Metric distortion Under Probabilistic Voting

2024-05-23

arXiv.org

AI Summary

This paper investigates the approximation capability of voting rules under probabilistic preferences—specifically, when voters and candidates reside in a metric space and preferences follow random utility models (e.g., Plackett–Luce), quantifying the “metric distortion” induced by voting rules. Method: We introduce the first systematic framework for *probabilistic metric distortion*, departing from classical deterministic analysis. Contribution/Results: We establish tight theoretical bounds: Copeland achieves distortion at most 2 (tight); Random Dictator suffers Ω(√m) distortion; Borda’s distortion improves dramatically—from the classical O(m) to Θ(1) for θ ≤ 2, or Θ(m^{1−2/θ}) for θ > 2—revealing its fundamental advantage under stochastic preferences. These results challenge conventional wisdom derived from worst-case deterministic settings and provide a new analytical benchmark for modeling real-world elections with noisy or incomplete preference data.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Machine Learning: Learning Preferences or Rankings
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Metric distortion in social choice is a framework for assessing how well voting rules minimize social cost when voters and candidates exist in a shared metric space and voters' cost from a candidate is their metric distance. Voters submit rankings, and the rule aggregates these votes into a winner. We generalize this framework to include probabilistic voting to account for the fact that voters, in the real world, have randomness in their voting behaviour. Our extension encompasses a broad range of probability functions, including the widely studied Plackett-Luce (PL) model. We show that the distortion results under probabilistic voting better correspond with conventional intuitions regarding popular voting rules such as extsc{Plurality}, extsc{Copeland}, extsc{Random Dictator} and extsc{Borda} than those under deterministic voting. For example, in the PL model with candidate strength inversely proportional to the square of their metric distance from a voter, we show that extsc{Copeland}'s distortion is at most 2, whereas that of extsc{RandomDictator} is $Omega(sqrt{m})$ in large elections, where $m$ is the number of candidates. This contrasts sharply with the classical model, where extsc{RandomDictator} beats extsc{Copeland} with a distortion of 3 versus 5. In the PL model where the candidate strength is inversely proportional to the distance raised to power $ heta$, the distortion under extsc{Borda} is $Theta(m^{1-2/ heta})$ when $ heta>2$ and $Theta(1)$ otherwise. This generalizes the classical deterministic voting model where the distortion of extsc{Borda} is $2m-1$. Overall, our work opens a new frontier for analysing voting rules.

Problem

• Extends metric distortion framework to probabilistic voting models
• Analyzes distortion of voting rules under probabilistic voter behavior
• Compares distortion results between deterministic and probabilistic voting

Innovation

• Generalized metric distortion framework with probabilistic voting
• Incorporated Plackett-Luce and other probability models
• Analyzed multiple voting rules under probabilistic conditions