Conditional Probability Spaces and the Structure of Agreement

Abstract

We use the machinery of a conditional probability space (R\'enyi, 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 CPSs 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.