Agreeing to disagree with lexicographic prior beliefs

The robustness of Aumann's seminal agreement theorem with respect to the common prior assumption is considered. More precisely, we show by means of an example that two Bayesian agents with almost identical prior beliefs can agree to completely disagree on their posterior beliefs. Besides, a more detailed agent model is introduced where posterior beliefs are formed on the basis of lexicographic prior beliefs. We then generalize Aumann's agreement theorem to lexicographic prior beliefs and show that only a slight perturbation of the common lexicographic prior assumption at some-even arbitrarily deep-level is already compatible with common knowledge of completely opposed posterior beliefs. Hence, agents can actually agree to disagree even if there is only a slight deviation from the common prior assumption.