We examine the limiting outcomes of a dynamic evolutionary process driven by stochastic learning and rare mutations. We first show that locally stable outcomes are subgame perfect and satisfy a forward induction property. To address cases in which locally stable outcomes fail to exist, we turn to a dynamic analysis. The limiting distribution of the dynamic process in a class of extensive form games with perfect information always includes the subgame perfect equilibrium outcome, but consists exclusively of that outcome only under stringent conditions. The limiting distribution in a class of outside option games satisfies a forward induction requirement.