Rational belief hierarchies
We consider agents whose language can only express probabilistic beliefs that attach a rationalnumber to every event. We call these probability measures rational. We introduce the notion of arational belief hierarchy, where the first order beliefs are described by a rational measure overthe fundamental space of uncertainty, the second order beliefs are described by a rational measureover the product of the fundamental space of uncertainty and the opponent''s first order rationalbeliefs, and so on. Then, we derive the corresponding (rational) type space model, thus providinga Bayesian representation of rational belief hierarchies. Our first main result shows that thistype-based representation violates our intuitive idea of an agent whose language expresses onlyrational beliefs, in that there are rational types associated with non-rational beliefs over thecanonical state space. We rule out these types by focusing on the rational types that satisfycommon certainty in the event that everybody holds rational beliefs over the canonical statespace. We call these types universally rational and show that they are characterized by a boundedrationality condition which restricts the agents'' computational capacity. Moreover, theuniversally rational types form a dense subset of the universal type space. Finally, we show thatthe strategies rationally played under common universally rational belief in rationalitygenerically coincide with those satisfying correlated rationalizability.
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