  We study multiplayer quantitative reachability games played on a finite
directed graph, where the objective of each player is to reach his target set
of vertices as quickly as possible. Instead of the well-known notion of Nash
equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE),
a refinement of NE well-suited in the framework of games played on graphs. It
is known that there always exists an SPE in quantitative reachability games and
that the constrained existence problem is decidable. We here prove that this
problem is PSPACE-complete. To obtain this result, we propose a new algorithm
that iteratively builds a set of constraints characterizing the set of SPE
outcomes in quantitative reachability games. This set of constraints is
obtained by iterating an operator that reinforces the constraints up to
obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be
represented by a finite graph of size at most exponential. A careful inspection
of the computation allows us to establish PSPACE membership.
