  We prove that any Bayesian game (\'a la Aumann) with a general state space,
compact metric action spaces, and nested information admits a Harsanyi
$\varepsilon$-equilibrium for every $\varepsilon> 0$. When, in addition, the
action spaces and the payoffs are discrete, there is a Bayesian
$\varepsilon$-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 $\varepsilon$-equilibrium in multi-stage Bayesian games with nested
information, and the canonical structure of the universal belief space when the
information structure is nested.
