A Canonical Self-Referring Type Space Representation Of The ’Context’ Of A Game

Following Aumann (1976) and Brandenburger-Friedenberg-Keisler (2008), the standard epistemic approach of strategic games imposes a given type structure to the players with the implicit idea that it is an objectively well-defined ’ commonly known environment or context’. But imposing such ’contexts’ has some important implications for the analysis e.g., under RCBR, it turns strategic uncertainty into the exogenous uncertainty of a (subjective) correlating device in a correlated equilibrium (Brandenburger and Dekel, 1987). Crucially, for a given game, there exists an infinite set of such possible ’contexts’ – each assigning a particular subset of types/ belief restrictions to the description of the game ( Battigalli and Friedenberg, 2009). So, the choice of one vs.another type structure remains an important open question. Here, we follow the suggestion in Brandenburger and Friedenberg (2010) and drop the standard assumption that a type structure (or ’context’ ) is exogenously assigned to the players. Our analysis treats the canonical case where each player is on a par with an analyst facing endogenous uncertainty with no underlying exogenous signals determining the players’ types i.e., no initial restricted subset of beliefs can be taken to be ’known’ by all the players. We show the existence of a ’meta- self-referring epistemicmodel’ where the notion of ’self-belief/knowledge’ arises as an ’intertwined/meta’ version of the ’self-evident events’ (Monderer and Samet [1989]): The subset of types at which the analyst believes possible a subset of states of the world–generating a certain subset of belief hierarchies for the players– must be those that agree with the subset of states at which the agent believes these types. In this self-referringmodel each ’context’–belief system– is an inherently subjective structure that always belongs to the mind of a single agent: An observer assigns subjectively a specific belief systemto the players if and only if he ’self-believes/knows’ that all the agents have the samemodel as his. Our results show that every such subjective player-specific type structure corresponds to a player-independent type structure whose types/beliefs are intertwined i.e., non-separable. In our central result we obtain that any notion of ’context’ of a game –wherein players are analysts allowed to choose their type-structures– is equivalent to the subjective specific type-structure of a meta-observer/analyst. These results suggest that the presence of endogenous uncertainty–wherein no exogenous ex ante stage is assumed– is incompatible with the existence of a ’player independent belief system’.