 However, all of these “no disagreement” results raise the same philosophical puzzle raised by Aumann’s original result: How are we to explain differences in belief? Aumann’s result leaves us with two options: (1) admit that at some level, common knowledge of the agents’ beliefs or how they form their beliefs fails, or (2) deny the CPA. Thus, even if agents do assign precise posterior probabilities to an event, Aumann shows that if they have merely first-order mutual knowledge of the posteriors, they can “agree to  disagree”.[18]  Another way Aumann’s result might fail is if agents do not have common knowledge that they update their beliefs by Bayesian conditionalization. Then clearly, agents can explain divergent opinions as the result of others having modified their beliefs in the “wrong” way. However, there are cases in which neither explanation will seem convincing and denying the requisite common knowledge seems a rather ad hoc move. Why should one think that such failures of common knowledge provide a general explanation for divergent beliefs?