# Almost Common Priors

Abstract:

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.

Date:

September, 2010

Paper:

Number:

560