The Common Prior Assumption in Belief Spaces: An Example

Robert Samuel Simon

With four persons there is an example of a probability space where 1) the space is generated by hierarchies of knowledge concerning a single proposition, 2) the subjective beliefs of the four persons are continuous regular conditional probability distributions of a common prior probability distribution (continuous with respect to the weak topology), and 3) for every subset that the four persons know in common there is no common prior probability distribution. Furthermore, for every measurable set, every person, and at every point in the space, the subjective belief in this measurable set is one of the quantities 0, 1/2 or 1. This example presents problems for understanding games of incomplete information through common priors.

December, 2000