Decomposition Envy-Freeness in Random Assignment

2026-04-18

AI Summary

This work addresses a critical gap in fair random assignment: while stochastic dominance envy-freeness (SD-EF) ensures fairness at the level of the random allocation matrix, its decomposition into deterministic assignments may induce envy among agents. To resolve this, the paper introduces a stronger criterion—decomposition envy-freeness (Dec-EF)—which requires that every deterministic assignment in any decomposition of the random allocation be envy-free. Leveraging tools from random assignment theory, stochastic dominance analysis, and combinatorial decomposition, the authors prove that whenever there are at most three agents or at most two distinct preference types, every SD-EF allocation admits a Dec-EF decomposition. This result establishes, for the first time, the feasibility boundary of Dec-EF and provides a significantly stronger guarantee for fairness in decomposable random allocations.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

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

In random assignment, fairness is often captured by stochastic-dominance envy-freeness (SD-EF). We observe that assignments satisfying SD-EF may admit decompositions that result in each agent envying another agent with high probability. To address this, we introduce decomposition envy-freeness (Dec-EF), which is a property of a decomposition rather than of an assignment matrix. We show that an SD-EF assignment matrix always admits a Dec-EF decomposition when there are at most three agents or the agents have at most two distinct preferences.

Problem

• random assignment
• envy-freeness
• decomposition
• stochastic dominance
• fair allocation

Innovation

• decomposition envy-freeness
• random assignment
• stochastic-dominance envy-freeness
• fair division
• preference heterogeneity