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On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations

Ziheng Chen, Bo Li, Zihan Luo, Jialin Zhang

TL;DR

The paper addresses fair allocation of $m$ indivisible goods among $n$ agents with binary XOS valuations, focusing on APS and WMMS under asymmetric entitlements. It proves the existence of a $ frac{1}{2}$-APS allocation and provides a polynomial-time algorithm to compute it, establishing tightness with known upper bounds. It also shows that a $ frac{1}{n}$-WMMS allocation exists for general XOS valuations, proves a matching upper bound showing the bound is tight, and gives an exact WMMS algorithm for binary additive valuations. Together, these results reveal a clear APS–WMMS landscape under asymmetry and establish tight benchmarks for binary XOS valuations, with practical implications for fair division among agents with weighted entitlements.

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.

On the Fair Allocation to Asymmetric Agents with Binary XOS Valuations

TL;DR

The paper addresses fair allocation of indivisible goods among agents with binary XOS valuations, focusing on APS and WMMS under asymmetric entitlements. It proves the existence of a -APS allocation and provides a polynomial-time algorithm to compute it, establishing tightness with known upper bounds. It also shows that a -WMMS allocation exists for general XOS valuations, proves a matching upper bound showing the bound is tight, and gives an exact WMMS algorithm for binary additive valuations. Together, these results reveal a clear APS–WMMS landscape under asymmetry and establish tight benchmarks for binary XOS valuations, with practical implications for fair division among agents with weighted entitlements.

Abstract

We study the problem of allocating indivisible goods among 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 -APS allocation can be found in polynomial time, and there exists an instance where no allocation is better than -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 for general XOS valuations. In this work, we focus on the asymmetric setting with binary XOS valuations, and further improve the approximation ratio to , 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 -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 -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.
Paper Structure (14 sections, 10 theorems, 38 equations, 1 table, 5 algorithms)

This paper contains 14 sections, 10 theorems, 38 equations, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Fairness Notions
  6. Valuation Functions
  7. APS with Binary XOS Valuations
  8. The Existence of $\frac{1}{2}$-APS Allocations
  9. A Polynomial-time Algorithm
  10. WMMS with General XOS Valuations
  11. Computing $\frac{1}{n}$-WMMS Allocations
  12. Upper Bound
  13. Binary Additive Valuations
  14. Conclusion Remarks

Key Result

Lemma 1

For some agent $a_i$ in a fair allocation instance $(N,M,(v_i)_{i \in [n]}, (b_i)_{i\in [n]})$ with binary XOS valuations, suppose $S\subseteq M$ is a subset of goods, then there exists a non-wasteful bundle $B\subseteq M\setminus S$, satisfying $v_i(B) = |B| = \mathsf{APS}_i - \lfloor b_i|S|\rfloor

Theorems & Definitions (26)

  • Definition 1: Maximin Share
  • Definition 2: Weighted Maximin Share
  • Definition 3: WMMS Partition
  • Definition 4: AnyPrice Share
  • Definition 5: Additive Function
  • Definition 6: XOS Function
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 16 more