Rational Belief in Hierarchies
Abstract:
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 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 first main result shows that this type-based representation
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 common
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.
Keywords: Epistemic game theory, bounded rationality, rational numbers, belief hierarchies,
type spaces, unawareness, computational capacity, common belief in rationality.