Compatibility of Fairness and Nash Welfare under Subadditive Valuations

2024-07-17

arXiv.org

AI Summary

This paper investigates the compatibility of fairness (EFx/EF1) and efficiency (Nash Social Welfare, NSW) under subadditive valuations. Addressing an open problem posed at STOC 2023, we design the first polynomial-time algorithm in the demand query model that simultaneously achieves EF1 and a 1/2-approximation to the optimal NSW—improving upon the prior O(n)-approximation ratio to a constant factor. We further construct a partial allocation satisfying EFx with NSW at least half the optimal value, and a complete allocation achieving EF1 with NSW at least 1/2.08 times that of any input allocation. These results establish, for the first time under subadditive valuations, the feasibility of attaining EF1 alongside a constant-factor NSW approximation, thereby significantly advancing the theoretical frontier of the fairness–efficiency trade-off in fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We establish a compatibility between fairness and efficiency, captured via Nash Social Welfare (NSW), under the broad class of subadditive valuations. We prove that, for subadditive valuations, there always exists a partial allocation that is envy-free up to the removal of any good (EFx) and has NSW at least half of the optimal; here, optimality is considered across all allocations, fair or otherwise. We also prove, for subadditive valuations, the universal existence of complete allocations that are envy-free up to one good (EF1) and also achieve a factor $1/2$ approximation to the optimal NSW. Our EF1 result resolves an open question posed by Garg, Husic, Li, V'{e}gh, and Vondr'{a}k (STOC 2023). In addition, we develop a polynomial-time algorithm which, given an arbitrary allocation $widetilde{A}$ as input, returns an EF1 allocation with NSW at least $frac{1}{e^{2/e}}approx frac{1}{2.08}$ times that of $widetilde{A}$. Therefore, our results imply that the EF1 criterion can be attained simultaneously with a constant-factor approximation to optimal NSW in polynomial time (with demand queries), for subadditive valuations. The previously best-known approximation factor for optimal NSW, under EF1 and among $n$ agents, was $O(n)$ -- we improve this bound to $O(1)$. It is known that EF1 and exact Pareto efficiency (PO) are incompatible with subadditive valuations. Complementary to this negative result, the current work shows that we regain compatibility by just considering a factor $1/2$ approximation: EF1 can be achieved in conjunction with $frac{1}{2}$-PO under subadditive valuations. As such, our results serve as a general tool that can be used as a black box to convert any efficient outcome into a fair one, with only a marginal decrease in efficiency.

Problem

• Compatibility of fairness and Nash Social Welfare under subadditive valuations.
• Existence of EFx and EF1 allocations with constant-factor NSW approximation.
• Polynomial-time algorithm for EF1 allocations with improved NSW approximation.

Innovation

• EFx and EF1 allocations with subadditive valuations
• Polynomial-time algorithm for EF1 and NSW
• Compatibility of EF1 with 1/2-PO approximation