Consider repeated two-player games with perfect monitoring and discounting. We provide an algorithm that computes the set V* of payoff pairs of all pure-strategy subgame perfect equilibria with public randomization. The algorithm provides signiﬁcant eﬃciency gains over the existing implementations of the algorithm from Abreu, Pearce and Stacchetti (1990). These eﬃciency gains arise from a better understanding of the manner in which extreme points of the equilibrium payoﬀ set are generated. An important theoretical implication of our algorithm is that the set of extreme points E of V* is finite. Indeed,≤ 3where A is the set of action proﬁles of the stage game.