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Existence of Fair and Efficient Allocation of Indivisible Chores

Ryoga Mahara

TL;DR

This paper resolves a central open question in fair division by proving the existence of EF1 and PO allocations for indivisible chores under additive cost functions, and also showing EF1 with fPO. The authors introduce a novel combination of a fixed-point (KKM) argument with discrete LP-based algorithms; they perturb costs to guarantee non-degeneracy, use LP duality to characterize fPO via a weighted social cost, and employ a price-envy framework with a FindpEF1 procedure to obtain EF1-like guarantees. They further show that EF1+PO allocations can be computed in polynomial time when the number of agents is constant and extend the framework to weighted EF1 (wEF1), preserving PO and fPO properties. These results advance both theory and methodology in fair division, opening doors to efficient algorithms under additive costs and weighted entitlements, with potential extensions to constraints and broader valuation classes.

Abstract

We study the problem of allocating indivisible chores among agents with additive cost functions in a fair and efficient manner. A major open question in this area is whether there always exists an allocation that is envy-free up to one chore (EF1) and Pareto optimal (PO). Our main contribution is to provide a positive answer to this question by proving the existence of such an allocation for indivisible chores under additive cost functions. This is achieved by a novel combination of a fixed point argument and a discrete algorithm, providing a significant methodological advance in this area. Our additional key contributions are as follows. We show that there always exists an allocation that is EF1 and fractional Pareto optimal (fPO), where fPO is a stronger efficiency concept than PO. We also show that an EF1 and PO allocation can be computed in polynomial time when the number of agents is constant. Finally, we extend all of these results to the more general setting of weighted EF1 (wEF1), which accounts for the entitlements of agents.

Existence of Fair and Efficient Allocation of Indivisible Chores

TL;DR

This paper resolves a central open question in fair division by proving the existence of EF1 and PO allocations for indivisible chores under additive cost functions, and also showing EF1 with fPO. The authors introduce a novel combination of a fixed-point (KKM) argument with discrete LP-based algorithms; they perturb costs to guarantee non-degeneracy, use LP duality to characterize fPO via a weighted social cost, and employ a price-envy framework with a FindpEF1 procedure to obtain EF1-like guarantees. They further show that EF1+PO allocations can be computed in polynomial time when the number of agents is constant and extend the framework to weighted EF1 (wEF1), preserving PO and fPO properties. These results advance both theory and methodology in fair division, opening doors to efficient algorithms under additive costs and weighted entitlements, with potential extensions to constraints and broader valuation classes.

Abstract

We study the problem of allocating indivisible chores among agents with additive cost functions in a fair and efficient manner. A major open question in this area is whether there always exists an allocation that is envy-free up to one chore (EF1) and Pareto optimal (PO). Our main contribution is to provide a positive answer to this question by proving the existence of such an allocation for indivisible chores under additive cost functions. This is achieved by a novel combination of a fixed point argument and a discrete algorithm, providing a significant methodological advance in this area. Our additional key contributions are as follows. We show that there always exists an allocation that is EF1 and fractional Pareto optimal (fPO), where fPO is a stronger efficiency concept than PO. We also show that an EF1 and PO allocation can be computed in polynomial time when the number of agents is constant. Finally, we extend all of these results to the more general setting of weighted EF1 (wEF1), which accounts for the entitlements of agents.

Paper Structure

This paper contains 32 sections, 33 theorems, 52 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Fair and efficient allocation for indivisible goods
  3. Fair and efficient allocation for indivisible chores
  4. Our Contributions
  5. Our techniques
  6. Novelty and significance
  7. Related Work
  8. Fair and efficient allocation for divisible items
  9. Fair and efficient allocation for indivisible items
  10. Fair and efficient allocation with constraints
  11. Weighted fair division
  12. Organization
  13. Preliminaries
  14. The fair division model for indivisible chores
  15. Fairness notions
  16. ...and 17 more sections

Key Result

Theorem 1.1

When each agent has an additive cost function, an EF1 and PO allocation always exists.

Figures (2)

  • Figure 1: A bipartite graph $G$ containing a cycle that prevents the existence of a pEF1 allocation.
  • Figure 2: Illustration of the FindpEF1 algorithm (Algorithm \ref{['alg:1']}). The left figure shows the state before applying a transfer operation, and the right figure shows the state after applying it. The black edges represent edges in $H$, while the gray edges represent allocation edges.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Lemma 2.1 in branzei2024algorithms
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Hall's theorem hall1935representatives
  • ...and 52 more