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Proportional and Pareto-Optimal Allocation of Chores with Subsidy

Jugal Garg, Eklavya Sharma, Xiaowei Wu

TL;DR

This paper addresses fair division of $m$ indivisible chores among $n$ agents with weights summing to 1, targeting proportionality and Pareto-optimality. It introduces a polynomial-time approach that achieves the best-known subsidy bound of $\frac{n}{3}-\frac{1}{6}$ while guaranteeing Pareto-optimality by first computing a proportionally-fair competitive equilibrium and then applying a low-cost rounding guided by a reduction to identical disutilities. The core novelty lies in preserving market-equilibrium structure to obtain a PROP-fPO fractional allocation with an acyclic consumption graph, followed by a decomposition-based rounding that yields a PO allocation with the same subsidy bound. The method is notably simpler than prior work and provides a practical algorithm for exact fairness with monetary transfers in chore allocations, with potential extensions to mixtures of divisible/indivisible items and other fairness notions.

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.

Proportional and Pareto-Optimal Allocation of Chores with Subsidy

TL;DR

This paper addresses fair division of indivisible chores among agents with weights summing to 1, targeting proportionality and Pareto-optimality. It introduces a polynomial-time approach that achieves the best-known subsidy bound of while guaranteeing Pareto-optimality by first computing a proportionally-fair competitive equilibrium and then applying a low-cost rounding guided by a reduction to identical disutilities. The core novelty lies in preserving market-equilibrium structure to obtain a PROP-fPO fractional allocation with an acyclic consumption graph, followed by a decomposition-based rounding that yields a PO allocation with the same subsidy bound. The method is notably simpler than prior work and provides a practical algorithm for exact fairness with monetary transfers in chore allocations, with potential extensions to mixtures of divisible/indivisible items and other fairness notions.

Abstract

We consider the problem of allocating indivisible chores among 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 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.

Paper Structure

This paper contains 20 sections, 15 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. PROP-Subsidy for Goods.
  5. EF-Subsidy.
  6. Relaxations of EF and PROP
  7. Mixed Divisible and Indivisible Items.
  8. Preliminaries
  9. Fair Division Instances and Allocations
  10. Fairness and Subsidy
  11. Lower bound on subsidy:
  12. Pareto-Optimality and Market Equilibria
  13. Technical Overview
  14. An Acyclic Proportional Market Equilibrium
  15. Low-Cost Rounding
  16. ...and 5 more sections

Key Result

Theorem 1

Let $(x, p)$ be a market equilibrium for the instance $\mathcal{I} := ([n], [m], (d_i)_{i=1}^n, (w_i)_{i=1}^n)$. Then $x$ is an fPO allocation.

Figures (2)

  • Figure 1: Decomposing a tree into 3 subtrees based on the partition $(\textcolor{textRed}{\{c_1\}}, \textcolor{textGreen}{\{c_2\}}, \textcolor{textBlue}{\{c_3, c_4\}})$ of the chores, and then rounding the subtrees independently. Circles represent agents and squares represent chores.
  • Figure 2: Types of trees. Circles are agents and squares are chores.

Theorems & Definitions (33)

  • Example 1
  • Example 2
  • Definition 1: Pareto-optimality
  • Definition 2: Market equilibrium
  • Theorem 1: First Welfare Theorem bogomolnaia2017competitive
  • proof
  • Theorem 2
  • proof : Proof sketch
  • Theorem 3
  • Example 3
  • ...and 23 more