  We revisit the recent breakthrough result of Gkatzelis et al. on
(single-winner) metric voting, which showed that the optimal distortion of 3
can be achieved by a mechanism called Plurality Matching. The rule picks an
arbitrary candidate for whom a certain candidate-specific bipartite graph
contains a perfect matching, and thus, it is not neutral (i.e, symmetric with
respect to candidates). Subsequently, a much simpler rule called Plurality Veto
was shown to achieve distortion 3 as well. This rule only constructs such a
matching implicitly but the winner depends on the order that voters are
processed, and thus, it is not anonymous (i.e., symmetric with respect to
voters).
  We provide an intuitive interpretation of this matching by generalizing the
classical notion of the (proportional) veto core in social choice theory. This
interpretation opens up a number of immediate consequences. Previous methods
for electing a candidate from the veto core can be interpreted simply as
matching algorithms. Different election methods realize different matchings, in
turn leading to different sets of candidates as winners. For a broad
generalization of the veto core, we show that the generalized veto core is
equal to the set of candidates who can emerge as winners under a natural class
of matching algorithms reminiscent of Serial Dictatorship.
  Extending these matching algorithms into continuous time, we obtain a highly
practical voting rule with optimal distortion 3, which is also intuitive and
easy to explain: Each candidate starts off with public support equal to his
plurality score. From time 0 to 1, every voter continuously brings down, at
rate 1, the support of her bottom choice among not-yet-eliminated candidates. A
candidate is eliminated if he is opposed by a voter after his support reaches
0. On top of being anonymous and neutral, this rule satisfies many other axioms
desirable in practice.
