Oikkonen, J. and J. Väänänen, Game-theoretic inductive definability, Annals of Pure and Applied Logic 65 265-306. We use game-theoretic ideas to define a generalization of the notion of inductive definability. This approach allows induction along non-well-founded trees. Our definition depends on an underlying partial ordering of the objects. In this ordering every countable ascending sequence is assumed to have a unique supremum which enables us to go over limits. We establish basic properties of this induction and examine examples where it (...) emerges naturally. In the main results we prove an abstract Kleene Theorem and restricted versions of the Stage-Comparison Theorem and the Reduction Theorem. (shrink)