Almost Common Priors

Ziv Hellman

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.

September, 2010