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

43 Pages Posted: 18 Jan 2010

See all articles by Cheng-Zhong Qin

Cheng-Zhong Qin

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

Chun-Lei Yang

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Date Written: January 14, 2010


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 the recent development of Bayesian game theory with higher-order uncertainty involve belief hierarchies of order 2.

We start with characterizing order-2 consistent priors for given payoff environments and show that they form a convex class that contains the convex hull of the naive and complete information priors.

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 extend the notion of consistent priors to arbitrary finite orders. We define an abstract common prior 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 derive finite-order projections" from a type space to get projection type spaces. The condition of global stability under uncertainty ensures the convergence of the Bayesian-Nash equilibria of a game with projection type spaces to those with the original type space as the order of projection increases.

We introduce an order-k total variation norm for priors based on order-k projections as an alternative to the product topology induced by the weak topology. By applying this norm, we generalize the Kajii and Morris's (1997) idea of equilibrium robustness and the Monderer and Samet's (1989) robustness results for complete information games to Bayesian 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 (January 14, 2010). Available at SSRN: or

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