The assessment of knowledge and learning progress in the context of game-based learning requires novel, noninvasive, and embedded approaches. In the present paper, we introduce a mathematical framework which relates the (problem solution) behavior of a learner in the game context to the learner's available and lacking competencies. We argue that a problem situation and its status at a certain point in time can be described by a set of game props and their current properties or states. In the course of the game, the learner can perform different actions to modify the props and, consequently, change the problem situation. Each action is evaluated with respect to its correctness or appropriateness for accomplishing a given task which, in turn, enables conclusions about the competence state of the learner. This assessment procedure serves as the basis for adaptive interventions, for instance, by providing the learner with guidance or feedback. (Contains 5 figures and 2 footnotes.)