Topologies on Types

Abstract

We de…ne and analyze a "strategic topology" on types in the Harsanyi-Mertens- Zamir universal type space, where two types are close if their strategic behavior is similar in all strategic situations. For a …xed game and action de…ne the distance be- tween a pair of types as the dierence between the smallest " for which the action is " interim correlated rationalizable. We de…ne a strategic topology in which a sequence of types converges if and only if this distance tends to zero for any action and game. Thus a sequence of types converges in the strategic topology if that smallest " does not jump either up or down in the limit. As applied to sequences, the upper-semicontinuity prop- erty is equivalent to convergence in the product topology, but the lower-semicontinuity property is a strictly stronger requirement, as shown by the electronic mail game. In the strategic topology, the set of "…nite types" (types describable by …nite type spaces) is dense but the set of …nite common-prior types is not. JEL classi…cation and keywords: C70, C72, rationalizability, incomplete informa- tion, common knowledge, universal type space, strategic topology.