The Hierarchical construction of a universal qualitative belief space

Abstract This paper constructs a canonical representation of players’ belief hierarchies—players’ beliefs over some exogenously given values such as their action profiles or payoff functions, their beliefs about their beliefs about exogenously given values, and so on ad infinitum—in the context of non-probabilistic beliefs, including knowledge. This paper demonstrates that the idea that any “possible” belief hierarchy of a player can be captured as the player’s type holds true regardless of whether players’ beliefs are probabilistic or qualitative. Formally, the first main result is to construct a universal qualitative belief space as the set of players’ belief hierarchies that can be induced by some qualitative belief space. The second is to show that the universal qualitative belief space coincides with the set of coherent belief hierarchies.