Topologies on Types

We de…ne and analyze a "strategic topology" on types in the Harsanyi-MertensZamir 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 between a pair of types as the di¤erence 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 property 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 information, common knowledge, universal type space, strategic topology. Previous versions were circulated under the title "Dense Types". We are grateful for comments from seminar participants at Stanford University, the University of Essex, the Cowles Foundation conference on Uncertainty in Economic Theory, the Third International PIER/IGIER Conference on Economic Theory, the 2005 NSF/CEME Mathematical Economics Conference at Berkeley and the Harvard/MIT theory workshop. We are grateful to Dirk Bergemann, Je¤Ely, Yossi Feinberg, Aviad Heifetz, Atsushi Kajii, Bart Lipman, Aldo Rustichini, Muhamet Yildiz and Shmuel Zamir for helpful discussions; Marcin Peski for detailed comments; and Satoru Takahashi for expert proof reading. We thank the referees and an editor for their detailed and helpful comments. This material is based upon work supported by the National Science Foundation under Grants No. 0426199, 0111830 and 0095321. Morris is grateful for …nancial support from the John Simon Guggenheim Foundation and the Center for Advanced Studies in Behavioral Economics.