  We study $n$-player turn-based games played on a finite directed graph. For
each play, the players have to pay a cost that they want to minimize. 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. We also study natural variants of SPE,
named weak (resp. very weak) SPE, where players who deviate cannot use the full
class of strategies but only a subclass with a finite number of (resp. a
unique) deviation step(s).
  Our results are threefold. Firstly, we characterize in the form of a Folk
theorem the set of all plays that are the outcome of a weak SPE. Secondly, for
the class of quantitative reachability games, we prove the existence of a
finite-memory SPE and provide an algorithm for computing it (only existence was
known with no information regarding the memory). Moreover, we show that the
existence of a constrained SPE, i.e. an SPE such that each player pays a cost
less than a given constant, can be decided. The proofs rely on our Folk theorem
for weak SPEs (which coincide with SPEs in the case of quantitative
reachability games) and on the decidability of MSO logic on infinite words.
Finally with similar techniques, we provide a second general class of games for
which the existence of a (constrained) weak SPE is decidable.
