We show that every Bayesian game with purely atomic types has a measurable Bayesian equilibrium when the common knowledge relation is smooth.
Conversely, for any common knowledge relation 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.