Utilitarian Distortion Under Probabilistic Voting

TL;DR

This paper analyzes the efficiency loss of various voting rules under a probabilistic model with noisy rankings, showing that Copeland and Borda achieve low, bounded distortion, aligning with normative voting principles.

Contribution

It provides tight bounds on the distortion of Copeland and Borda under probabilistic voting, resolving longstanding tensions and demonstrating the impact of noise on voting rule efficiency.

Findings

Copeland and Borda have bounded distortion independent of candidate number

Plurality and top-choice rules have significantly higher distortion

Probabilistic modeling aligns voting rule efficiency with normative expectations

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.