# Fair allocation strategies for opioid settlements

## TL;DR

This paper proposes a fair way to distribute opioid settlement funds to counties in the U.S., using mathematical models to balance fairness and interpretability.

## Contribution

The paper introduces two fairness measures (deviation and maximum regret) and formulates fair allocation as convex optimization problems.

## Key findings

- Rural, low-income counties with poor health factors face lower fairness in allocation strategies.
- Interpretability constraints increase maximum regret more than deviation.
- The deviation-regret frontier shows the trade-off between the two fairness measures.

## Abstract

Multi-billion-dollar opioid settlement agreements have been reached with pharmaceutical manufacturers and distributors to address their liability in contributing to the opioid epidemic in the United States. These agreements stipulate that within the state, the settlement funds must be directly allocated to local government (e.g., counties) and used for abatement activities to remediate the harm of the opioid epidemic in communities. This naturally leads to an important question of how the funds should be distributed to meet the diverse needs of the counties consistently across all counties to be deemed fair. Although there exist various definitions of fairness in the literature, it remains unclear how to empirically quantify the fairness of settlement allocation based on data, which is crucial for developing evidence-based allocation policies. To fill this gap, we define two allocation fairness measures, deviation and maximum regret, and formulate the fair settlement allocation as convex optimization problems. To further enhance the interpretability of the allocation policies, we restrict the allocation to a weighted sum of the given empirical metrics. We apply our analytical framework in a case study of the settlement allocation in Pennsylvania using real-world empirical metrics. We identify the frontiers of the non-dominated allocation policies between min-deviation and minimax-regret allocations, which dominate all alpha fairness-based and formula-based allocation policies. All allocation policies show lower fairness (with higher deviation or maximum regret) in counties that are rural, low-income, and with lower-ranking health factors. The price of interpretability is more significant in terms of maximum regret compared with deviation.

The online version contains supplementary material available at 10.1007/s10729-025-09716-8.

We develop an analytical framework for fair allocation of the opioid settlement, which defines two context-specific allocation fairness measures, namely, deviation and maximum regret, and formulates the fair allocation decision questions as convex optimization problems.We present the deviation-regret frontier showing the trade-off between the two fairness measures and the boundary beyond which the two fairness measures cannot be further improved simultaneously by any other feasible allocation strategy.We examine the disparities in allocation fairness across counties by several social determinants of health, and find that counties in rural areas, with low income, and with lower-ranking health factors may be disadvantaged with lower allocation fairness measures across all allocation strategies.Incorporating interpretability constraints affects the maximum regret more than the deviation and moves the frontier further away from the origin, demonstrating the trade-off between interpretability and fairness when choosing allocation strategies.

We develop an analytical framework for fair allocation of the opioid settlement, which defines two context-specific allocation fairness measures, namely, deviation and maximum regret, and formulates the fair allocation decision questions as convex optimization problems.

We present the deviation-regret frontier showing the trade-off between the two fairness measures and the boundary beyond which the two fairness measures cannot be further improved simultaneously by any other feasible allocation strategy.

We examine the disparities in allocation fairness across counties by several social determinants of health, and find that counties in rural areas, with low income, and with lower-ranking health factors may be disadvantaged with lower allocation fairness measures across all allocation strategies.

Incorporating interpretability constraints affects the maximum regret more than the deviation and moves the frontier further away from the origin, demonstrating the trade-off between interpretability and fairness when choosing allocation strategies.

The online version contains supplementary material available at 10.1007/s10729-025-09716-8.