Technical Summary: Learning Long-Term Educational Investment Policies under Residential Sorting

Problem Statement

Allocating public-school investment effectively and fairly is complicated by the fact that school access is often determined by residential location. When high-quality schools are concentrated in specific communities, government investments intended to improve school quality can inadvertently raise local housing demand and prices. This dynamic creates a feedback loop where school improvements may reshape enrollment and demographic composition, potentially limiting access for lower-income households.

Existing literature typically studies school funding, household choice, and housing markets in isolation or employs static models that fail to capture the interconnected, long-term effects of these systems. Specifically, there is a gap in understanding how a government should allocate resources when school quality, residential sorting, housing prices, enrollment, and educational access co-evolve over time. The core challenge lies in the endogenous responses of households and housing markets to policy interventions, which unfold gradually and interact simultaneously.

Methodology

The authors propose a dynamic multi-agent framework that links government investment, household sorting, housing prices, population turnover, enrollment, and evolving school quality. The system is modeled as a sequential decision-making problem where the government acts as a planner and households act as agents responding to economic and educational constraints.

1. Environment and Agents

• Households: Modeled as heterogeneous agents with types defined by income, parental education, and child ability. They make residential choices based on a utility function that balances school quality, housing costs, moving costs, and neighborhood amenities. Households enter the school system at age six and remain in their chosen community for a fixed duration.
• Schools: Modeled as education-production components. School quality updates based on government per-student investment and the peer composition (average ability) of enrolled students.
• Housing Market: A market-clearing mechanism determines housing prices. Vacancies are released through population turnover. Prices adjust to equate expected demand (from school-entry movers and in-migrants) with available supply, subject to a price floor.
• Government: The government observes the system state (summarized school quality, occupancy, and demographics) and allocates a fixed annual budget across schools. It cannot directly assign households to schools.

2. Equilibrium Computation (Lower Level)

Household sorting is treated as a convex optimization problem. Given school quality and housing prices, households choose communities to maximize utility. The resulting demand for housing units and the clearing of the housing market (matching demand to vacancies) are characterized as a complementarity system. The authors prove that the sorting equilibrium corresponds to the minimizer of a convex potential function and provide an efficient projected gradient descent algorithm (Algorithm 1) to compute equilibrium prices and choice probabilities.

3. Learning Allocation Policy (Upper Level)

The government's problem is formulated as a Partially Observable Markov Decision Process (POMDP).

• State: The full system state includes household microstates, but the government only observes a summarized state (school qualities, occupancy rates, mean income/ability).
• Action: A budget allocation vector across schools.
• Objective: Maximize a discounted sum of welfare, defined by an Atkinson-style inequality-averse function of accessed school quality, while penalizing abrupt changes in allocation shares.
• Algorithm: The authors use Proximal Policy Optimization (PPO) to learn a policy that maps observations to budget allocations. The policy is parameterized as a Dirichlet distribution to ensure budget constraints are satisfied.

Key Contributions

1. Dynamic Multi-Agent Framework: Development of a computational framework that explicitly captures the education–housing feedback loop, linking government investment, school quality, household sorting, and educational access in a unified dynamic system.
2. Convex Characterization of Sorting: Characterization of household sorting as a convex program and the development of efficient algorithms to compute the resulting equilibrium, enabling the integration of market-clearing dynamics into the RL loop.
3. RL-Based Allocation Approach: An approach that enables a government planner to learn multi-objective investment policies under delayed effects and endogenous household responses, balancing aggregate effectiveness and equity.
4. Comprehensive Simulation and Analysis: Extensive simulations comparing the learned policy against representative baselines (equal-split, enrollment-proportional, and compensatory funding), including sensitivity analyses on key parameters (inequality aversion, housing price sensitivity, choice randomness) and scalability evaluations.

Results

In simulations involving 12 communities and 4 schools:

• Performance: The RL-based policy (PPO) achieved the highest Mean Access (0.4780) and the second-lowest Access Gini Coefficient (0.0164) among all tested baselines.
• Equity-Efficiency Balance: The policy demonstrated a favorable balance, achieving high overall access while maintaining significantly lower inequality compared to equal-split and compensatory funding strategies. Human Capital outcomes were comparable to the best-performing baselines.
• Socioeconomic Stratification: While the policy improved the overall distribution of school quality access, it did not eliminate the association between household income and school quality (Income-Access Gap remained near zero, but Income-Quality correlation persisted). This highlights that equalizing school quality alone may not fully eliminate residential mechanisms that sustain income-based disparities.
• Mechanism Importance: Ablation studies revealed that both student-composition feedback (peer effects) and endogenous housing-price formation are critical. Removing price formation artificially lowered segregation metrics but obscured the channel through which demand for better schools affects residential costs.
• Scalability: Increasing the number of communities significantly reduced average access, indicating that spatial dispersion poses challenges to accessibility. Conversely, increasing the number of schools without adjusting spatial structure tended to increase inequality and segregation, suggesting that simply expanding school quantity does not guarantee equitable outcomes.

Significance and Claims

The paper claims that its framework provides a computational foundation for evaluating educational investments whose effectiveness and distributional consequences evolve through household behavior over time. By making the education–housing feedback explicit, the framework supports long-term analysis of how school investment shapes educational opportunity.

The authors emphasize that their approach allows for the explicit trade-off between effectiveness and equity in policy design. They argue that ignoring the endogenous responses of the housing market and household sorting can lead to suboptimal or inequitable long-term outcomes. The results suggest that while RL-based policies can significantly improve the balance between access and equity, they operate within the constraints of the existing residential sorting system, where income differences continue to influence access patterns. The framework is presented as a tool to support future evaluations where income-based fairness or housing access can be incorporated more explicitly into policy design.