Every abstract type of a belief-closed type space corresponds to an infinite belief hierarchy. But only finite order of beliefs is necessary for most applications. As we demonstrate, many important insights from recent development in the theory of Bayesian games with higher-order uncertainty involve belief hierarchies of order 2.
We start with characterizing order 2 "consistent priors" and show that they form a convex set and contain the convex hull of both the naïve and complete-information type spaces. We establish conditions for private-value heterogeneous naïve priors to be embedded in order-2 consistent priors, so as to retro-fit the Harsanyi doctrine of having nature generate all fundamental uncertainties in a game at the very beginning.
We then extend the notion of consistent priors to arbitrary finite order k. We define an abstract belief-closed space to be of order k if it can be mapped via a type morphism into the "canonical representation" of an order-k consistent prior. We show that order-k type spaces are those in which any two types of each player must be either identical implying one of them is redundant or separable by their order (k-1) belief hierarchies. Finite type spaces are always of finite orders. We consider "finite-order projection" or a type space and show that they are finite-order type spaces themselves. The condition of global stability under uncertainty ensures the convergence of the Bayesian-Nash equilibria with the projection type spaces to those with the original type space.
By defining a total variation norm based on finite-order projections, we generalize Kajii and Morris's (1997) idea of equilibrium robustness to Bayesian games. We then establish the robustness of Bayesian-Nash equilibria that generalizes the robustness results of Monderer and Samet (1989) for complete-information games.
We apply our framework of finite-order type spaces or consistent priors to review several important models in the literature and illustrate some new insights.