We prove the existence of a subgame-perfect "-equilibrium, for every " > 0, in a class of multi-player games with perfect information, which we call free transition games. The novelty is that a non-trivial class of perfect information games is solved for subgame- perfection, with multiple non-terminating actions, in which the payo structure is generally not semi-continuous. Due to the lack of semi-continuity, there is no general rule of comparison between the payo s that a player can obtain by deviating a large but nite number of times or, respectively, in nitely many times. We introduce new techniques to overcome this di culty.