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A Polynomial-Time Algorithm for Fair and Efficient Allocation with a Fixed Number of Agents

Ryoga Mahara

TL;DR

A polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed is proposed.

Abstract

We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuation functions. Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods, while Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely recognized efficiency criteria. Although each property is straightforward to achieve individually, simultaneously ensuring both fairness and efficiency is challenging. Caragiannis et al.~\cite{caragiannis2019unreasonable} established the surprising result that maximizing Nash social welfare yields an allocation that is both EF1 and PO; however, since maximizing Nash social welfare is NP-hard, this approach does not provide an efficient algorithm. To overcome this barrier, Barman, Krishnamurthy, and Vaish~\cite{barman2018finding} designed a pseudo-polynomial time algorithm to compute an EF1 and PO allocation, and showed the existence of EF1 and fPO allocations. Nevertheless, the latter existence proof relies on a non-constructive convergence argument and does not directly yield an efficient algorithm for finding EF1 and fPO allocations. Whether a polynomial-time algorithm exists for finding an EF1 and PO (or fPO) allocation remains an important open problem. In this paper, we propose a polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed. Our primary idea is to avoid processing the entire instance at once; instead, we sequentially add agents to the instance and construct an allocation that satisfies EF1 and fPO at each step.

A Polynomial-Time Algorithm for Fair and Efficient Allocation with a Fixed Number of Agents

TL;DR

A polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed is proposed.

Abstract

We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuation functions. Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods, while Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely recognized efficiency criteria. Although each property is straightforward to achieve individually, simultaneously ensuring both fairness and efficiency is challenging. Caragiannis et al.~\cite{caragiannis2019unreasonable} established the surprising result that maximizing Nash social welfare yields an allocation that is both EF1 and PO; however, since maximizing Nash social welfare is NP-hard, this approach does not provide an efficient algorithm. To overcome this barrier, Barman, Krishnamurthy, and Vaish~\cite{barman2018finding} designed a pseudo-polynomial time algorithm to compute an EF1 and PO allocation, and showed the existence of EF1 and fPO allocations. Nevertheless, the latter existence proof relies on a non-constructive convergence argument and does not directly yield an efficient algorithm for finding EF1 and fPO allocations. Whether a polynomial-time algorithm exists for finding an EF1 and PO (or fPO) allocation remains an important open problem. In this paper, we propose a polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed. Our primary idea is to avoid processing the entire instance at once; instead, we sequentially add agents to the instance and construct an allocation that satisfies EF1 and fPO at each step.

Paper Structure

This paper contains 27 sections, 12 theorems, 12 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Fair and efficient allocation for indivisible goods
  4. Fair and efficient allocation for indivisible chores
  5. Approximating Nash social welfare
  6. Our Contributions
  7. Our Techniques
  8. Organization
  9. Preliminaries
  10. The fair division model
  11. Fairness notions
  12. Efficiency notions
  13. Nash social welfare
  14. Fisher market
  15. Price envy-freeness, minimum spenders, maximum violators
  16. ...and 12 more sections

Key Result

Theorem 1.1

When each agent has an additive valuation function and the number of agents is fixed, an EF1 and fPO allocation can be computed in polynomial time.

Figures (2)

  • Figure 1: MBB graph $G(\mathbf{p})$ and augmented MBB graph $G(\mathbf{x}, \mathbf{p})$. In $G(\mathbf{x},\mathbf{p})$, the black edges represent MBB edges, while the gray edges represent allocation edges.
  • Figure 2: Illustration of the Transfer algorithm (Algorithm \ref{['alg:3']}). The left figure shows the state before applying the Transfer algorithm, and the right figure shows the state after applying it. The black edges represent MBB edges, while the gray edges represent allocation edges in the augmented MBB graph $G(\mathbf{x}, \mathbf{p})$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Example 2.3
  • Lemma 2.4
  • Remark 3.1
  • proof
  • Lemma 3.3
  • proof
  • ...and 13 more