Whereof one cannot speak, thereof one must be silent: A comment on the existence of universal structures.

Abstract.

It is a recurrent issue in epistemic game theory whether the models of information and higher-order information that are used make substantial assumptions about what the players know about each other, and in particular about how information is imparted to them. For Harsanyi type spaces, which are probabilistic models of the players' (higher-order) beliefs, the issue has been ``solved'' by showing that there exists a so-called universal type space, in which every possible coherent system of beliefs and higher-order beliefs are represented or, alternatively, in which every other type space can be mapped in an belief-preserving way. On the other hand, for Aumann or Kripke structures, the most widely used alternative to Harsanyi type spaces, it has been shown in a number of papers that no such universal structure exists.

In this talk I want to qualify this point of view, by arguing that the existence of a universal Aumann / Kripke structure depends on the language one uses to describe them. The argument will be threefold. I will first look at the notion of information-invariance which underpins the existing proofs of the non-existence of a universal knowledge structure. I will show that it is much stronger than the notions of invariance one gets from normal modal logic, PDL or the mu-calculus, and that once one substitutes the former for the latter, the counter-examples indeed disappear. Second, I will point out that, up to these notions of invariance, there is a structure that is universal, in both senses mentioned above. Finally, I will observe that the issue of language-dependency bears also on Harsanyi type spaces, as universal type spaces cease to exists once one lifts certain constraints on the language of probability calculus that is used to construct them.

(Work in progress with Eric Pacuit (TiLPS, Tilburg)).