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

Abstract

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