  We study the classic mechanism design problem of locating a public facility
on a real line. In contrast to previous work, we assume that the agents are
unable to fully specify where their preferred location lies, and instead only
provide coarse information---namely, that their preferred location lies in some
interval. Given such partial preference information, we explore the design of
robust deterministic mechanisms, where by robust mechanisms we mean ones that
perform well with respect to all the possible unknown true preferred locations
of the agents. Towards this end, we consider two well-studied objective
functions and look at implementing these under two natural solution concepts
for our setting i) very weak dominance and ii) minimax dominance. We show that
under the former solution concept, there are no mechanisms that do better than
a naive mechanism which always, irrespective of the information provided by the
agents, outputs the same location. However, when using the latter, weaker,
solution concept, we show that one can do significantly better, and we provide
upper and lower bounds on the performance of mechanisms for the objective
functions of interest. Furthermore, we note that our mechanisms can be viewed
as extensions to the classical optimal mechanisms in that they perform
optimally when agents precisely know and specify their preferred locations.
