We show that every Bayesian game with purely atomic
types has a measurable Bayesian equilibrium when the common knowl-
edge relation is smooth. Conversely, for any common knowledge rela-
tion that is not smooth, there exists a type space that yields this common
knowledge relation and payoffs such that the resulting Bayesian game
will not have any Bayesian equilibrium. We show that our smoothness
condition also rules out two paradoxes involving Bayesian games with
a continuum of types: the impossibility of having a common prior on
components when a common prior over the entire state space exists, and
the possibility of interim betting/trade even when no such trade can be
supported
ex ante
