Conditional belief types

We study type spaces where a player’s type at a state is a conditional probability on the space. We axiomatize these type spaces using conditional belief operators, and examine three additional axioms of increasing strength. First, introspection, which requires the agent to be unconditionally certain of her beliefs. Second, echo, according to which the unconditional beliefs implied by the condition must be held given the condition. Third, determination, which says that the conditional beliefs are the unconditional beliefs that are conditionally certain. The echo axiom implies that conditioning on an event is the same as conditioning on the event being certain, which formalizes the standard informal interpretation of conditioning in probability theory. The echo axiom also implies that the conditional probability given an event is a prior of the unconditional probability. The game-theoretic application of our model, which we treat in the context of an example, sheds light on a number of basic issues in the analysis of extensive form games. Type spaces are closely related to the sphere models of counterfactual conditionals and to models of hypothetical knowledge, and we discuss these relationships in detail.