Fair Allocation of Indivisible Goods: Improvement and Generalization (1704.00222v3)

Abstract: We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procaccia and Wang (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1-MMS allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a $1/2$-MMS allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a $2/3$-MMS allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of [Procaccia and Wang] to a $3/4$ factor in the additive setting. The main idea for our $3/4$-MMS allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time.

• The paper improves fairness guarantees by proving a 3/4-MMS allocation is always achievable in the additive setting.
• It introduces a reducibility framework that simplifies complex allocation instances without compromising maxmin share properties.
• The study extends algorithmic approaches to submodular, XOS, and subadditive valuations, broadening its real-world applicability.

Overview of Fair Allocation of Indivisible Goods

This paper investigates the problem of fair allocation concerning indivisible goods, leveraging the maxmin share (MMS) paradigm to measure fairness of allocation. Over the years, several fairness notions like proportionality and envy-freeness were proposed; however, they didn't provide guarantees when goods were indivisible. Budish introduced the concept of maxmin share as a more fitting measure for indivisible items, which has since become a focal point in this area of paper.

Key Contributions and Results

1. Improvement of Approximation Guarantees: Procaccia and Wang had previously demonstrated that a 2/3-MMS allocation is achievable but left any better guarantees as an open problem. This paper improves the best known results by showing that a 3/4-MMS allocation is always possible in the additive setting. This is significant because previous methods were thought to be tight, indicating that current techniques were inadequate for surpassing a 2/3 approximation guarantee.
2. Reducibility Framework: The authors developed a reducibility argument that simplifies the problem by reducing instances without losing generality. They define reducibility as identifying subsets of items and agents where allocations can satisfy some agents while preserving the maxmin share capabilities of the remaining items for the other agents. This framework aids in simplifying complex instances to more manageable subproblems.
3. Algorithmic Advancement: For the additive setting, a practical algorithm is presented which finds a (3/4-ε)-MMS allocation in polynomial time, leveraging existing techniques and the new insights from the reducibility framework. Given the known hardness (NP) of computing exact MMS values, the algorithm's efficiency marks a significant advancement.
4. Extension to Submodular, XOS, and Subadditive Valuations: The paper goes beyond additive valuations, extending results to submodular, XOS, and subadditive settings which are often more representative of real-world scenarios. They show constant approximation guarantees for submodular and XOS valuations, and logarithmic guarantees for subadditive settings, thus broadening the applicability of their results.
5. Upper Bound Construction: They also establish upper bounds across different valuation settings, presenting instances where no allocation can guarantee more than a specified fraction, reaffirming the boundaries of achievable fairness.

Implications and Speculation

The implications of this research are vast, affecting both theoretical aspects and practical applications. By improving the guaranteed fairness level for allocations and providing computationally efficient methods, this work brings closer the possibility of equitable resource distribution even in highly competitive environments. The adaptability of these results to non-additive valuations opens doors to apply them in more complex economic and social scenarios, such as resource allocation in collaborative projects or storage management in shared databases.

Future Developments

Moving forward, further investigation into multi-agent allocations with more complex constraints or utility functions could still deepen understanding and refine the results. Additionally, exploring the impact of dynamic valuations in evolving environments could provide insight into application for real-time systems. As AI continues to permeate decision-making, enhancing fairness and optimality in allocations will be crucial to many domains, including economics, logistics, and beyond.