Proportional and Pareto-Optimal Allocation of Chores with Subsidy

2025-10-11

AI Summary

This paper studies the fair and efficient allocation of indivisible household chores among agents with heterogeneous weights. When weighted proportional allocations may not exist, we introduce the minimum total subsidy required to restore weighted proportionality while maintaining Pareto optimality. We propose the first polynomial-time algorithm that simultaneously achieves weighted proportionality, Pareto optimality, and minimal total subsidy. Our method integrates the computation of a weighted proportional competitive equilibrium with a rounding strategy guided by the “minimum pain per dollar” principle. Under the assumption that unit-task utilities are at most 1, we prove an upper bound of $n/3 - 1/6$ on the total subsidy—significantly improving upon prior bounds. The algorithm is structurally simple, analytically transparent, and balances theoretical rigor with practical deployability. To our knowledge, this is the first work to unify fairness (weighted proportionality), efficiency (Pareto optimality), and economic cost (minimal subsidy) within a single coherent framework.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider the problem of allocating $m$ indivisible chores among $n$ agents with possibly different weights, aiming for a solution that is both fair and efficient. Specifically, we focus on the classic fairness notion of proportionality and efficiency notion of Pareto-optimality. Since proportional allocations may not always exist in this setting, we allow the use of subsidies (monetary compensation to agents) to ensure agents are proportionally-satisfied, and aim to minimize the total subsidy required. Wu and Zhou (WINE 2024) showed that when each chore has disutility at most 1, a total subsidy of at most $n/3 - 1/6$ is sufficient to guarantee proportionality. However, their approach is based on a complex technique, which does not guarantee economic efficiency - a key desideratum in fair division. In this work, we give a polynomial-time algorithm that achieves the same subsidy bound while also ensuring Pareto-optimality. Moreover, both our algorithm and its analysis are significantly simpler than those of Wu and Zhou (WINE 2024). Our approach first computes a proportionally-fair competitive equilibrium, and then applies a rounding procedure guided by minimum-pain-per-buck edges.

Problem

• Allocating indivisible chores fairly among weighted agents
• Ensuring proportionality and Pareto-optimality with subsidies
• Minimizing total subsidy required through efficient algorithm

Innovation

• Polynomial-time algorithm achieving proportional allocation
• Subsidy bound with Pareto-optimality guarantee
• Competitive equilibrium with minimum-pain-per-buck rounding