Rational belief hierarchies

We consider agents who attach a rational probability to every Borel event. We call these Borel probability measures rational, and introduce the notion of a rational belief hierarchy, where the first order beliefs are described by a rational measure over the fundamental space of uncertainty, the second order beliefs are described by a rational measure over the product of the fundamental space of uncertainty and the opponent's first order rational beliefs, and so on. Then, we derive the corresponding rational type space model, thus providing a Bayesian representation of rational belief hierarchies. Our main result shows that this type-based representation has the counterintuitive property that some rational types are associated with non-rational beliefs over the product of the fundamental space of uncertainty and the opponent's types, thus implying that the agent may attach an irrational probability to some Borel event even if she has a rational belief hierarchy.