An Explicit Approach to Modeling Finite-Order Type Spaces and Applications

43 Pages Posted: 5 Dec 2009

See all articles by Cheng-Zhong Qin

Cheng-Zhong Qin

University of California, Santa Barbara (UCSB) - Department of Economics

Chun-Lei Yang

Multiple version iconThere are 2 versions of this paper

Date Written: December 3, 2009

Abstract

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 naive and complete-information type spaces. We establish conditions for private-value heterogeneous naive 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 type 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 projections" of 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 (1997) notion of robustness to general Bayesian games. We then establish the robustness of Bayesian-Nash equilibria for a Bayesian game 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.

Suggested Citation

Qin, Cheng-Zhong and Yang, Chun-Lei, An Explicit Approach to Modeling Finite-Order Type Spaces and Applications (December 3, 2009). Available at SSRN: https://ssrn.com/abstract=1508742 or http://dx.doi.org/10.2139/ssrn.1508742

Cheng-Zhong Qin (Contact Author)

University of California, Santa Barbara (UCSB) - Department of Economics ( email )

2127 North Hall
Santa Barbara, CA 93106
United States

No contact information is available for Chun-Lei Yang

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