  We consider the following well studied problem of metric distortion in social
choice. Suppose we have an election with $n$ voters and $m$ candidates who lie
in a shared metric space. We would like to design a voting rule that chooses a
candidate whose average distance to the voters is small. However, instead of
having direct access to the distances in the metric space, each voter gives us
a ranked list of the candidates in order of distance. Can we design a rule that
regardless of the election instance and underlying metric space, chooses a
candidate whose cost differs from the true optimum by only a small factor
(known as the distortion)?
  A long line of work culminated in finding deterministic voting rules with
metric distortion $3$, which is the best possible for deterministic rules and
many other classes of voting rules. However, without any restrictions, there is
still a significant gap in our understanding: Even though the best lower bound
is substantially lower at $2.112$, the best upper bound is still $3$, which is
attained even by simple rules such as Random Dictatorship. Finding a rule that
guarantees distortion $3 - \varepsilon$ for some constant $\varepsilon $ has
been a major challenge in computational social choice.
  In this work, we give a rule that guarantees distortion less than $2.753$. To
do so we study a handful of voting rules that are new to the problem. One is
Maximal Lotteries, a rule based on the Nash equilibrium of a natural zero-sum
game which dates back to the 60's. The others are novel rules that can be
thought of as hybrids of Random Dictatorship and the Copeland rule. Though none
of these rules can beat distortion $3$ alone, a careful randomization between
Maximal Lotteries and any of the novel rules can.
