Topologies on Types

We define and analyze strategic topologies on types, under which two types are close if their strategic behavior will be similar in all strategic situations. To operationalize this idea, we adopt interim rationalizability as our solution concept, and define a metric topology on types in the Harsanyi-Mertens-Zamir universal type space. This topology is the coarsest metric topology generating upper and lower hemicontinuity of rationalizable outcomes. While upper strategic convergence is equivalent to convergence in the product topology, lower strategic convergence is a strictly stronger requirement, as shown by the electronic mail game. Nonetheless, we show that the set of finite types (types describable by finite type spaces) are dense in the lower strategic topology