  We study facility location problems where agents control multiple locations
and when reporting their locations can choose to hide some locations (hiding),
report some locations more than once (replication) and lie about their
locations (manipulation). We fully characterize all facility location
mechanisms that are anonymous, efficient, and strategyproof with respect to the
richer strategic behavior for this setting. We also provide a characterization
with respect to manipulation only. This is the first, to the best of our
knowledge, characterization result for the strategyproof facility location
mechanisms where each agent controls multiple locations.
