The positive foundation of the common prior assumption

The existence of a common prior is a property of the state space used to model the players' incomplete information. We show that this property is not just a technical artifact of the model, but that it is immanent to the players' beliefs. To this end, we devise a condition, phrased solely in terms of the players' mutual beliefs about the basic, objective issues of possible uncertainty, which is equivalent to the existence of a common prior. This condition specifies a countable sequence of questions addressed to the players, which detects when there is no common prior among them.