Almost common priors

What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type space has δ prior distance, then for any bet f it cannot be common knowledge that each player expects a positive gain of δ times the sup-norm of f, thus extending no betting results under common priors. Furthermore, as more information is obtained and partitions are refined, the prior distance, and thus the extent of common knowledge disagreement, can only decrease. We derive an upper bound on the number of refinements needed to arrive at a situation in which the knowledge space has a common prior, which depends only on the number of initial partition elements.