Proportional Fairness for Harmful Decisions

2026-07-28

AI Summary

This study addresses the lack of suitable proportionality-based fairness criteria for the equitable allocation of divisible public bads—such as pollution or risk—where traditional core concepts fail. The work proposes a novel proportional fairness framework for randomized allocations of public bads, introducing two new core notions and leveraging Lindahl equilibria alongside a public-good-to-bad transformation technique. Under certain structural conditions, it demonstrates that zero-respecting Lindahl equilibria satisfy the proposed core definitions and achieve stronger fairness guarantees. In general settings, however, an impossibility result arises; nevertheless, partial core guarantees can be recovered by reducing the problem to a public good allocation setting. This research establishes a foundational theoretical link between fairness and efficiency in the allocation of public bads.

Abstract

We study allocation of (divisible) public bads, where agents incur costs for alternatives and the goal is to pick a lottery over the alternatives. We show that the traditional definitions of the core, a central criterion of proportional representation for allocation of public goods, private goods, and private bads (chores), do not make sense for allocation of public bads. We introduce two formalizations of the core tailored to public bads. Under a structural condition which subsumes allocation of private bads, we show that zero-respecting Lindahl equilibria satisfy both formalizations, exhibit additional fairness guarantees, and strictly generalize competitive equilibria from equal incomes (CEEI) for allocation of private bads. Without this structural condition, we show that Lindahl equilibria exhibit undesirable behaviors, prove sharp impossibility results separating public bads from public goods, but show that a rule using a reduction to public goods recovers one of our formalizations of the core. Our results lay the groundwork for studying fair allocation of public bads, an overlooked yet fundamental problem, and highlight several structural and algorithmic directions that remain open.

Problem

• public bads
• proportional fairness
• core
• allocation
• Lindahl equilibrium

Innovation

• public bads
• proportional fairness
• Lindahl equilibrium
• core
• CEEI