[Paper Review] On Fair Allocation of Indivisible Goods to Submodular Agents

Gilad Ben Uziahu, Uriel Feige|arXiv (Cornell University)|Mar 22, 2023

Game Theory and Voting Systems4 citations

TL;DR

This paper presents polynomial-time algorithms for fair allocation of indivisible goods to submodular agents, achieving a 10/27-approximation of the maximin share (MMS) for equal entitlements and a 1/3-approximation of the anyprice share (APS) for arbitrary entitlements. The approach leverages a bidding game framework to ensure agents receive bundles of value at least a constant fraction of their respective share, improving prior results in submodular valuation settings.

ABSTRACT

We consider the problem of fair allocation of indivisible goods to agents with submodular valuation functions, where agents may have either equal entitlements or arbitrary (possibly unequal) entitlements. We focus on share-based fairness notions, specifically, the maximin share (MMS) for equal entitlements and the anyprice share (APS) for arbitrary entitlements, and design allocation algorithms that give each agent a bundle of value at least some constant fraction of her share value. For the equal entitlement case (and submodular valuations), Ghodsi, Hajiaghayi, Seddighin, Seddighin, and Yami [EC 2018] designed a polynomial-time algorithm for $\frac{1}{3}$-maximin-fair allocation. We improve this result in two different ways. We consider the general case of arbitrary entitlements, and present a polynomial time algorithm that guarantees submodular agents $\frac{1}{3}$ of their APS. For the equal entitlement case, we improve the approximation ratio and obtain $\frac{10}{27}$-maximin-fair allocations. Our algorithms are based on designing strategies for a certain bidding game that was previously introduced by Babaioff, Ezra and Feige [EC 2021].

Motivation & Objective

• To address fair allocation of indivisible goods to agents with submodular valuation functions under equal and arbitrary entitlements.
• To improve upon prior approximation ratios for maximin share (MMS) fairness in submodular settings, particularly for equal entitlements.
• To extend fairness guarantees to arbitrary entitlements using the anyprice share (APS) notion.
• To design polynomial-time algorithms that ensure each agent receives a bundle of value at least a constant fraction of their share.
• To analyze the limitations of bidding strategies and establish tight bounds on achievable approximation ratios.

Proposed method

• The authors employ a bidding game framework previously introduced by Babaioff, Ezra, and Feige, adapting it to submodular valuations.
• For equal entitlements, the algorithm uses a proportional bidding strategy to achieve a 10/27-MMS approximation, improving over the prior 1/3 bound.
• For arbitrary entitlements, a modified bidding strategy ensures each agent receives at least 1/3 of their APS, even with unequal entitlements.
• The method relies on constructing adversarial bidding runs to test the robustness of approximation guarantees and establish tight bounds.
• Theoretical analysis proves that no constant ρ > 1/3 can guarantee a better approximation than 1/3 for APS, and that 10/27 is optimal for MMS under the proposed framework.
• The paper uses negative examples with structured item sets and substitute valuations to demonstrate the tightness of the bounds.

Experimental results

Research questions

• RQ1 Can a polynomial-time algorithm achieve a better than 1/3-approximation of the maximin share (MMS) for submodular agents with equal entitlements?
• RQ2 Can the anyprice share (APS) fairness notion be approximated within a constant factor for submodular agents with arbitrary entitlements?
• RQ3 What is the best possible approximation ratio achievable via bidding strategies in the submodular setting, and can it be improved beyond 1/3 for APS or 1/3 for MMS?
• RQ4 Do the proposed bidding strategies fail to extend to broader valuation classes such as XOS, and if so, why?
• RQ5 Can the theoretical limits of approximation be demonstrated via adversarial bidding game constructions?

Key findings

• The paper achieves a 10/27-approximation of the maximin share (MMS) for submodular agents with equal entitlements, improving upon the prior 1/3 bound.
• For agents with arbitrary entitlements, the algorithm guarantees each agent a bundle of value at least 1/3 of their anyprice share (APS).
• The authors prove that no constant ρ > 1/3 can guarantee a better than 1/3-approximation for APS fairness, establishing a tight bound.
• A negative example demonstrates that the 10/27 bound for MMS is tight under the proposed bidding strategy framework, and cannot be improved beyond this ratio.
• For XOS valuations, the paper shows that no bidding strategy can guarantee more than a 1/k-fraction of MMS when n ≥ 4k², indicating fundamental limitations in this class.
• The analysis confirms that the proposed bidding game strategies do not extend to improve approximation ratios beyond 1/3 for APS or 10/27 for MMS, even with modifications.