On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations

Ziheng Chen; Bo Li; Zihan Luo; Jialin Zhang

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arXiv · 2601.09299

On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations

Ziheng Chen·Bo Li·Zihan Luo·Jialin Zhang

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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.

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BibTeX RIS

Ziheng Chen, Bo Li, Zihan Luo, Jialin Zhang. 2026-01-14. On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations. https://arxiv.org/abs/2601.09299

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