Rational belief hierarchies

We consider agents whose language can only express probabilistic beliefs that attach a rational number to every event. We call these probability measures rational. We intro- duce 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 un- certainty 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 first main result shows that this type-based representa- tion violates our intuitive idea of an agent whose language expresses only rational beliefs, in that there are rational types associated with non-rational beliefs over the canonical state space. We rule out these types by focusing on the rational types that satisfy com- mon certainty in the event that everybody holds rational beliefs over the canonical state space. We call these types universally rational and show that they are characterized by a bounded rationality condition which restricts the agents’ computational capacity. Moreover, the universally rational types form a dense subset of the universal type space. Finally, we show that the strategies rationally played under common universally rational belief in rationality generically coincide with those satisfying correlated rationalizability.