Abstract. Economic models employ assumptions about agents'
infinite hierarchies of belief. We might hope to achieve
reasonable approximations by specifying only finitely many
levels in the hierarchy. However, it is well known since
Rubinstein (1989) that the behaviors of some fully specified
hierarchies can be very different from the behavior of such
finite approximations. Examples and earlier results in the
literature suggest that these critical types are characterized
by some strong assumptions on higher-order beliefs. We formalize
this connection. We define a critical type to be any hierarchy
at which the rationalizable correspondence exhibits a
discontinuity. We show that critical types are precisely those
types for which there is common belief in a certain class of
event. All types from finite type spaces and almost all types in
common prior type spaces are critical. On the other hand, we
show that regular types, i.e. types which exhibit no
discontinuities, are generic. In particular they form a residual
set in the product topology. This second result strengthens a
previous one due to Weinstein and Yildiz (2006) in two ways.
First, while Weinstein and Yildiz (2006) considered a xed game,
our regular types have continuous behavior across all games.
Second, our result applies to an arbitrary space of basic
uncertainty and does not require the rich-fundamentals
assumption employed by Weinstein and Yildiz (2006). Our proofs
involve a novel characterization of the strategic topology first
introduced by Dekel, Fudenberg, and Morris (2006a).
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