Subgame perfection in recursive perfect information games

We consider sequential multi-player games with perfect information and with deterministic transitions. The players receive a reward upon termination of the game, which depends on the state where the game was terminated. If the game does not terminate, then the rewards of the players are equal to zero. We prove that, for every game in this class, a subgame perfect $$\varepsilon $$-equilibrium exists, for all $$\varepsilon > 0$$. The proof is constructive and suggests a finite algorithm to calculate such an equilibrium.