Conditional belief types

- We axiomatize type spaces where a player's type at a state is a conditional probability on the space.
- We examine three additional axioms of increasing strength: introspection, echo, determination.
- Echo implies that conditioning on an event is conditioning on the event being certain.
- We illustrate the game-theoretic application of the model.
- Formal connections to the sphere models of counterfactual conditionals and to models of hypothetical knowledge are discussed.

We study type spaces where a player's type at a state is a conditional probability on the space. We axiomatize these spaces using conditional belief operators, examining 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. Echo implies that conditioning on an event is the same as conditioning on the event being certain, which formalizes the standard informal interpretation of conditional probability. The game-theoretic application of our model, discussed within an example, sheds light on a number of 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.