  We consider competitive facility location as a two-stage multi-agent system
with two types of clients. For a given host graph with weighted clients on the
vertices, first facility agents strategically select vertices for opening their
facilities. Then, the clients strategically select which of the opened
facilities in their neighborhood to patronize. Facilities want to attract as
much client weight as possible, clients want to minimize congestion on the
chosen facility.
  All recently studied versions of this model assume that clients can split
their weight strategically. We consider clients with unsplittable weights but
allow mixed strategies. So clients may randomize over which facility to
patronize. Besides modeling a natural client behavior, this subtle change
yields drastic changes, e.g., for a given facility placement, qualitatively
different client equilibria are possible.
  As our main result, we show that pure subgame perfect equilibria always exist
if all client weights are identical. For this, we use a novel potential
function argument, employing a hierarchical classification of the clients and
sophisticated rounding in each step. In contrast, for non-identical clients, we
show that deciding the existence of even approximately stable states is
computationally intractable. On the positive side, we give a tight bound of $2$
on the price of anarchy which implies high social welfare of equilibria, if
they exist.
