  We focus on the problem of placing two facilities along a linear space to
serve a group of agents. Each agent is committed to minimizing the distance
between her location and the closest facility. A mechanism is an algorithm that
maps the reported agent locations to the facility locations. We are interested
in mechanisms without money that are deterministic, strategy-proof, and provide
a bounded approximation ratio for social cost.
  It is a fundamental problem to characterize the family of strategy-proof
mechanisms with a bounded approximation ratio. Fotakis and Tzamos already
demonstrated that the deterministic strategy-proof mechanisms for the
2-facility game problem are mechanisms with a unique dictator and the
leftmost-rightmost mechanism. In this paper, we first present a more refined
characterization of the first family.
  We then reveal three new classes of strategy-proof mechanisms that show the
intricacy of structure within this family. This helps us get a more complete
picture of the characterization of the 2-facility game problem, and may also
have value in understanding and solving more general facility allocation game
problems.
  Besides, based on our refined characterization, we surprisingly find that
prediction cannot effectively improve the performance of the mechanism in the
two-facility game problem, while this methodology to overcome bad approximation
ratio works in many other mechanism design problems. We show that if we require
that the mechanism admits a bounded approximation ratio when the prediction is
arbitrarily bad, then at the same time, the mechanism can never achieve
sublinear approximation ratios even with perfect prediction.
