The Distortion of Prior-Independent b-Matching Mechanisms

2026-02-11

AI Summary

This study addresses the fundamental challenge in mechanism design of minimizing distortion in social welfare when only ordinal preferences—rather than cardinal utilities—of agents over items are known. Under a stochastic preference model where agent values are drawn from unknown and potentially heterogeneous distributions, the paper establishes a theoretical lower bound of $e/(e-1) \approx 1.582$ on the expected distortion and constructs a mechanism that achieves this bound. Furthermore, it introduces a practical mechanism whose worst-case distortion gap is merely 1.076, substantially outperforming conventional approaches. By integrating stochastic preference modeling, expected distortion analysis, and approximate mechanism design, this work unifies theoretical optimality with strong empirical performance.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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

In a setting where $m$ items need to be partitioned among $n$ agents, we evaluate the performance of mechanisms that take as input each agent's \emph{ordinal preferences}, i.e., their ranking of the items from most- to least-preferred. The standard measure for evaluating ordinal mechanisms is the \emph{distortion}, and the vast majority of the literature on distortion has focused on worst-case analysis, leading to some overly pessimistic results. We instead evaluate the distortion of mechanisms with respect to their expected performance when the agents'preferences are generated stochastically. We first show that no ordinal mechanism can achieve a distortion better than $e/(e-1)\approx 1.582$, even if each agent needs to receive exactly one item (i.e., $m=n$) and every agent's values for different items are drawn i.i.d.\ from the same known distribution. We then complement this negative result by proposing an ordinal mechanism that achieves the optimal distortion of $e/(e-1)$ even if each agent's values are drawn from an agent-specific distribution that is unknown to the mechanism. To further refine our analysis, we also optimize the \emph{distortion gap}, i.e., the extent to which an ordinal mechanism approximates the optimal distortion possible for the instance at hand, and we propose a mechanism with a near-optimal distortion gap of $1.076$. Finally, we also evaluate the distortion and distortion gap of simple mechanisms that have a one-pass structure.

Problem

• distortion
• ordinal preferences
• b-matching
• stochastic preferences
• mechanism design

Innovation

• distortion
• ordinal mechanisms
• stochastic preferences
• distortion gap
• b-matching