Critical Types

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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Jeff Ely