Beliefs and rationalizability in games with complementarities

We propose two characteristics of players' beliefs and study their role in shaping the set of rationalizable strategy profiles in games with incomplete information. The first characteristic, type-sensitivity, is related to how informative a player thinks his type is. The second characteristic, optimism, is related to how "favorable" a player expects the outcome of the game to be. The paper has two main results: the first result provides an upper bound on the size of the set of rationalizable strategy profiles, the second gives a lower bound on the change of location of this set. These bounds have explicit and relatively simple expressions that feature type-sensitivity, optimism, and properties of the payoffs. Our results generalize and clarify the well-known uniqueness result of global games (Carlsson and van Damme (1993)). They imply new uniqueness results and allow to study rationalizability in new environments. We provide applications to supermodular mechanism design (Mathevet (2010)) and non-Bayesian updating (Epstein (2006)).