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Rational belief hierarchies

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Working paper
Issue number: 
RM/12/004
Publisher: 
Maastricht University
Year: 
2012
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.
Developed by Paolo Gittoi