On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations

2026-01-14

AI Summary

This study addresses the problem of fair allocation of indivisible goods among asymmetric agents with binary XOS valuations, focusing on the AnyPrice Share (APS) and Weighted Maximin Share (WMMS) fairness criteria. By leveraging structural properties of XOS valuations and combinatorial optimization techniques, the authors design a polynomial-time algorithm that improves the APS approximation ratio from 1/6 to 1/2, matching the known upper bound. Furthermore, they extend the existence guarantee for WMMS allocations from additive to general binary XOS valuations, proving that a 1/n-WMMS allocation always exists and is tight. The work also uncovers a fundamental distinction between binary XOS and binary additive valuations regarding the attainability of WMMS fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of allocating $m$ indivisible goods among $n$ agents, where each agent's valuation is fractionally subadditive (XOS). With respect to AnyPrice Share (APS) fairness, Kulkarni et al. (2024) showed that, when agents have binary marginal values, a $0.1222$-APS allocation can be found in polynomial time, and there exists an instance where no allocation is better than $0.5$-approximate APS. Very recently, Feige and Grinberg (2025) extended the problem to the asymmetric case, where agents may have different entitlements, and improved the approximation ratio to $1/6$ for general XOS valuations. In this work, we focus on the asymmetric setting with binary XOS valuations, and further improve the approximation ratio to $1/2$, which matches the known upper bound. We also present a polynomial-time algorithm to compute such an allocation. Beyond APS fairness, we also study the weighted maximin share (WMMS) fairness. Farhadi et al. (2019) showed that, a $1/n$-WMMS allocation always exists for agents with general additive valuations, and that this approximation ratio is tight. We extend this result to general XOS valuations, where a $1/n$-WMMS allocation still exists, and this approximation ratio cannot be improved even when marginal values are binary. This shows a sharp contrast to binary additive valuations, where an exact WMMS allocation exists and can be found in polynomial time.

Problem

• fair allocation
• asymmetric agents
• binary XOS valuations
• AnyPrice Share
• weighted maximin share

Innovation

• asymmetric agents
• binary XOS valuations
• AnyPrice Share (APS)
• weighted maximin share (WMMS)
• fair allocation