Bayesian games with nested information

We prove that any Bayesian game (\'a la Aumann) with a general state space, compact metric action spaces, and nested information admits a Harsanyi -equilibrium for every > 0. When, in addition, the action spaces and the payoffs are discrete, there is a Bayesian -equilibrium. To this end, we develop a new finite approximation of information structures, which has independent interest. We also put forth several open problems, including the existence of a 0-equilibrium (Harsanyi or Bayesian) in Bayesian games with nested information, the existence of a Harsanyi -equilibrium in multi-stage Bayesian games with nested information, and the canonical structure of the universal belief space when the information structure is nested.