On the structure of rationalizability for arbitrary spaces of uncertainty

Weinstein and Yildiz (Econometrica, 2007) have shown that only very weak predictions are robust to mispecifications 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_{i}^{∞} of actions that are uniquely rationalizable for some hierarchy of beliefs. The main result shows that for any type t_{i}, and any action a_{i} rationalizable for t_{i}, if a_{i} belongs to A_{i}^{∞} and is justified by conjectures concentrated on A_{-i}^{∞}, then there exists a sequence of types converging to t_{i} for which a_{i} is uniquely rationalizable. This result significantly generalizes Weinstein and Yildiz's. Some of its implications are discussed in the context of auctions, equilibrium refinements and in connection with the literature on global games.