When do type structures contain all hierarchies of beliefs?

Friedenberg, Amanda

doi:10.1016/j.geb.2009.05.005

“…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.…”

Section: Introductionmentioning

confidence: 87%

When Do Types Induce the Same Belief Hierarchy?

Perea

¹

,

Kets

²

2016

Games

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Type structures are a simple device to describe higher-order beliefs. However, how can we check whether two types generate the same belief hierarchy? This paper generalizes the concept of a type morphism and shows that one type structure is contained in another if and only if the former can be mapped into the other using a generalized type morphism. Hence, every generalized type morphism is a hierarchy morphism and vice versa. Importantly, generalized type morphisms do not make reference to belief hierarchies. We use our results to characterize the conditions under which types generate the same belief hierarchy.

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“…13 It is possible to define a sense in which these two measurable space are equal from a measurable perspective. This is captured by the following definition.…”

Section: Preliminariesmentioning

confidence: 99%

“…Building on previous work by [38] and [29], this is the path chosen by [22] to construct the topology-free terminal type structure with unawareness. An alternative definition of terminality for topological settings can be found in [13], where a terminal type structure T is defined as a type structure such that, for every player j ∈ I, for every type t j ∈ T j in an arbitrary type structure T , there is a type t j ∈ T j in T such that t j and t j induce the same coherent hierarchy of beliefs.…”

Section: Definition 38 (Universal Type Structure)mentioning

confidence: 99%

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

Guarino

¹

2017

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We construct the universal type structure for conditional probability systems without any topological assumption, namely a type structure that is terminal, belief-complete, and non-redundant. In particular, in order to obtain the belief-completeness in a constructive way, we extend the work of Meier [An Infinitary Probability Logic for Type Spaces. Israel Journal of Mathematics, 192, by proving strong soundness and strong completeness of an infinitary conditional probability logic with truthful and non-epistemic conditioning events.

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“…8 See Section 5 for the definition of non-redundancy, which is the notion that does not appear in the original taxonomy [35]. However, recent contributions have stressed the importance of this notion (e.g., [13]). 9 See [23, Chapter 3] for an introduction to the topic.…”

Section: Preliminariesmentioning

confidence: 99%

“…Building on previous work by [38] and [29], this is the path chosen by [22] to construct the topology-free terminal type structure with unawareness. An alternative definition of terminality for topological settings can be found in [13], where a terminal type structure T is defined as a type structure such that, for every player j ∈ I, for every type t j ∈ T j in an arbitrary type structure T , there is a type t j ∈ T j in T such that t j and t j induce the same coherent hierarchy of beliefs.…”

Section: Large Type Structuresmentioning

confidence: 99%

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

Guarino

¹

2017

Electron. Proc. Theor. Comput. Sci.

1

0

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We construct the universal type structure for conditional probability systems without any topological assumption, namely a type structure that is terminal, belief-complete, and non-redundant. In particular, in order to obtain the belief-completeness in a constructive way, we extend the work of Meier [An Infinitary Probability Logic for Type Spaces. Israel Journal of Mathematics, 192, by proving strong soundness and strong completeness of an infinitary conditional probability logic with truthful and non-epistemic conditioning events.2 See Section 1.1 for an explanation of why we choose to call such space "terminal". 3 Conditional probability systems have been introduced in the game-theoretic literature by [30] building on the notion of conditional probability space of [32] (see [16] for an analysis of this and related notions). 4 Observe that it is the fact that this construction is indeed explicit (i.e., performed via infinitary probability logic) that makes it more informative than constructions performed via coalgebraic methods, since both constructions ensure that the type structure obtained is both terminal and belief-complete ((see Section 3.3 for an explanation of these notions). Indeed, as noticed by [22], proving the terminality of a type structure via coalgebraic methods ensures a fortiori also the belief-completeness of this type structure thanks to a standard result of category theory from [24] known as Lambek's Lemma. 5 There is an alternative notion of "universality" that can be found in the literature, for example in [8], [13], and [12]. According to this notion, a type structure is universal if there is an ordinal number α such that, for every ordinal β > α, the β -belief order is the same as the α-belief order, that is the α-belief order determines all subsequent belief-orders (all the explicit constructions starting from a topological space satisfy this definition with α := ω, where ω is the ordinal counterpart of N).In [20] this idea is used to prove that there is no universal (in the sense above) structure for knowledge spaces. See [2] for a treatment of knowledge spaces. 6 A preference structure is a structure that takes preferences and not beliefs as primitive objects, building on the idea of [34] that beliefs can be derived from preferences. See [11], [10] for explicit constructions of large preference structures with

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When do type structures contain all hierarchies of beliefs?

Cited by 34 publications

References 25 publications

When Do Types Induce the Same Belief Hierarchy?

When Do Types Induce the Same Belief Hierarchy?

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

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