🤖 AI Summary
This study addresses the challenge of eliciting complete preference information from agents in high-dimensional matching markets by proposing a cognitively lightweight yet efficient matching mechanism. The approach leverages agents’ reported preferences over object features, applies singular value decomposition (SVD) to identify dominant directions of variation in the feature space, and reduces the high-dimensional matching problem to a one-dimensional ranking task. By integrating geometric distributionally robust optimization with a one-dimensional sorting algorithm, the mechanism computes a solution in \(O(N \log N)\) time. Theoretical analysis uncovers a novel connection between Nash social welfare and distributionally robust optimization, guaranteeing approximate welfare optimality, distributional truthfulness, and symmetry. Empirical results demonstrate that the method achieves 99% of optimal welfare while running three orders of magnitude faster than direct optimization, making it suitable for applications such as school choice, labor markets, and course allocation.
📝 Abstract
This paper proposes a computationally efficient mechanism for multi-dimensional matching markets where agents report preferences over object features rather than complete utility assessments. We use Singular Value Decomposition (SVD) to identify the principal direction of variation in feature space and match agents to objects along this dimension, reducing a complex multi-dimensional problem to an effectively one-dimensional problem solvable in $O(N \log N)$ time.
We show that when data exhibit low effective dimensionality, our mechanism approximately maximizes Nash Social Welfare, satisfies distributional truthfulness, and achieves symmetry. We establish a novel connection between Nash Social Welfare and Geometric Distributionally Robust Optimization, providing robustness guaranties. Numerical experiments demonstrate that our approach achieves 99\% optimal welfare while running three orders of magnitude faster than direct optimization. The framework applies naturally to school choice, labor markets, and course allocation, where feature-based elicitation reduces the cognitive burden on agents.