Tight distortion bounds for distributed metric voting on a line

We consider a voting problem where agents and alternatives are on the line of real numbers, the agents are partitioned into disjoint districts, and the goal is to choose one alternative using a distributed voting mechanism. Such mechanisms select a representative alternative for each district and then choose one of them as the winner. We design simple mechanisms with distortion at most 2 + √5 for the average-of-max and max-of-average cost objectives, matching the corresponding lower bound shown in previous work.