On the structure of rationalizability for arbitrary spaces of uncertainty

Weinstein and Yildiz (2007) show that only very weak predictions are robust to misspecifications of higher order beliefs. Whenever a type has multiple rationalizable actions, any of these actions is uniquely rationalizable for some arbitrarily close type. Hence, refinements of rationalizability are not robust. This negative result is obtained under a richness condition, which essentially means that all common knowledge assumptions on payoffs are relaxed. In many settings, this condition entails an unnecessarily demanding robustness test. It is, therefore, natural to explore the structure of rationalizability when arbitrary common knowledge assumptions are relaxed (i.e., without assuming richness). For arbitrary spaces of uncertainty and for every player i, I construct a set A{script}i ∞ of actions that are uniquely rationalizable for some hierarchy of beliefs. The main result shows that for any type ti and any action ai rationalizable for ti, if ai belongs to A{script}i ∞ and is justified by conjectures concentrated on A{script}-i ∞, then there exists a sequence of types converging to ti for which ai is uniquely rationalizable. This result significantly generalizes Weinstein and Yildiz's. Some of its implications are discussed in the context of auctions and equilibrium refinements, and in connection with the literature on global games.