  Metric distortion in social choice provides a framework for assessing how
well voting rules minimize social cost in scenarios where voters and candidates
exist in a shared metric space, with voters submitting rankings and the rule
outputting a single winner. We expand this framework to include probabilistic
voting. Our extension encompasses a broad range of probability functions,
including widely studied models like Plackett-Luce (PL) and Bradley-Terry, and
a novel "pairwise quantal voting" model inspired by quantal response theory.
  We demonstrate that distortion results under probabilistic voting better
correspond with conventional intuitions regarding popular voting rules such as
Plurality, Copeland, and Random Dictator (RD) than those under deterministic
voting. For example, in the PL model with candidate strength inversely
proportional to the square of their metric distance, we show that Copeland's
distortion is at most 2, whereas that of RD is $\Omega(\sqrt{m})$ in large
elections, where $m$ is the number of candidates. This contrasts sharply with
the classical model, where RD beats Copeland with a distortion of 3 versus 5
[1].
