Rational belief 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.

Universally Rational Belief Hierarchies

“…Henceforth, whenever we write "rational belief hierarchies" or "rational types", we implicitly refer to elements of T Q , thus omitting to explicitly say that they satisfy coherency and common certainty in coherency. Following Brandenburger and Dekel (1993), Tsakas (2013) proved the existence of a terminal type space model of rational belief hierarchies, implying that every rational belief hierarchy is identified by a Borel probability measure on Θ × T Q , via the injective function…”

“…The latter is not surprising, as one can easily see that there exist probability measures π ∈ ∆(Θ × T Q ) with marg Θ π / ∈ ∆ Q (Θ), e.g., a measure with π({θ} × T Q ) = √ 2/2. More interestingly, Tsakas (2013) showed that there exist types that are mapped via g to nonrational probability measures over Θ × T Q . Throughout the paper, we call the rational types that do not exhibit this property universally rational, i.e., a type t ∈ T Q is universally rational if and only if g(t) ∈ ∆ Q (Θ × T Q ).…”

“…In a recent paper, Tsakas (2013) restricted attention to probabilistic beliefs that can take only rational values, e.g., he ruled out beliefs of the form "E occurs with probability √ 2/2". Such beliefs are modeled by Borel probability measures that attach a rational number to every Borel event.…”

Pairwise Stable Matching in Large Economies

“…1616 It can be shown that a finite measure that takes on only rational values can take on only finitely many values; see, for example, Tsakas (2014, Proposition 1). In the present context, such a measure must have a finite support.…”