  In the one-dimensional facility assignment problem, m facilities and n agents
are positioned along the real line. Each agent will be assigned to a single
facility to receive service. Each facility incurs a building cost, which is
shared equally among the agents utilizing it. Additionally, each agent
independently bears a connection cost to access a facility. Thus, an agent's
cost is the sum of the connection cost and her portion of the building cost.
The social cost is the total cost of all agents. Notably, the optimal
assignment that minimizes the social cost can be found in polynomial time. In
this paper, we study the problem from two game-theoretical settings regarding
the strategy space of agents and the rule the assignment. In both settings,
agents act strategically to minimize their individual costs.
  In our first setting, the strategy space of agents is the set of facilities,
granting agents the freedom to select any facility. Consequently, the
self-formed assignment can exhibit instability, as agents may deviate to other
facilities. We focus on the computation of an equilibrium assignment, where no
agent has an incentive to unilaterally change her choice. We show that we can
compute a pure Nash equilibrium in polynomial time.
  In our second setting, agents report their positions to a mechanism for
assignment to facilities. The strategy space of agents becomes the set of all
positions. Our interest lies in strategyproof mechanisms. It is essential to
note that the preference induced by the agents' cost function is more complex
as it depends on how other agents are assigned. We establish a strong lower
bound against all strategyproof and anonymous mechanisms: none can achieve a
bounded social cost approximation ratio. Nonetheless, we identify a class of
non-trivial strategyproof mechanisms for any n and m that is unanimous and
anonymous.
