Common Knowledge and Disparate Priors: When it is O.K. to Agree to Disagree

Abandoning the oft-presumed common prior assumption, partitioned type spaces with disparate priors are studied. It is shown that in the two-player case, a unique fundamental pair of priors can be identified in each type space, from whose properties boundaries on the possible ranges of expected values under common knowledge can be derived. In the limit as the elements of this pair approach each other,a common prior is identified, and standard results stemming from the common prior assumption are recapitulated. It is further shown that this two-player fundamental pair of priors is a special case of the n-player situation, where a representative n-tuple of fundamentally associated priors can be selected, out of at most n-1 such n-tuples, to play an analogous role.