  We study a non-cooperative two-sided facility location game in which
facilities and clients behave strategically. This is in contrast to many other
facility location games in which clients simply visit their closest facility.
Facility agents select a location on a graph to open a facility to attract as
much purchasing power as possible, while client agents choose which facilities
to patronize by strategically distributing their purchasing power in order to
minimize their total waiting time. Here, the waiting time of a facility depends
on its received total purchasing power. We show that our client stage is an
atomic splittable congestion game, which implies existence, uniqueness and
efficient computation of a client equilibrium. Therefore, facility agents can
efficiently predict client behavior and make strategic decisions accordingly.
Despite that, we prove that subgame perfect equilibria do not exist in all
instances of this game and that their existence is NP-hard to decide. On the
positive side, we provide a simple and efficient algorithm to compute
3-approximate subgame perfect equilibria.
