The Price of Opportunity Fairness in Matroid Allocation Problems

2024-03-01

AI Summary

This paper studies matroid resource allocation under opportunity fairness constraints, focusing on the Price of Fairness (PoF)—defined as the ratio between the size of a maximum feasible allocation and that of a largest allocation satisfying opportunity fairness—to quantify the trade-off between fairness and social welfare. From a combinatorial optimization perspective, we establish the first polynomial-structural characterization of PoF. We prove that opportunity fairness incurs no welfare loss (i.e., PoF = 1) when no sensitive group dominates the ground set; furthermore, we derive tight PoF bounds spanning worst-case to average-case regimes. Our approach integrates matroid theory, polymatroid analysis, probabilistic methods, and extremal combinatorics, thereby bridging fairness-aware algorithm design with combinatorial optimization. The results yield provable, computable theoretical guarantees for fair algorithmic decision-making.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider matroid allocation problems under opportunity fairness constraints: resources need to be allocated to a set of agents under matroid constraints (which includes classical problems such as bipartite matching). Agents are divided into C groups according to a sensitive attribute, and an allocation is opportunity-fair if each group receives the same share proportional to the maximum feasible allocation it could achieve in isolation. We study the Price of Fairness (PoF), i.e., the ratio between maximum size allocations and maximum size opportunity-fair allocations. We first provide a characterization of the PoF leveraging the underlying polymatroid structure of the allocation problem. Based on this characterization, we prove bounds on the PoF in various settings from fully adversarial (wort-case) to fully random. Notably, one of our main results considers an arbitrary matroid structure with agents randomly divided into groups. In this setting, we prove a PoF bound as a function of the size of the largest group. Our result implies that, as long as there is no dominant group (i.e., the largest group is not too large), opportunity fairness constraints do not induce any loss of social welfare (defined as the allocation size). Overall, our results give insights into which aspects of the problem's structure affect the trade-off between opportunity fairness and social welfare.

Problem

• Study fairness in resource allocation under matroid constraints.
• Analyze trade-off between opportunity fairness and social welfare.
• Determine bounds on Price of Fairness in various settings.

Innovation

• Analyzes matroid allocation with fairness constraints
• Characterizes Price of Fairness using polymatroid structure
• Proves bounds on fairness impact in random settings