Pareto refinements of pure-strategy equilibria in games with public and private information

In a Bayesian framework with public and private information that allows countably many players and infinitely many actions, we provide two sufficient conditions that ensure the existence of Pareto-undominated and socially-maximal pure-strategy Bayes–Nash equilibria under the usual diffuseness and independence assumptions: every player has (i) a countable action set, or (ii) a relatively-diffuse strategy-relevant private information space conditioned on a public signal. Our results rely on the theory of distributions of correspondences with infinite-dimensional range and draw on notions of nowhere equivalence, relative saturation, and saturation.