  We study the problem of designing voting rules that take as input the ordinal
preferences of $n$ agents over a set of $m$ alternatives and output a single
alternative, aiming to optimize the overall happiness of the agents. The input
to the voting rule is each agent's ranking of the alternatives from most to
least preferred, yet the agents have more refined (cardinal) preferences that
capture the intensity with which they prefer one alternative over another. To
quantify the extent to which voting rules can optimize over the cardinal
preferences given access only to the ordinal ones, prior work has used the
distortion measure, i.e., the worst-case approximation ratio between a voting
rule's performance and the best performance achievable given the cardinal
preferences.
  The work on the distortion of voting rules has been largely divided into two
worlds: utilitarian distortion and metric distortion. In the former, the
cardinal preferences of the agents correspond to general utilities and the goal
is to maximize a normalized social welfare. In the latter, the agents' cardinal
preferences correspond to costs given by distances in an underlying metric
space and the goal is to minimize the (unnormalized) social cost. Several
deterministic and randomized voting rules have been proposed and evaluated for
each of these worlds separately, gradually improving the achievable distortion
bounds, but none of the known voting rules perform well in both worlds
simultaneously.
  In this work, we prove that one can achieve the best of both worlds by
designing new voting rules, that simultaneously achieve near-optimal distortion
guarantees in both distortion worlds. We also prove that this positive result
does not generalize to the case where the voting rule is provided with the
rankings of only the top-$t$ alternatives of each agent, for $t<m$.
