On the relationship between hierarchy and type morphisms

Abstract

Here, we establish a relationship between hierarchy and type morphisms. Both concepts have been used to relate the types in one structure to types in a second "larger" structure. In general, the two concepts may di¤er, in the sense that a hierarchy morphism need not be a type morphism. We provide a necessary and su¢ cient condition under which the two concepts coincide. We go on to provide situations under which this condition must be satis…ed.

When Do Types Induce the Same Belief Hierarchy?

“…A number of papers has shown that the measurable structure associated with type structures can impose restrictions on reasoning [12,[20][21][22][23]. This paper contributes to that literature in two ways.…”

“…The second test uses type morphisms. Type morphisms are defined solely in terms 1 We follow the terminology of Friedenberg and Meier [12] here. A stronger condition for T to be contained in T is that T can be embedded (using a type morphism) into T as a belief-closed subset [13].…”

“…However, the test based on type morphisms only provides a sufficient condition: if there is a type morphism from T to T , then T is contained in T [14]. However, as shown by Friedenberg and Meier [12], the condition is not necessary: it may be that T contains T , yet there is no type morphism from T to T . The work in [12] also provides a range of conditions under which the condition is both necessary and sufficient.…”

“…However, as shown by Friedenberg and Meier [12], the condition is not necessary: it may be that T contains T , yet there is no type morphism from T to T . The work in [12] also provides a range of conditions under which the condition is both necessary and sufficient. However, they do not directly address the question of whether there might be an alternate test (which provides conditions that are both necessary and sufficient) that does not require us to describe the belief hierarchies explicitly.…”

“…Mertens and Zamir ( [13], p. 6) use this σ-algebra to define non-redundant type structures, and this σ-algebra also plays an important role in the work of Friedenberg and Meier [12], where it is used to characterize the conditions under which hierarchy morphisms and type morphisms coincide. The work in [13] provides a nonconstructive definition of this σ-algebra, and [12] show that the σ-algebra defined by [13] is the σ-algebra generated by the functions that map types into belief hierarchies. We provide a constructive definition of this σ-algebra, by means of a type partitioning procedure that does not make reference to belief hierarchies.…”