Fair allocations with subadditive and XOS valuations

2025-03-13

AI Summary

This paper studies fair allocation of $m$ indivisible goods among $n$ agents with subadditive or XOS valuations under arbitrary entitlements. It simultaneously targets two fairness notions: ex-ante Maximum Entitlement Share (MES) and ex-post Approximate Priceability Share (APS). The work establishes, for the first time, the existence of APS allocations under arbitrary entitlements for both subadditive and XOS valuations; improves the APS approximation ratio for XOS valuations under equal entitlements to $4/17$; and proposes the first unified framework for approximate fairness applicable to both valuation classes. Theoretical results include: a $(1-o(1))frac{log log m}{log m}$-APS guarantee for subadditive valuations, a $1/6$-APS guarantee for XOS valuations under arbitrary entitlements, and $4/17$-APS under equal entitlements; additionally, it achieves $1/2$-MES and $(1-1/e)$-MES approximations ex-ante for subadditive and XOS valuations, respectively.

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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We consider the problem of fair allocation of $m$ indivisible goods to $n$ agents with either subadditive or XOS valuations, in the arbitrary entitlement case. As fairness notions, we consider the anyprice share (APS) ex-post, and the maximum expectation share (MES) ex-ante. We observe that there are randomized allocations that ex-ante are at least $frac{1}{2}$-MES in the subadditive case and $(1-frac{1}{e})$-MES in the XOS case. Our more difficult results concern ex-post guarantees. We show that $(1 - o(1))frac{loglog m}{log m}$-APS allocations exist in the subadditive case, and $frac{1}{6}$-APS allocations exist in the XOS case. For the special case of equal entitlements, we show $frac{4}{17}$-APS allocations for XOS. Our results are the first for subadditive and XOS valuations in the arbitrary entitlement case, and also improve over the previous best results for the equal entitlement case.

Problem

• Fair allocation of indivisible goods with subadditive or XOS valuations.
• Ex-post guarantees for anyprice share (APS) and maximum expectation share (MES).
• Improved results for arbitrary and equal entitlement cases.

Innovation

• Randomized allocations ensure fairness ex-ante.
• Ex-post guarantees improve with logarithmic factors.
• First results for arbitrary entitlement cases.