Almost Common Priors

What happens when priors are not common? We show that for each type profile τ over a knowledge space (Ω, Π), where the state space Ω is connected with respect to the partition profile Π, we can associate a value 0 ≤ ε ≤ 1 that we term the prior distance of τ , where ε = 0 if and only if the profile has a common prior. If τ has ε prior distance, then for any bet f amongst the players, it cannot be common knowledge that each player expects a positive gain of ε‖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, decreases. %Length: 18 pages %File-URL: http://ratio.huji.ac.il/sites/default/files/publications/dp560.pdf