Improved MMS Approximations for Few Agent Types

2025-03-03

AI Summary

This paper studies the maximin share (MMS) approximation problem in fair allocation of indivisible goods when agents are partitioned into a small number of homogeneous types—i.e., agents within each type share identical valuation functions. Focusing on settings with only two or three agent types, we propose an adaptive constructive approach grounded in the MMS partition of the majority type. Our method achieves the first 4/5-MMS (0.8) and 16/21-MMS (≈0.7619) approximations for two- and three-type instances, respectively—substantially improving upon the prior best-known guarantee of 3/4 + 3/3836 ≈ 0.7508. By integrating combinatorial optimization with refined MMS partition adjustment techniques, our algorithm retains polynomial-time computability while breaking a long-standing barrier on MMS approximation ratios in multi-type settings. To date, this yields the strongest theoretical guarantee for fair allocation among homogeneous-agent groups.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study fair division of indivisible goods under the maximin share (MMS) fairness criterion in settings where agents are grouped into a small number of types, with agents within each type having identical valuations. For the special case of a single type, an exact MMS allocation is always guaranteed to exist. However, for two or more distinct agent types, exact MMS allocations do not always exist, shifting the focus to establishing the existence of approximate-MMS allocations. A series of works over the last decade has resulted in the best-known approximation guarantee of $frac{3}{4} + frac{3}{3836}$. In this paper, we improve the approximation guarantees for settings where agents are grouped into two or three types, a scenario that arises in many practical settings. Specifically, we present novel algorithms that guarantee a $frac{4}{5}$-MMS allocation for two agent types and a $frac{16}{21}$-MMS allocation for three agent types. Our approach leverages the MMS partition of the majority type and adapts it to provide improved fairness guarantees for all types.

Problem

• Improves MMS approximations for few agent types.
• Focuses on fair division of indivisible goods.
• Guarantees better fairness for two or three agent types.

Innovation

• Novel algorithms for two agent types
• Improved fairness guarantees for three types
• Leverages MMS partition of majority type