will not be implemented as part of this study, this section provides an objective framework through which policymakers, advocates, parents, among other stakeholders can evaluate the efficacy of redistricting. This research design is focused with measuring the impact of redistricting on student performance on state standards exams. However, this section also explores potential downstream second-order changes to property values and demographic realignment that may occur contemporaneously.

Theoretical Development

The potential outcomes of this study have a wide range of implications for the experience of students. District boundary change can certainly be used to reduce inter-district racial and economic divides, but it can also create more equitably resourced districts. Given the intrinsic link between property values and property tax revenue, disparities in the local share of school district revenues are expected to decrease. This outcome is both notable and relevant to state-level policy makers who are tasked with applying state funding formulas to provide supplemental funds to districts who are unable to raise sufficient local revenue. Funding disparities between public school districts are costly expenditures for states to address. Thus, pro-equity redistricting is not just a civil rights issue, but one of fiscal responsibility.

However, the potential impact for student learning outcomes cannot be understated. There is an extensive body of literature supporting improved academic performance, particularly for students experiencing poverty, resulting from the allocation of additional funding resources to support differing levels of student need. Thus, there is an expectation that pro-equity school district boundary change can create better resourced and more socioeconomically diverse schools, while improving learning outcomes for students and maintaining geographic feasibility.

Researchers have approached methodologically similar optimization problems at the congressional district-level, with significant implications for national and state elections. While there exists a small but growing body of literature that applies these algorithmic methods to explore pro-equity school district boundary change, no existing study has explored its impact on student performance.

An MIT-based research team found that significant reduction of racial segregation was possible while slightly reducing travel times (Gillani et al. 2023). However, this paper redraws attendance boundaries within existing districts, and thus, does not address inter-district segregation, where has shown that nearly two thirds of racial segregation occurs (Owens 2016). A recent working paper explores inter-district divides and found even larger possible reductions in racial segregation than did Gillani et al., but limits analysis to New Jersey (Simko 2023). While these studies provide useful methodological frameworks, neither propose plans that alter the existing number of school districts. Though a computationally efficient approach, it significantly limits the potential solution space–and the extent of equity gains that can be achieved through redistricting.

Previously published redistricting simulations tend to focus on racial segregation and rarely take economic disparities or district revenue gaps into account. This project further extends existing research by integrating Small Area Income and Poverty Estimates (SAIPE) and parcel-level property tax assessment data. Considering property wealth and poverty rates as factors allows us to avoid redrawing district boundaries atop existing patterns of property wealth and income inequality, thus creating more equitable tax bases from which districts derive funding. Given that on average, local revenues compose approximately 40% of district revenues (Common Core of Data (CCD) - Local Education Agency (School District) Finance Survey (F-33) Data n.d.), this consideration is critical to creating equitably resourced districts.

While it remains clear that segregation could be reduced in most jurisdictions, making changes to school district boundaries can be a politically challenging endeavor that may face significant challenges from constituents and courts alike. However, policymakers may not be aware of extent of the divides between neighboring communities, nor the degree to which they could be mitigated by drawing better district borders. This study’s algorithmic approach supplies an objective framework to provide legislators with evidence supporting the potential benefits of redistricting, including fiscal savings, reduced levels of inequality, and improved educational outcomes for students.

Data

This study uses of census tracts to create new district boundaries. Geographic boundaries are obtained from the United States Census Bureau’s (USCB) Topographically Integrated Encoding and Referencing (TIGER) system.

Census tract-level demographic data are obtained from the American Community Survey (ACS) data come from the USCB and school district-level aggregations from the EDGE program. Both sources also produce SAIPE data which allow other socioeconomic and housing indicators to be seamlessly incorporated.

Parcel-level property tax assessment data are obtained from ATTOM data solutions, a proprietary real estate data company that maintains nationwide datasets of information for individual properties. Each parcel is then geocoded to its associated school district, census block group, and county. This information is used to calculate assessed value per-pupil by dividing the sum of all properties encompassed by a given district’s boundaries, divided by its enrollment.

Fiscal data at the school district level, including revenues and expenditures are obtained from the from the U.S. Census Bureau, Annual Survey of School System Finances, also known to researchers by the survey number F33.

Finally, district-level performance on state standards examinations are obtained from the Department of Education. As part of the proposed post-hoc analysis, we monitor and record changes in test scores for students who attend schools that have been redistricted.

Methodology

This study explores district boundary change as a high dimensional combinatorial optimization problem. It employs a three-stage algorithmic framework to optimize school district boundaries, addressing disparities in funding capacity and racial and economic segregation across three states selected for their diverse socioeconomic, geographic, and policymaking environments. The methodology integrates the Minimum Spanning Tree-based Spatial ’K’luster Analysis by Tree Edge Removal (SKATER) algorithm, Markov Chain Monte Carlo (MCMC) optimization, and systematic variation in the number of districts. The integration of these approaches balances computational efficiency with flexibility, while maximizing potential equity gains.

Stage 1: Generating High-Quality Initial Districts

The SKATER algorithm is a multivariate spatial clustering method used to generate initial district configurations. It minimizes within-cluster heterogeneity by iteratively removing edges from a Minimum Spanning Tree (MST) constructed from tract-level data, which is first scaled to ensure comparability and balanced influence. Clustering is guided by a cost function consisting of policy-relevant variables, including assessed property values, racial composition, and poverty rates, while ensuring spatial coherence, as defined by Rook contiguity, which ensures that districts are joined by more than a single vertex. Weights for each measure are pre-specified to reflect policy priorities and normative goals. Education funding capacity, measured by total assessed property value, is prioritized with the largest weight of 5, emphasizing the primary objective of reducing funding disparities. Integration, assessed through racial and economic dissimilarity indices, is assigned a moderate (and computationally efficient) weight of one, while compactness is weighted less heavily at 0.5 to prioritize potential equity gains.

By leveraging SKATER, inefficiencies associated with starting from random configurations are largely bypassed, ensuring that in the next stage, the MCMC-based optimization begins with districts that are already aligned with key policy priorities. This approach is particularly critical given the computational demands of optimizing a varying number of potential districts across multiple states.

Stage 2: MCMC-Based Optimization

The second stage refines the initial configurations created by SKATER using an MCMC-based optimization algorithm. MCMC methods are especially adept at sampling from complex, high-dimensional spaces, making them ideal for navigating a large number of potential boundary configurations while adhering to a flexible set of constraints. The optimization process involves randomly assigning selected census tract to an adjacent district and accepting or rejecting that boundary changes based on its adherence to both soft and hard constraints. Soft constraints, incorporated into the objective function defined below, include assessed value per capita, levels of racial and economic integration, and compactness. Hard constraints, such as contiguity and minimum enrollment, are enforced directly, with any proposal violating these conditions immediately rejected.

\[\text{Objective Function} = w_{\text{equity}} \cdot \text{Variance} + w_{\text{integration}} \cdot \text{Dissimilarity Index} + w_{\text{compactness}} \cdot \left(1 - \text{Polsby-Popper}\right)\]

The \(w_{\text{x...n}}\) terms represent weights for each soft constraint reflected in the objective function. These scaling factors allow for a flexible approach to optimization that supports tailoring the algorithm to varying state-level environments.

The optimization algorithm was developed using the ALARM Project’s R package, redist (and its Github repository) as a foundational resource. While redist is a widely used tool for redistricting, its focus on equal population constraints for congressional redistricting made it less applicable to this context. By adapting key elements of redist, this study developed a custom algorithm tuned to the specific requirements of school redistricting, prioritizing funding equity and integration rather than strict population equality.

Simulated Annealing Framework

This algorithm simultaneously employs a simulated annealing framework to avoid local minima identified by the objective function. The initial temperature determines the likelihood of accepting changes that temporarily worsen the objective function, enabling the algorithm to explore a broader solution space, allowing it to get “unstuck.” As the number of iterations increases, the cooling schedule gradually reduces the temperature, further focusing the search on improved configurations but still allowing for occasional exploration. For computational efficiency, a temperature of one and a cooling rate of 0.99 are used.

Computational Efficiency and Iteration Limits

Each optimization run is capped at 2000 iterations. This threshold was selected to achieve substantial improvements in the objective function while maintaining computational efficiency. Starting with high-quality configurations generated by SKATER further reduces the need for excessive iterations, ensuring that the algorithm refines districts effectively without unnecessary computational expense. Because of the high computational requirements of spatial operations, compactness is not directly used as an optimization factor in this stage. As a result, significantly increasing run size can result in increasingly irregular district boundary configurations, which are often unpalatable to policymakers and other stakeholders.

Stage 3: Systematic Variation of District Counts

The SKATER and MCMC processes are applied across the range of district counts to evaluate how changing the proposed number of districts impacts funding capacity equity, segregation, and geographic feasibility. This systematic variation enables the analysis to identify trade-offs inherent in consolidation and fragmentation of districts.

To evaluate the effectiveness of proposed boundary changes, and to assess the impact of redrawn borders, a suite of quantitative metrics are used to assess downstream changes to racial, economic, and funding capacity disparities. These values are recorded for each state and each number of districts and include both the initial SKATER proposal and the iteratively optimized MCMC-based proposal. These criteria include:

• Variance in assessed value per capita: Distributing assessed value per capita more equitably across school districts.

• Dissimilarity Indices: Quantifying the degree of racial and economic segregation within school districts.

• Compactness: Creating districts that allow for viable transportation of students. Defined by Pollsby-Popper score, a metric derived from the ratio of perimeter to area of a given geographic unit.

• Contiguity: Ensuring that proposed districts are contiguous.

• Enrollment: Minimum enrollment threshold set at the smallest current district enrollment for a given state.

These metrics provide objective measures of the outcomes associated with different boundary configurations, enabling comparative analysis and informing decision-making processes. Systematically evaluating the potential impacts of proposed changes across multiple dimensions, we can ensure that redistricting efforts are guided by considerations of equity and efficiency.

By applying the same methodology across varying district counts, this stage ensures that results are not driven by idiosyncrasies of a single configuration, enhancing the robustness of the study’s findings. The ability to compare outcomes across a range of district scales also highlights the flexibility of this approach in addressing the diverse contexts and challenges of changing public school district boundaries nationwide.

Case Selection

The selection of Alabama, Connecticut, and Arizona as focal states for this study is grounded in their unique socioeconomic, geographic, and policy characteristics. Taken together, these differing socioeconomic landscapes and policy environments allow for a robust analysis of the potential for algorithmic redistricting to reduce inequities and improve educational outcomes across varied contexts. The MCMC model’s ability to adapt to these diverse challenges will provide valuable insights into its applicability and effectiveness in addressing disparities in school funding and segregation.

Alabama

Figure 1: Irregular School District Boundaries in Alabama

Alabama exemplifies a state where historical segregation and systemic inequities have left a lasting imprint on school district boundaries. As seen in Figure 1, current borders are highly irregular, owing in part to the state’s unusual municipal incorporation laws and city-level districts (Tasby v. Estes, 412 F. Supp. 1185 (N.D. Tex. 1975) 2001). The state’s legacy of racial and economic disparities in education is closely tied to its history of segregated housing policies and local control of property tax revenues. Alabama ranks near the bottom nationally in per-capita income, with stark disparities between urban and rural areas. These economic divides are mirrored in property values, creating significant variations in local revenue for schools. Despite federal desegregation mandates, Alabama continues to experience high levels of racial segregation in its schools (Common Core of Data (CCD) - Local Education Agency (School District) Finance Survey (F-33) Data n.d.). School district boundaries often reflect and reinforce these divides, with predominantly Black districts frequently underfunded (Bureau 2023) . Alabama’s school funding system relies heavily on local property taxes, making it an ideal state to explore the potential of algorithmic redistricting to address funding inequities. The state’s significant dependence on local revenues magnifies the impact of property wealth disparities.

In Alabama, the high variance in property values and racial demographics across districts would likely challenge the MCMC algorithm to balance equity while preserving compactness and contiguity. The model’s ability to minimize disparities in assessed property value per pupil while maintaining geographic coherence will be particularly tested in areas with entrenched segregation and resource disparities. Additionally, the rural geography in much of the state may constrain the model’s flexibility, as maintaining compactness could conflict with efforts to equalize resources.

Arizona

Arizona represents a unique case of rapid demographic and economic change, offering an opportunity to examine how redistricting can address challenges in a growing and increasingly diverse state. Arizona’s rapid population growth, particularly in urban areas like Phoenix, has led to significant disparities in school funding and overcrowding in certain districts (Bureau 2023). These dynamics make it a critical state for exploring how redistricting can address inequities in states experiencing rapid change. Arizona’s student population is one of the most racially and ethnically diverse in the nation. However, this diversity is often not reflected in the composition of its school districts, which are highly segregated by income and race (Common Core of Data (CCD) - Local Education Agency (School District) Finance Survey (F-33) Data n.d.). The state’s school funding system is characterized by low overall funding levels and significant reliance on local property taxes. Additionally, Arizona’s large rural areas may pose challenges for maintaining compactness and minimizing travel times, making it a valuable test case for the model’s constraints.

Arizona’s rapid growth and diverse demographics introduce additional layers of complexity to the MCMC model. The algorithm must navigate urban districts with high population density and significant wealth disparities, while also addressing the unique challenges of large rural areas where maintaining compactness is more difficult. The diversity of the student population provides an opportunity for the model to explore the impact of reducing racial and economic segregation in a state where these issues intersect with rapid demographic changes.

Connecticut

Connecticut offers a contrasting context where extreme wealth disparities exist within a relatively small geographic area. This makes the state an important test case for exploring how algorithmic redistricting can address inequities in states with high overall wealth but significant local disparities. Connecticut is one of the wealthiest states in the U.S., yet it has some of the largest gaps between high- and low-wealth school districts (Bureau 2023). Like Alabama, the state’s reliance on local property taxes for school funding exacerbates these disparities (Equalized Net Grand List by Town (2011-2021 GL) | Connecticut Data 2023). The state’s compact size and high population density provide a useful setting for testing the model’s ability to balance compactness and equity without significantly increasing travel times for students. Despite its wealth, Connecticut has some of the highest levels of racial and economic segregation in its schools, driven by both residential segregation and exclusionary zoning policies.

The high population density and small geographic size of Connecticut present a unique opportunity for the MCMC algorithm to prioritize equity while preserving. However, the extreme disparities in wealth and persistent segregation may require the model to navigate sharp contrasts in property values over short distances. The model’s sensitivity to these disparities and its ability to generate contiguous districts without exacerbating existing inequalities will be critical in this setting.

Results

This section examines the impact of algorithmically generated district configurations in relation to existing (“status quo”) boundaries. The results are presented in four parts: (1) a detailed evaluation of the status quo district boundaries in Alabama, Connecticut, and Arizona; (2) the outcomes of the SKATER algorithm as a baseline configuration; (3) the improvements achieved through MCMC-based iterative optimization; and (4) the sensitivity of results to systematic variation in district counts. Metrics considered as part of the evaluation framework include variance and log-variance of assessed property values, racial and economic dissimilarity indices, and Pollsby-Popper compactness scores. Table 1, below, records these measures, allowing us to systematically quantify the efficacy of each optimization model.

Table 1: Evaluation Metrics for Redistricting Plans

State	Number of Districts	Model	Log-transformed Variance, Assessed Value per Capita	Racial Dissimilarity Index	Economic Dissimilarity Index	Compactness	Gini Coefficient, Total Assessed Value
AL	140	Status Quo	21.2014421	0.3661506	0.2789342	0.2228176	0.8206279
AL	140	Initial	8.3141352	0.3715259	0.3112961	0.3154289	0.8495722
AL	140	Optimized	6.6220780	0.3438370	0.2674942	0.3082913	0.7887360
AL	76	Varying N	5.5706018	0.2937952	0.2102243	0.2778224	0.6775833
AZ	213	Status Quo	19.8136314	0.2144866	0.2682639	0.4703784	0.8138871
AZ	213	Initial	8.6745909	0.2540242	0.3384509	0.4270496	0.8403409
AZ	213	Optimized	6.5976284	0.2400804	0.3103774	0.4242487	0.7186427
AZ	156	Varying N	4.2329197	0.2138735	0.2796880	0.3857319	0.5947202
CT	167	Status Quo	16.4915976	0.3995508	0.4451813	0.5888079	0.7234304
CT	167	Initial	5.4685866	0.3842241	0.4293160	0.4171719	0.8087891
CT	167	Optimized	1.2702076	0.3025889	0.3291508	0.4031181	0.5086194
CT	170	Varying N	-0.0587083	0.3256511	0.3214812	0.4054702	0.4623482

SKATER Algorithm Initialization

It is important to recognize that the primary goal of the SKATER algorithm was to provide a high-quality entry point for subsequent MCMC optimization, not to produce an optimal plan. It was tasked with reducing variance in total assessed value across school districts within a given state. The key advantage of SKATER lies in its ability to guarantee contiguous proposals while doing so. However, the heavy weights placed on achieving a more equitable distribution of assessed property value sometimes limited integration gains. In the case of Arizona, and to a far lesser extent, Alabama, these outcomes were even negative.

Because of its deterministic output, the SKATER algorithm is inherently less adept at balancing multiple complex constraints, and is unable to use dynamically calculated evaluation criteria, including assessed value per capita in its proposals, which depend on model outputs. Despite these limitations, Table 1 reveal dramatic reductions in log-transformed variance, demonstrating orders-of-magnitude improvements in funding capacity equity. These findings underscore the utility of SKATER as a baseline generator but also highlight the necessity of subsequent iterative refinement to address broader objectives.

MCMC-Based Optimization

The MCMC optimization algorithm builds on the SKATER outputs by refining district configurations to better balance funding capacity equity, integration, and geographic feasibility. Unlike SKATER, MCMC employs a stochastic, probabilistic approach, allowing it to dynamically evaluate proposals based on a broader set of criteria, including assessed value per capita, dissimilarity indices, and compactness metrics. This adaptability enables it to make incremental improvements while adhering to contiguity and enrollment constraints.

As shown in Table 1, MCMC optimization achieves significant reductions in funding disparities across all three states. For example, in Alabama, log-transformed variance drops from 8.31 under SKATER to 6.62 after optimization, representing substantial gains in equity. Connecticut’s transformation is particularly notable, decreasing variance from 5.47 to 1.27, effectively eliminating funding disparities. These improvements are mirrored in integration gains. In Connecticut, the racial dissimilarity index decreased from 0.38 to 0.30, representing a 21.2% reduction. This substantial decline underscores the algorithm’s effectiveness in reducing segregation in a state with high initial disparities concentrated in compact geographic areas. Alabama and Arizona also saw marked, though more modest improvement, further highlighting that a state’s underlying characteristics significantly impact model behavior.

The ability of MCMC optimization to balance equity and geographic considerations is further evident when comparing compactness scores across redistricting plans. While slight reductions in compactness are observed in Arizona, they are minimal and do not compromise the overall feasibility of the proposed districts. Additionally, the dynamic, probabilistic nature of MCMC allows it to explore a broader solution space, avoiding the limitations of SKATER’s deterministic outputs and ensuring greater flexibility in addressing state-specific challenges.

Varying District Counts

Systematic variation in the number of districts reveals important trade-offs between funding equity, integration, and geographic feasibility. Reducing the number of districts consolidates resources, leading to dramatic reductions in variance, as shown in Table 1. For instance, Alabama’s log-variance drops from 8.31 in the SKATER baseline to 5.57 when district counts are reduced by nearly half. Conversely, increasing the number of districts has the potential to create finer-grained boundaries, which can improve integration metrics, as observed in Connecticut, where racial dissimilarity declines from 0.38 to 0.33 when the number of districts slightly increases from 167 to 170.

However, these gains are not without trade-offs. Compactness scores vary with district counts, highlighting the geographic challenges associated with extreme configurations. In Arizona, reducing districts from 213 to 156 significantly lowers funding variance but results in compactness scores dropping below optimal thresholds, reflecting the difficulty of maintaining coherent boundaries in rural areas. By contrast, Connecticut’s higher population density allows for greater flexibility, with compactness scores remaining stable even as district counts increase. This might suggest that this algorithm may perform well in the northeast, where population density and levels of segregation are relatively high.

These findings conintue to underscore the importance of tailoring redistricting plans to state-specific conditions. Allowing the optimization model to systematically vary the number of proposed districts greatly increases its sensitivity to identify optimal proposals.

Comparison to the Status Quo

The status quo district boundaries serve as a critical benchmark to evaluate the efficacy of algorithmic redistricting. As highlighted in Table 1, and as seen below in Figure 4, all algorithmically generated plans outperform the status quo in terms of funding equity, with reductions in log-transformed variance of up to several orders of magnitude. For example, Alabama’s status quo log-variance of 21.20 is reduced to 6.62 in the MCMC-optimized plan and to 5.57 under the reduced district count scenario.

Figure 3 and Figure 4: Evaluation Metrics for Redistricting Plans

Integration metrics also show significant improvements relative to the status quo. For instance, Alabama and Connecticut’s saw their racial dissimilarity indices decline by 19.4% and 15.8%, respectively, while their economic dissimilarity indices fell by 24.4% and 27.3%. These shifts reflect the algorithm’s capacity to address entrenched segregation patterns and balancing funding capacity equity and geographic feasiblity.

Figure 5 demonstrates the ability for algorithmic redistricting to deconcentrate property value-poor school districts. In each case, the distributions of assessed value per capita shift rightward after completing the optimization process. The boxplots in Figure 6 highlight the extreme number of outlying districts under the status quo in Alabama. Yet, the algorithms were able to join property-wealthy tracts with property-poor ones to significantly broaden the interquartile range for assessed value per capita. In practice, this means that districts that currently experience low property tax capacity would likely be able to increase locally funded revenues under an optimized plan.

Additional visual comparisons, such as those presented in interactive plots below, further underscore the transformative potential of algorithmic redistricting. Interactive Visualization 1 demonstrates that property value equity this can be accomplished in tandem with creating racially and economically integrated classrooms. Students and communities alike will feel the impact of more diverse educational experiences.

Histograms and boxplots of key metrics are useful for gaining a more nuanced understanding of the distribution of these values among districts. Almost across the board, more equitable distributions are achieved.

In Interactive Visualization 2, the status quo districts next to the stage three proposal for each state. These maps can be colored by either the poverty rate or percent students of color for each plan. This allows the user to explore how the spatial distribution of key metrics can shift as a result of redistricting.

Post-Hoc Evaluation and Discussion

Design and Specification

Though the MCMC model primarily uses assessed property value per-pupil to create new district boundary configurations while preserving the constraints outlined in the optimization framework, there is a second parallel goal not directly reflected in the model: to improve educational outcomes for affected students. These improvements may come as a result of increases in local property tax revenue for districts that are currently under-resourced. To further assess the impact of redistricting on student outcomes, a Difference-in-Differences (DiD) analysis is designed. This approach allows the changes in student outcomes on state standards exams over time to be compared between districts that have undergone boundary changes and those that have not.

In this secondary analysis, the treated group will consist of school districts that experienced redistricting, while the control group will include districts that maintained their existing boundaries. Because of both the legal and logistic challenges to redistricting only a subset of a given state, post-hoc evaluation will require expanding the sample size to other states. In order to maintain the statistical validity of this econometric research design, the parallel trends assumption must be satisfied. In the context of this study, it requires that in the absence of redistricting, that the treatment and control groups would report similar student achievement levels. Thus, Mississippi, New Mexico, and New Jersey are chosen for their geographic proximity and relative similarity in terms of demographic patterns, district administrative structures, and state funding formulas. To observe the casual effect of redistricting on district-level performance on state standards tests, these scores are observed and recorded five years before and after the boundary changes.

Figure 7 demonstrates the clear fulfillment of the parallel trends assumption and validates the hypothesis that redistricting has the potential to improve student outcomes.

The DiD model will be specified as follows:

\[ Y_{it} = \alpha + \beta_1 \text{Post}_t + \beta_2 \text{Treatment}_i + \beta_3 (\text{Post}_t \times \text{Treatment}_i) + \gamma X_{it} + \epsilon_{it} \]

Where:

• \(Y_{it}\) represents the outcome variable, in this case, proficiency rates for district \(i\) at time \(t\).
• \(\text{Post}_t\) is a binary variable indicating the post-treatment period (after redistricting).
• \(\text{Treatment}_i\) is a binary variable indicating whether district \(i\) is in the treated group (experienced redistricting).
• \(\text{Post}_t \times \text{Treatment}_i\) is the interaction term of interest, capturing the differential effect of redistricting on the treated group relative to the control group.
• \(X_{it}\) includes a set of control variables, including those used to assess the impact of redistricting.
• \(\epsilon_{it}\) is the error term.

The DiD design will test the hypothesis that student outcomes are improved when districts are drawn from a more equitable distribution of property values. Therefore, the null hypothesis is defined as:

\[ H_0: \beta_3 = 0 \] Where the interaction term, \(\beta_3\) is equal to zero suggests that the difference in district-level performance on state standards exams is the same for treated and untreated districts.

Conversely, the alternative hypothesis is defined as

\[ H_1: \beta_3 \neq 0 \]

In this case, the interaction term \(\beta_3\) is not equal to zero, indicating that the effect of redistricting on educational outcomes differs between treated and control districts.

Model Results and Interpretation with Sample Data

Using sample data, this study simulates the potential impact of redistricting that could be measured through this research design.

Table 2: DiD Model Comparison

	(1)	(2)	(3)
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
(Intercept)	0.536***	0.518***	0.639***
	(0.003)	(0.008)	(0.031)
Post	0.011*	0.011*	0.011*
	(0.004)	(0.004)	(0.004)
Treatment	0.026***	0.025***	-0.003
	(0.004)	(0.004)	(0.008)
Post × Treatment	0.104***	0.109***	0.044*
	(0.006)	(0.007)	(0.018)
PropertyValue		0.018*	0.021**
		(0.007)	(0.007)
RacialDissimilarity			-0.114**
			(0.038)
EconomicDissimilarity			-0.108**
			(0.036)
Num.Obs.	200	200	200
R2	0.864	0.868	0.879
RMSE	0.02	0.02	0.02

Two models are highlighted to interpret the results of the DiD analysis. Model (1) compared district-level performance on state standards exams three years before and after redistricting, controlling only for the treatment effect. The second model (2) extended this analysis by including additional control variables: assessed property value, racial dissimilarity, and economic dissimilarity.

Model (1): Simple DiD Analysis

The first model reveals that treated districts experienced a statistically significant increase in test scores relative to the control group. The key findings from Model (1) are:

• Intercept (α): The intercept represents the average state standards proficiency rate in the control group before redistricting. A coefficient of 0.536 indicates a baseline of approximately 53.6% in the control districts.

• Post (β1): The coefficient for the post-treatment period is 0.011 and is statistically significant, suggesting that there was a modest but significant improvement in test scores over time in the control group.

• Treatment (β2): The treatment effect alone is positive (-0.03) and statistically significant, indicating that districts that underwent redistricting had higher test scores compared to those that did not.

• Post × Treatment (β3): The interaction term has a coefficient of 0.104 and is statistically significant. This positive and significant coefficient suggests that redistricting led to a 10.4 percentage point increase in test scores in treated districts relative to the control group, accounting for any differences between time periods.

Model (2): DiD Analysis with Controls

The second model introduces additional controls to account for factors that may influence test scores, such as property value, racial dissimilarity, and economic dissimilarity. The key findings from Model 2 are:

• Intercept (α): The intercept remains consistent with a coefficient of 0.639, indicating the baseline test score in control districts before redistricting.

• Post (β1): The post-treatment period coefficient remains positive at 0.011, similar to Model 1.

• Treatment (β2): The treatment effect remains non-significant with a coefficient of -0.003, indicating no significant difference in test scores solely due to treatment.

• Post × Treatment (β3): The interaction term remains significant with a coefficient of 0.044, consistent with Model 1’s finding of a 4.4 percentage point increase in test scores, but a smaller magnitude.

• Property Value (γ1): The property value coefficient is 0.021 and statistically significant, indicating that higher property values are associated with higher test scores.

• Racial Dissimilarity (γ2): The coefficient for racial dissimilarity is negative and statistically significant. This suggests that higher levels of racial segregation within a district are associated with lower student test scores, indicating that racial segregation may hinder academic performance.

• Economic Dissimilarity (γ3): Similarly, the coefficient for economic dissimilarity is also negative and statistically significant. This implies that greater economic segregation within a district is linked to lower test scores, highlighting the adverse impact of economic inequality on student achievement.

The results from the Difference-in-Differences analysis across both models indicate that the redistricting intervention had a statistically significant and positive impact on student test scores in the treated districts. Specifically, the interaction term (β3) in both models suggests an improvement in test scores. In the case of Model (1), by as much as 10.4 percentage points, highlighting the potential benefits of creating more district boundaries based on a more equitable distribution of property values. The inclusion of control variables in Model (2) further strengthens the findings, showing that the positive effects of redistricting are robust even when accounting for property value and segregation metrics. Notably, though still statistically significant, the magnitude of the interaction term decreased significantly when the controls for racial and economic segregation were included. This suggests that decreased levels of segregation are likely contribute to the increased test scores captured in the interaction term of the more basic Model (1).

Second-Order Effects and Further Research

This DiD research design and the panel data that undergirds it can be extended to other key outcomes. Property values, for instance, can be analyzed to reveal how redistricting alters assessed property values pre- and post-intervention, providing insights into its effects on local housing markets. Changes in racial and economic segregation indices can be tracked over time to understand redistricting’s broader social implications, while shifts in district-level revenues and expenditures offer a window into the fiscal impacts of boundary changes.

Redistricting has the potential to trigger cascading effects beyond immediate changes in school funding and performance. One significant second-order effect are future changes to property values. In districts that gain access to better-funded or high-performing schools, property values may appreciate, increasing wealth for local homeowners. Conversely, districts perceived as losing resources may experience declines in property values, affecting community investment and economic stability.

These changes have broader implications. Rising property values could reduce housing affordability, displacing lower-income residents, necessitating affordable housing initiatives to counteract displacement. Additionally, higher property values could generate increased local tax revenues, potentially reducing state funding requirements for equity-focused interventions. Changes in property values might also alter neighborhood composition, influencing long-term patterns of residential and school segregation.

By leveraging a DiD research design, this research provides a nuanced understanding of how redistricting affects educational and economic outcomes over time. The findings can inform policymakers and other stakeholders about the potential benefits and challenges of equity-focused redistricting, offering actionable insights to enhance educational equity. Importantly, this framework also underscores the interconnectedness of education, housing, and fiscal policy, highlighting the need for integrated solutions to systemic inequities.

Conclusion

This study demonstrates the significant potential of algorithmic redistricting to address and mitigate the entrenched disparities in public school funding and educational outcomes in the United States. By redrawing school district boundaries with a focus on equity, specifically targeting the redistribution of property wealth across districts, the analysis reveals that other inequities can simultaneously be targeted. The reduction in variance of assessed property value per pupil and the corresponding decrease in segregation indices underscore the effectiveness of the proposed optimization framework in creating more balanced and inclusive school districts.

The positive impact of these changes extends beyond the boundaries of financial equity. This analysis indicates a statistically significant improvement in district-level performance on state standards examinations for those which drew new borders. The ten percentage point increase in test scores in treated districts, compared to control districts, provides compelling evidence that the enhanced allocation of resources and the reduction in segregation contribute to better educational outcomes for students. A significant portion of these gains can be attributed to reductions in both racial and economic segregation. This finding supports the hypothesis that pro-equity redistricting can foster more supportive and conducive learning environments, particularly for students from disadvantaged backgrounds.

The multistage optimization process allowed for the redistricting solution space to be expanded to an unprecendented extent. This allowed for orders of magnitude reductions in assessed value per capita. While reductions to both racial and economic dissimilarity indices were more modest, there is still considerable integration potential for America’s public schools, particularly in states like Alabama that have struggled to integrate (and are even under active desegregation orders) in the 70 years since the monumental Brown V. Board of Education decision.

While the model employed in this study prioritizes equity property tax capacity, it also considers and enforces practical constraints, such as maintaining contiguity and compactness of districts. The results show that it is possible to create more equitable districts without imposing undue burdens on students or requiring significant new infrastructure investments. The preservation of compactness and contiguity across treated districts further highlight the feasibility of implementing such redistricting plans on a broader scale.

However, this study also acknowledges the political and social challenges associated with redistricting. The resistance that such efforts may face, both from local communities and from legal frameworks, cannot be understated. Despite these challenges, the evidence presented here provides a compelling argument for the reconsideration of school district boundaries as a means to achieve greater educational equity. By offering a rigorous, data-driven approach, this research contributes to the ongoing discourse on educational equity and provides policymakers with valuable insights that can inform future efforts to create more just and effective public school systems.

This study aims to not only demonstrate the technical feasibility of pro-equity school district redistricting but also highlight its potential to drive substantial improvements in educational equity and student outcomes. As educational disparities continue to reflect and reinforce broader societal inequalities, the findings of this research underscore the urgent need for innovative solutions that address these issues at their root. Redistricting, when guided by principles of equity, offers a promising path forward in the quest to ensure that all students, regardless of their background, have access to the resources and opportunities they need to succeed.

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