Utilitarian Distortion Under Probabilistic Voting

2026-02-11

AI Summary

This work addresses the long-standing tension between normative desirability and efficiency in social choice by analyzing distortion under the Plackett–Luce probabilistic voting model, where voters’ rankings are generated from utility values perturbed by noise controlled by a temperature parameter β. Unlike deterministic models—under which classic rules like Copeland and Borda suffer unbounded distortion—this study establishes for the first time that both Copeland and Borda achieve a distortion bound of β(1+e⁻ᵝ)/(1−e⁻ᵝ), which is O(β) and independent of the number of candidates, closely approaching the theoretical lower bound of (1−ε)β. In contrast, rules relying solely on top-choice information, such as Plurality, exhibit distortion that grows with either the number of candidates m or eᵝ, thereby significantly underperforming. These results reconcile normative appeal with efficiency in a behaviorally plausible setting.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The utilitarian distortion framework evaluates voting rules by their worst-case efficiency loss when voters have cardinal utilities but express only ordinal rankings. Under the classical model, a longstanding tension exists: Plurality, which suffers from the spoiler effect, achieves optimal $\Theta(m^2)$ distortion among deterministic rules, while normatively superior rules like Copeland and Borda have unbounded distortion. We resolve this tension under probabilistic voting with the Plackett-Luce model, where rankings are noisy reflections of utilities governed by an inverse temperature parameter $\beta$. Copeland and Borda both achieve at most $\beta\frac{1+e^{-\beta}}{1-e^{-\beta}}$ distortion, independent of the number of candidates $m$, and within a factor of 2 of the lower bound for randomized rules satisfying the probabilistic Condorcet loser criterion known from prior work. This improves upon the prior $O(\beta^2)$ bound for Borda. These upper bounds are nearly tight: prior work establishes a $(1-o(1))\beta$ lower bound for Borda, and we prove a $(1-\epsilon)\beta$ lower bound for Copeland for any constant $\epsilon>0$. In contrast, rules that rely only on top-choice information fare worse: Plurality has distortion $\Omega(\min(e^\beta, m))$ and Random Dictator has distortion $\Theta(m)$. Additional `veto'information is also insufficient to remove the dependence on $m$; Plurality Veto and Pruned Plurality Veto have distortion $\Omega(\beta \ln m)$. We also prove a lower bound of $(\frac{5}{8}-\epsilon)\beta$ (for any constant $\epsilon>0$) for all deterministic finite-precision tournament-based rules, a class that includes Copeland and any rule based on pairwise comparison margins rounded to fixed precision. Our results show that the distortion framework aligns with normative intuitions once the probabilistic nature of real-world voting is taken into account.

Problem

• utilitarian distortion
• probabilistic voting
• voting rules
• Plackett-Luce model
• efficiency loss

Innovation

• utilitarian distortion
• probabilistic voting
• Plackett-Luce model
• Copeland rule
• Borda count