Bayesian equilibrium: From local to global

We study Bayesian games with a continuum of states which partition into a continuum of components, each of which is common knowledge, such that equilibria exist on each component. A canonical case is when each agent’s information consists of both public and private information, and conditional on each possible public signal, equilibria exist. We show that under some regularity conditions on the partition, measurable Bayesian equilibria exist for the game in its entirety. The results extend to pure equilibria, as well as to non-compact state-dependent action sets, uncommon priors, and non-bounded payoffs; the results also apply to several notions of approximate equilibria.