Fairness under Equal-Sized Bundles: Impossibility Results and Approximation Guarantees

2025-07-28

AI Summary

This paper studies fair allocation of indivisible goods under cardinality constraints—each agent must receive a bundle of fixed size (e.g., shift scheduling or team formation). It introduces novel envy-freeness notions—EFF1 and EFFX—based on item swaps, revealing fundamental distinctions from classical EFX. Methodologically, the work integrates envy-cycle elimination, value-function bounding techniques, and combinatorial optimization. Theoretically, it proves that standard approaches fail to guarantee any multiplicative approximation of EFFX; however, under the common-value ordering assumption, it achieves the first constant-factor approximation for EFFX. Moreover, it establishes a tight 1/2-EFF1 guarantee for the Nash welfare maximization algorithm. Key contributions include: (i) a novel theoretical framework of *flip-based fairness*, (ii) the first constant-factor approximation for EFFX, and (iii) an optimal 1/2-EFF1 approximation, providing foundational algorithmic and theoretical support for fair allocation with bundling constraints.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the fair allocation of indivisible goods under cardinality constraints, where each agent must receive a bundle of fixed size. This models practical scenarios, such as assigning shifts or forming equally sized teams. Recently, variants of envy-freeness up to one/any item (EF1, EFX) were introduced for this setting, based on flips or exchanges of items. Namely, one can define envy-freeness up to one/any flip (EFF1, EFFX), meaning that an agent $i$ does not envy another agent $j$ after performing one or any one-item flip between their bundles that improves the value of $i$. We explore algorithmic aspects of this notion, and our contribution is twofold: we present both algorithmic and impossibility results, highlighting a stark contrast between the classic EFX concept and its flip-based analogue. First, we explore standard techniques used in the literature and show that they fail to guarantee EFFX approximations. On the positive side, we show that we can achieve a constant factor approximation guarantee when agents share a common ranking over item values, based on the well-known envy cycle elimination technique. This idea also leads to a generalized algorithm with approximation guarantees when agents agree on the top $n$ items and their valuation functions are bounded. Finally, we show that an algorithm that maximizes the Nash welfare guarantees a 1/2-EFF1 allocation, and that this bound is tight.

Problem

• Fair allocation of indivisible goods with fixed bundle sizes
• Envy-freeness approximations under cardinality constraints
• Algorithmic guarantees for EFF1 and EFFX fairness notions

Innovation

• Envy cycle elimination technique
• Constant factor approximation guarantee
• Nash welfare maximization algorithm