On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations

TL;DR

This paper improves the approximation ratio for fair allocation of indivisible goods among agents with binary XOS valuations in asymmetric settings, matching the known upper bound, and extends WMMS fairness results to XOS valuations.

Contribution

It introduces a polynomial-time algorithm achieving a 1/2-approximate APS fairness for asymmetric binary XOS valuations, matching the upper bound, and extends WMMS fairness existence to XOS valuations.

Findings

Achieved a 1/2-approximate APS allocation for asymmetric binary XOS valuations.

Extended the existence of 1/n-WMMS allocations to general XOS valuations.

Provided polynomial-time algorithms for fair allocation under these conditions.

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.