Metric distortion Under Probabilistic Voting
Mohak Goyal, Sahasrajit Sarmasarkar
TL;DR
This paper extends metric distortion to probabilistic voting, integrating randomness in voter rankings via the Plackett-Luce framework and axiomatizing admissible marginal distributions. It develops a general Dist^{(g)} framework with key constants ${\hat g}_{\textsc{mid}}$ and ${\hat g}_{\textsc{out}}$, and analyzes how Plurality, Copeland, RandomDictator, and Borda perform under this stochastic setting, deriving orderwise tight bounds. Notably, under PL with strength $s_{i,j}=d(i,j)^{-\theta}$, Copeland achieves constant distortion, RandomDictator can scale as $\Theta(m^{1/\theta})$, and Borda exhibits a phase transition with distortion $\Theta(m^{\max(1-2/\theta,0)})$, depending on the randomness parameter $\theta$. These results reconcile metric distortion with intuitive expectations about popular voting rules in realistic, noisy voting environments and motivate randomness-aware design of aggregation rules for spatial voting contexts.
Abstract
Metric distortion in social choice is a framework for evaluating how well voting rules minimize social cost when both voters and candidates exist in a shared metric space, with a voter's cost defined by their distance to a candidate. Voters submit rankings, and the rule aggregates these rankings to determine a winner. We extend this framework to incorporate probabilistic voting, recognizing that real-world voters exhibit randomness in how they vote. Our extension includes various probability functions, notably 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 \textsc{Plurality}, \textsc{Copeland}, \textsc{Random Dictator} and \textsc{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 \textsc{Copeland}'s distortion is at most 2, whereas that of \textsc{RandomDictator} is $Ω(\sqrt{m})$ in large elections (i.e., number of voters $n \rightarrow \infty$), where $m$ is the number of candidates. This contrasts sharply with the classical model, where \textsc{RandomDictator} beats \textsc{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 $θ$, the distortion under \textsc{Borda} is $Θ(m^{1-2/θ})$ when $θ>2$ and $Θ(1)$ otherwise. This generalizes the classical deterministic voting model where the distortion of \textsc{Borda} is $2m-1$. The proof uses a novel variant of asymptotic duality where we choose the Lagrange multiplier via asymptotically maximizing the derivative of the objective function. Overall, our work opens a new frontier for analyzing voting rules.