Direct Representations for Interim Correlated Rationalizability

We study direct representations of information for interim correlated rationalizability. For a fixed finite payoff structure, each type induces a hierarchy of surviving action sets. Pushing the common prior through this map projects the information structure onto the solution concept's output language. When best-response regions are convex, this representation is direct: the solution concept applied to the hierarchy seen as a type is the identity. The induced distributions are characterized by level-by-level obedience constraints. Terminal ICR sets alone do not have this property. For arbitrary finite payoff structures, we refine each hierarchy level with a tag identifying a convex cell of its best-response region. Augmented hierarchies provide a direct representation and project onto the ordinary hierarchy. Full augmented hierarchies may form a continuum, but retaining only the tags at the boundaries of constant stretches of the ordinary hierarchy yields an exact countable representation with finitely many obedience constraints per type. Finite-type models are dense in terminal rationalizability outcome distributions.