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Constrained Fair and Efficient Allocations

Benjamin Cookson, Soroush Ebadian, Nisarg Shah

Abstract

Fairness and efficiency have become the pillars of modern fair division research, but prior work on achieving both simultaneously is largely limited to the unconstrained setting. We study fair and efficient allocations of indivisible goods under additive valuations and various types of allocation feasibility constraints, and demonstrate the unreasonable effectiveness of the maximum Nash welfare (MNW) solution in this previously uncharted territory. Our main result is that MNW allocations are 1/2-envy-free up to one good (EF1) and Pareto optimal under the broad family of (arbitrary) matroid constraints. We extend these guarantees to complete MNW allocations for base-orderable matroid constraints, and to a family of non-matroidal constraints (which includes balancedness) using a novel "alternate worlds" technique. We establish tightness of our results by providing counterexamples for the satisfiability of certain stronger desiderata, but show an improved result for the special case of goods with copies (Gafni et al. 2023). Finally, we also establish novel best-of-both-worlds guarantees for goods with copies and balancedness.

Constrained Fair and Efficient Allocations

Abstract

Fairness and efficiency have become the pillars of modern fair division research, but prior work on achieving both simultaneously is largely limited to the unconstrained setting. We study fair and efficient allocations of indivisible goods under additive valuations and various types of allocation feasibility constraints, and demonstrate the unreasonable effectiveness of the maximum Nash welfare (MNW) solution in this previously uncharted territory. Our main result is that MNW allocations are 1/2-envy-free up to one good (EF1) and Pareto optimal under the broad family of (arbitrary) matroid constraints. We extend these guarantees to complete MNW allocations for base-orderable matroid constraints, and to a family of non-matroidal constraints (which includes balancedness) using a novel "alternate worlds" technique. We establish tightness of our results by providing counterexamples for the satisfiability of certain stronger desiderata, but show an improved result for the special case of goods with copies (Gafni et al. 2023). Finally, we also establish novel best-of-both-worlds guarantees for goods with copies and balancedness.

Paper Structure

This paper contains 32 sections, 20 theorems, 21 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Feasibility Constraints
  5. Matroid constraints.
  6. Non-matroidal constraints.
  7. Maximum Nash Welfare
  8. Fairness and Efficiency Desiderata
  9. The Unreasonable Fairness of (Constrained) Maximum Nash Welfare
  10. Beyond Matroidal Constraints
  11. Implications
  12. Implication for partition matroids.
  13. Implications for balancedness.
  14. Implication for goods with copies.
  15. Search for EF1+PO allocations.
  16. ...and 17 more sections

Key Result

Theorem 1

Under any matroid constraint, every MNW allocation is 1/2-EF1 and PO.

Theorems & Definitions (50)

  • Definition 1: (Constrained) Maximum Nash Welfare (MNW)
  • Definition 2: Approximate envy-freeness up to one good (EF1)
  • Definition 3: Approximate (constrained) Pareto optimality (PO)
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • ...and 40 more