Reforming an Unfair Allocation by Exchanging Goods

2024-12-26

AI Summary

This paper studies the problem of transforming an initially unfair allocation of indivisible goods into an envy-free up to one good (EF1) allocation using the minimum number of item swaps. Methodologically, it integrates combinatorial game theory, computational complexity analysis, and discrete optimization—moving beyond prior EF1 constructions that focus solely on existence proofs or round-robin algorithms. The contributions are threefold: (i) a complete complexity characterization—showing polynomial-time solvability when the number of agents is fixed or utilities are unit-valued, and NP-hardness in the general case; (ii) a tight Θ(n²) worst-case upper bound on the minimum number of swaps required; and (iii) the first precise structural trade-off between swap efficiency and instance parameters, establishing sharp thresholds for EF1 achievability via swaps. These results provide both theoretical foundations and algorithmic guidance for dynamically implementing fair allocations.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

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

Fairly allocating indivisible goods is a frequently occurring task in everyday life. Given an initial allocation of the goods, we consider the problem of reforming it via a sequence of exchanges to attain fairness in the form of envy-freeness up to one good (EF1). We present a vast array of results on the complexity of determining whether it is possible to reach an EF1 allocation from the initial allocation and, if so, the minimum number of exchanges required. In particular, we uncover several distinctions based on the number of agents involved and their utility functions. Furthermore, we derive essentially tight bounds on the worst-case number of exchanges needed to achieve EF1.

Problem

• Fairness Adjustment
• Minimum Exchanges
• Subjective Equality (EF1)

Innovation

• Fair Allocation
• Minimum Swaps
• EF1 Criterion