Tight Bounds on Ordinal Maximin Share Guarantees for Fair Division Among Groups

Core Concepts

We establish tight asymptotic bounds on the smallest integer p such that an MMS1-out-of-p allocation always exists when allocating indivisible items among groups of agents.

Abstract

The paper investigates fairness in the allocation of indivisible items among groups of agents using the notion of maximin share (MMS). Previous work has shown that no nontrivial multiplicative MMS approximation can be guaranteed in this setting for general group sizes. The authors focus on ordinal relaxations of MMS, where each agent partitions the items into k bundles and receives the best bundle. They derive tight asymptotic bounds on the smallest integer p such that an MMS1-out-of-p allocation always exists, where p depends on the relative sizes of the groups. The key results are: For the "balanced" case where the group sizes are not too unbalanced, the authors show that pMMS(n1, ..., ng) = Θ(g log(n/g)), where n is the total number of agents and g is the number of groups. For the "unbalanced" case where one group is much larger than the others, they prove that pMMS(n1, ..., ng) = Θ(log n/ log log n), where n1 is the size of the large group. The authors also provide non-asymptotic results for the case of two groups and establish the tightness of their bounds. Additionally, they present efficient randomized algorithms to find the desired MMS1-out-of-p allocations.

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Ordinal Maximin Guarantees for Group Fair Division

by Pasin Manura... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.11543.pdf

Deeper Inquiries

How would the results change if the agents within each group have different utility functions, rather than additive and non-negative utilities

In the scenario where agents within each group have different utility functions, the results of the study would likely be impacted. The assumption of additive and non-negative utilities simplifies the analysis and allows for clear comparisons between agents based on their utility values. When agents have different utility functions, the complexity of the problem increases significantly. The notion of maximin share fairness, which is central to the study, may need to be redefined or adapted to account for varying utility functions. The algorithmic approaches and constructions used in the paper may need to be modified to accommodate the diverse preferences and valuations of agents within each group. This could involve developing new techniques to ensure fairness in allocations that consider the individual utility functions of agents. The results may also differ in terms of the guarantees provided for fair division among groups with heterogeneous utility functions. The bounds and algorithms derived in the paper may need to be adjusted to address the challenges posed by differing utility functions and ensure equitable outcomes for all agents involved.

Can the authors' techniques be extended to other fairness notions beyond maximin share, such as envy-freeness or proportionality

The techniques and methodologies presented in the paper can potentially be extended to address other fairness notions beyond maximin share, such as envy-freeness or proportionality. While the focus of the study is on maximin share fairness in group fair division, the underlying principles and algorithmic approaches can be adapted to incorporate different fairness criteria. For envy-freeness, which ensures that no agent envies another agent's allocation, the algorithms could be modified to guarantee envy-free allocations within groups or across groups. This may involve developing new constructions and proofs tailored to the specific requirements of envy-freeness while maintaining efficiency and optimality in the allocation process. Similarly, for proportionality, which aims to allocate resources such that each agent receives a fair share based on their contribution or valuation, the techniques used in the paper can be leveraged to ensure proportional allocations among groups of agents. Adjustments to the bounds and algorithms may be necessary to accommodate the constraints and objectives of proportionality in group fair division scenarios. By extending the authors' techniques to encompass a broader range of fairness notions, the study can contribute to the development of comprehensive and versatile solutions for group fair division problems that cater to various fairness criteria.

Are there practical applications or real-world scenarios where the group fair division problem considered in this paper would be particularly relevant

The group fair division problem addressed in this paper has significant relevance to various practical applications and real-world scenarios where resources need to be allocated fairly among groups of agents. Some potential applications include: Resource Allocation in Organizations: In corporate settings, fair division among departments, teams, or project groups is crucial for maintaining harmony and productivity. The techniques discussed in the paper can be applied to allocate resources, budgets, or rewards equitably among different groups within an organization. Community Resource Distribution: In community settings, such as neighborhood associations or local councils, there is often a need to distribute resources or benefits fairly among different groups of residents. The study's findings can be utilized to ensure fair and efficient allocation of resources based on the preferences and needs of each group. Government Funding Allocation: Government agencies responsible for distributing funding or grants to various sectors or regions can benefit from the insights provided in the paper. By applying the principles of group fair division, policymakers can make informed decisions on resource allocation to promote equity and fairness across different groups in society. Supply Chain Management: In supply chain networks involving multiple stakeholders and entities, fair division of costs, profits, or resources is essential for maintaining sustainable and collaborative relationships. The techniques developed in the study can be adapted to optimize resource allocation and ensure fairness in supply chain operations. Overall, the group fair division problem addressed in the paper has broad applications in diverse domains where equitable distribution of resources among groups is a critical concern. By implementing the methodologies and algorithms proposed in the study, organizations and decision-makers can enhance fairness and efficiency in resource allocation processes.