Reflections on the APD iBudget Algorithm Study

In a public budgeting system, an algorithm is never just a formula. It is a bridge between law, data, institutions, and human lives.

I am pleased to share the public release of the Florida Agency for Persons with Disabilities iBudget Algorithm Study, together with the scientific report behind it. The study examines Florida’s iBudget system, a Medicaid Home and Community-Based Services waiver mechanism that allocates individualized budgets for Floridians with developmental disabilities.

The dream of many mathematicians is to receive 𝘵𝘩𝘦 phone call: “𝘞𝘦 𝘯𝘦𝘦𝘥 𝘺𝘰𝘶 𝘵𝘰 𝘱𝘳𝘰𝘥𝘶𝘤𝘦 𝘢 𝘮𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘤𝘢𝘭 𝘮𝘰𝘥𝘦𝘭… ” It happened to me and it turned out to be a fascinating data science project in algorithmic government. The goal: To optimize a $2.2B legislative appropriation of the State of Florida to serve the most vulnerable in society.

The phrase “allocation algorithm” may sound administrative. It is not. In this setting, a mathematical model helps determine the financial resources available for services and supports. Those resources affect how people live, what care they can access, how families plan, and how public agencies translate legislative appropriations into individualized decisions.

That is what makes this work a case study in algorithmic government.

The problem

The existing iBudget algorithm was built around a regression model developed about a decade ago. That model was important historically. It provided a standardized method for allocating budgets and supported a large public system over many years.

But statistical models age. Populations change. Service costs change. Regulations change. Data systems change. When a model trained on older patterns is used in a new environment, its coefficients may no longer represent the relationships they were intended to capture. In a person-centered planning system, that is not a technical inconvenience. It is a governance problem.

The legislative mandate created an opportunity to ask a direct mathematical question: Can we design an allocation architecture that is more accurate, more transparent, more inclusive of the full population, and better aligned with person-centered planning?

The architecture

The study develops a two-stage approach.

The first stage is prediction. A Random Forest model is trained on recent assessment and expenditure data to estimate individualized service needs. This model is designed to retain full client inclusion rather than removing high-need cases as statistical outliers. That choice matters. In health and disability services, extreme observations are often not errors. They are people with complex support needs.

The model is paired with explainability tools, especially SHAP values, so that predictions can be decomposed into variable-level contributions. In a high-stakes public system, accuracy alone is insufficient. The model must also support explanation, auditability, and review.

The second stage is allocation. Once predicted needs are estimated, the problem becomes one of constrained optimization: how should a legislative appropriation be distributed subject to statutory floors, program rules, and fairness constraints?

The report analyzes both a linear programming formulation and a nonlinear CRRA formulation. The linear program makes the constraint structure transparent and connects naturally to KKT conditions. The nonlinear formulation allows the allocation to reflect diminishing marginal benefit and personalized welfare tradeoffs. In that setting, dual variables become more than mathematical artifacts. They act as shadow prices that quantify the welfare cost of underfunding.

This is one of the reasons I find the project mathematically compelling. The model does not merely produce numbers. It exposes tradeoffs. It makes constraints visible. It clarifies what is being optimized, what is being protected, and what remains scarce.

The role of LLMs

A distinctive feature of this work was the use of large language models as active collaborators throughout development.

LLMs helped accelerate Python coding, LaTeX report generation, proof specification, documentation, and implementation checks. They compressed the distance between mathematical formulation and working computational artifacts. They made it easier to move quickly among statistical modeling, optimization, regulatory language, and written explanation.

But the project also made clear what LLMs cannot replace.

They do not replace human validation. They do not replace domain expertise. They do not replace legal review, stakeholder engagement, ethical judgment, or accountability. In a system that affects thousands of people, no model output is self-justifying. Every computational claim must be inspected, tested, documented, and situated within the institutional process that gives it meaning.

The larger lesson

Mathematics in government is not only about prediction. It is about public reasoning under constraints.

A budget allocation system must contend with limited resources, statutory requirements, heterogeneous needs, imperfect data, and the moral seriousness of individual lives. The value of mathematics is not that it eliminates hard choices. It is that it can make those choices explicit, testable, and open to scrutiny.

That is the central lesson I take from this study.

Algorithmic government should not mean black-box automation. It should mean accountable mathematical infrastructure: models that are validated, constraints that are visible, explanations that are available, and decisions that remain subject to human responsibility.

Reports

Executive Report

Scientific report