  We consider elections where both voters and candidates can be associated with
points in a metric space and voters prefer candidates that are closer to those
that are farther away. It is often assumed that the optimal candidate is the
one that minimizes the total distance to the voters. Yet, the voting rules
often do not have access to the metric space $M$ and only see preference
rankings induced by $M$.Consequently, they often are incapable of selecting the
optimal candidate. The distortion of a voting rule measures the worst-case loss
of the quality being the result of having access only to preference rankings.
We extend the idea of distortion to approval-based preferences. First, we
compute the distortion of Approval Voting. Second, we introduce the concept of
acceptability-based distortion---the main idea behind is that the optimal
candidate is the one that is acceptable to most voters. We determine
acceptability-distortion for a number of rules, including Plurality, Borda,
$k$-Approval, Veto, the Copeland's rule, Ranked Pairs, the Schulze's method,
and STV.
