  We consider the distributed single-winner metric voting problem on a line,
where agents and alternative are represented by points on the line of real
numbers, the agents are partitioned into disjoint districts, and the goal is to
choose a single winning alternative in a decentralized manner. In particular,
the choice is done by a distributed voting mechanism which first selects a
representative alternative for each district of agents and then chooses one of
these representatives as the winner. In this paper, we design simple
distributed mechanisms that achieve distortion at most $2+\sqrt{5}$ for the
average-of-max and the max-of-average social cost objectives, matching the
corresponding lower bound shown in previous work for these objectives.
