Conditional Probability Spaces and the Structure of Agreement

Abstract: We use the machinery of a conditional probability space (Rényi, 1955) to obtain an Agreement Theorem (Aumann, 1976) under general conditions. A conditional probability space (CPS) is a family of probability measures defined relative to a family of conditioning events that satisfies concentration and a chain rule. Using this apparatus, we derive an Agreement Theorem that dispenses with the traditional assumptions of a common prior, information partitions, positivity of measure, and knowledge operators. Our treatment can be viewed as ``deconstructing" the classic Agreement Theorem, by showing how it can be built up from local probabilistic-epistemic ingredients. The main technical contribution is to define an augmentation procedure for CPS's that adds into the conditioning family all (sub)events that receive probability $1$ -- thereby achieving consistency between an agent's information and subjective certainty of events.