The existence of universal qualitative belief spaces

This paper constructs a canonical representation of players' interactive beliefs, irrespective of natures of beliefs: whether beliefs are qualitative, truthful (i.e., knowledge), or probabilistic (e.g., countably-additive, finitely-additive, or non-additive). The canonical model is the “largest” interactive belief model to which any particular model can be mapped in a unique belief-preserving way. The key insight for the construction is the need to specify players' possible depth of reasoning up to which they can interactively reason about their beliefs (e.g., their beliefs, their beliefs about their beliefs, their beliefs about their beliefs about their beliefs, and so on). The possible depth of reasoning may be a transfinite level (beyond any finite level) when beliefs are qualitative. The specification of possible depth of reasoning also has game-theoretic implications for characterizations of some solution concepts using the canonical space. For instance, for any strategic game with ordinal payoffs, there exists a canonical interactive belief model which characterizes iterated elimination of strictly dominated actions as an implication of common belief in rationality.