Bayesian games with a continuum of states

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

Three Variations on Money Pump, Common Prior, and Trade

We consider finite information structures, and quest for the answer of the question: What is the proper definition of prior? In the single player setting we conclude that a probability distribution is a prior if it is disintegrable, because this definition excludes money pump. In the multiplayer setting our analysis does not boil down to one proper notion of common prior (the multiplayer version of prior). The appropriate notion is a choice of the modeller in this setting. We consider three variants of money pump, each "defines" a notion of common prior. Furthermore, we also consider three variants of trade, each correspond to one of the money pump variants, hence to one of the common prior variants