Metric Distortion with Preference Intensities

2026-01-05

arXiv.org

AI Summary

This work addresses the limitation of traditional ordinal voting, which fails to capture the intensity of voters’ preferences between alternatives, thereby distorting social choice outcomes. Within the metric distortion framework, the paper introduces— for the first time—intensity-aware ranked ballots and proposes a class of position-scoring matching rules that can be optimized to minimize distortion. By formulating the design of such rules as a zero-sum game and solving for its equilibrium, the authors derive an optimal rule that achieves a distortion upper bound strictly below 3 when preference intensities are incorporated. In contrast, ignoring intensity leads to substantial performance degradation, thereby quantitatively demonstrating the critical role of preference intensity in enhancing the quality of social choice.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions

Abstract

In voting with ranked ballots, each agent submits a strict ranking of the form $a \succ b \succ c \succ d$ over the alternatives, and the voting rule decides on the winner based on these rankings. Although this ballot format has desirable characteristics, there is a question of whether it is expressive enough for the agents. Kahng, Latifian, and Shah address this issue by adding intensities to the rankings. They introduce the ranking with intensities ballot format, where agents can use both $\succ\!\!\succ$ and $\succ$ in their rankings to express intensive and normal preferences between consecutive alternatives in their rankings. While they focus on analyzing this ballot format in the utilitarian distortion framework, in this work, we look at the potential of using this ballot format from the metric distortion viewpoint. We design a class of voting rules coined Positional Scoring Matching rules, which can be used for different problems in the metric setting, and show that by solving a zero-sum game, we can find the optimal member of this class for our problem. This rule takes intensities into account and achieves a distortion lower than $3$. In addition, by proving a bound on the price of ignoring intensities, we show that we might lose a great deal in terms of distortion by not taking the intensities into account.

Problem

• metric distortion
• preference intensities
• ranked ballots
• voting rules
• social choice

Innovation

• metric distortion
• preference intensities
• positional scoring matching
• zero-sum game
• voting rules