  We consider a social choice setting with agents that are partitioned into
disjoint groups, and have metric preferences over a set of alternatives. Our
goal is to choose a single alternative aiming to optimize various objectives
that are functions of the distances between agents and alternatives in the
metric space, under the constraint that this choice must be made in a
distributed way: The preferences of the agents within each group are first
aggregated into a representative alternative for the group, and then these
group representatives are aggregated into the final winner. Deciding the winner
in such a way naturally leads to loss of efficiency, even when complete
information about the metric space is available. We provide a series of (mostly
tight) bounds on the distortion of distributed mechanisms for variations of
well-known objectives, such as the (average) total cost and the maximum cost,
and also for new objectives that are particularly appropriate for this
distributed setting and have not been studied before.
