We study the strategic impact of players’ higher-order uncertainty over the observability of actions in general two-player games. More specifically, we consider the space of all belief hierarchies generated by the uncertainty over whether the game will be played as a static game or with perfect information. Over this space, we characterize the correspondence of a solution concept which captures the behavioral implications of Rationality and Common Belief in Rationality (RCBR), where `rationality’ is understood as sequential whenever a player moves second. We show that such a correspondence is generically single-valued, and that its structure supports a robust refinement of rationalizability, which often has very sharp implications. For instance: (i) in a class of games which includes both zero-sum games with a pure equilibrium and coordination games with a unique efficient equilibrium, RCBR generically ensures efficient equilibrium outcomes (eductive coordination); (ii) in a class of games which also includes other well-known families of coordination games, RCBR generically selects components of the Stackelberg profiles (Stackelberg selection); (iii) if it is commonly known that player 2’s action is not observable (e.g., because 1 is commonly known to move earlier, etc.), in a class of games which includes all of the above RCBR generically selects the equilibrium of the static game most favorable to player 1 (pervasiveness of first-mover advantage).