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Uniform Topology on Types and Strategic Convergence
Abstract: We study the
continuity of the correspondence of interim epsilon-rationalizable
actions in incomplete information games. We introduce a topology
on types, called uniform weak topology, under which two types of
a player are close if they have similar first-order beliefs,
attach similar probabilities to other players having similar
first-order beliefs, and so on, where the degree of similarity
is uniform over the levels of the belief hierarchy. This notion
of proximity of types is an extension of the concept of common
p-belief due to Monderer and Samet (1989). We show that given
any finite game every action that is interim rationalizable for
a finite type t remains interim epsilon-rationalizable for all
types sufficiently close to t in the uniform-weak topology.
Conversely, given any finite type t there exist epsilon > 0 and
a finite game such that some interim rationalizable action for t
fails to be interim epsilon-rationalizable for every type that
is not close to t in the uniform-weak topology. Our results thus
establish the equivalence between the uniform-weak topology and
the strategic topology of Dekel, Fudenberg, and Morris (2006)
around finite types.
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