Finding Fair and Efficient Allocations (1707.04731v2)

Abstract: We study the problem of allocating a set of indivisible goods among a set of agents in a fair and efficient manner. An allocation is said to be fair if it is envy-free up to one good (EF1), which means that each agent prefers its own bundle over the bundle of any other agent up to the removal of one good. In addition, an allocation is deemed efficient if it satisfies Pareto optimality (PO). While each of these well-studied properties is easy to achieve separately, achieving them together is far from obvious. Recently, Caragiannis et al. (2016) established the surprising result that when agents have additive valuations for the goods, there always exists an allocation that simultaneously satisfies these two seemingly incompatible properties. Specifically, they showed that an allocation that maximizes the Nash social welfare (NSW) objective is both EF1 and PO. However, the problem of maximizing NSW is NP-hard. As a result, this approach does not provide an efficient algorithm for finding a fair and efficient allocation. In this paper, we bypass this barrier, and develop a pseudopolynomial time algorithm for finding allocations that are EF1 and PO; in particular, when the valuations are bounded, our algorithm finds such an allocation in polynomial time. Furthermore, we establish a stronger existence result compared to Caragiannis et al. (2016): For additive valuations, there always exists an allocation that is EF1 and fractionally PO. Another contribution of our work is to show that our algorithm provides a polynomial-time 1.45-approximation to the NSW objective. This improves upon the best known approximation ratio for this problem (namely, the 2-approximation algorithm of Cole et al. (2017)). Unlike many of the existing approaches, our algorithm is completely combinatorial.

• The paper introduces a pseudopolynomial time algorithm that finds allocations satisfying both EF1 and Pareto Efficiency (PO) for additive valuations.
• It proves the existence of allocations that are both EF1 and fractionally Pareto Efficient, reinforcing foundational theory in fair division.
• A combinatorial approach is applied to yield a 1.45-approximation for maximizing Nash Social Welfare while ensuring fairness and efficiency.

An Algorithmic Study on Fair and Efficient Allocations

The paper "Finding Fair and Efficient Allocations" addresses a significant problem in the field of fair division, focusing on the allocation of indivisible goods among agents with the dual aim of fairness and efficiency. Specifically, it seeks to concurrently achieve Envy-Freeness up to one good (EF1) and Pareto Efficiency (PO) in allocations when agents have additive valuation functions.

Core Contributions

This research presents several key contributions that advance the theoretical and practical understanding of fair division problems:

1. Pseudopolynomial Time Algorithm for EF1 and PO Allocations: The authors develop a novel pseudopolynomial time algorithm to find allocations that are both EF1 and PO. When valuations are bounded, this algorithm operates in polynomial time. This circumvents the complexity barrier posed by the NP-hard nature of maximizing Nash Social Welfare (NSW), which emerges as both EF1 and PO but lacks an efficient computation method.
2. Existence of EF1 and Fractionally Pareto Efficient Allocations: Beyond merely providing an algorithm, the paper strengthens theoretical foundations by proving that EF1 and fractionally Pareto efficient (fPO) allocations exist for additive valuations. This is a stronger claim than earlier results and suggests alternative pathways towards addressing similar problem structures in theoretical computer science and algorithm design.
3. Improved Approximation of Nash Social Welfare: The research also creates a combinatorial approach resulting in a polynomial-time 1.45-approximation algorithm for the NSW maximization problem. This represents an improvement over previous approximation ratios, matching the integrality gap lower bound established in prior literature while also ensuring outcomes are fair and efficient.

Methodological Highlights

An innovative aspect of this paper is the use of integral Fisher markets and price envy-freeness, which are leveraged during the algorithmic process. The algorithm maintains equilibrium conditions to ensure Pareto efficiency at each step, rigorously ironing out price envy to achieve the desired fairness criterion. This methodology distinguishes itself from previous relax-and-round processes, offering a purely combinatorial strategy that could inspire future algorithmic solutions in similar contexts.

Implications and Future Directions

Practically, these developments provide more feasible approaches to real-world fair division applications, such as resource allocation in auctions or course assignments in educational institutions. Theoretically, they suggest new complexity classes for exploring total functions in NP-like conditions, expanding the tools available to researchers facing similar challenges in other allocation and optimization problems.

While the paper has succeeded in providing substantial advancements, questions remain about broader applicability across varied valuation models beyond additivity, such as submodular or non-linear valuations. Future work might explore whether these algorithms and existence results can be adapted or extended into these more general valuation settings. Additionally, the quest for strongly polynomial algorithms for achieving EF1 and PO allocations persists as an intriguing challenge for subsequent investigations.

Overall, this paper makes significant strides in the intersection of fair division, algorithm design, and economic theory, providing both a detailed theoretical foundation and practical algorithmic solutions that promise to impact the field extensively.