  We study multi-player turn-based games played on (potentially infinite)
directed graphs. An outcome is assigned to every play of the game. Each player
has a preference relation on the set of outcomes which allows him to compare
plays. We focus on the recently introduced notion of weak subgame perfect
equilibrium (weak SPE). This is a variant of the classical notion of SPE, where
players who deviate can only use strategies deviating from their initial
strategy in a finite number of histories. Having an SPE in a game implies
having a weak SPE but the contrary is generally false.
  We propose general conditions on the structure of the game graph and on the
preference relations of the players that guarantee the existence of a weak SPE,
that additionally is finite-memory. From this general result, we derive two
large classes of games for which there always exists a weak SPE: (i) the games
with a finite-range outcome function, and (ii) the games with a finite
underlying graph and a prefix-independent outcome function. For the second
class, we identify conditions on the preference relations that guarantee
memoryless strategies for the weak SPE.
