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

Listed author(s):
  • Qin, Cheng-Zhong
  • Yang, Chun-Lei
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    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.

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    Paper provided by Department of Economics, UC Santa Barbara in its series University of California at Santa Barbara, Economics Working Paper Series with number qt8hq7j89k.

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    Date of creation: 03 Dec 2009
    Handle: RePEc:cdl:ucsbec:qt8hq7j89k
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    3. Liu, Qingmin, 2009. "On redundant types and Bayesian formulation of incomplete information," Journal of Economic Theory, Elsevier, vol. 144(5), pages 2115-2145, September.
    4. Atsushi Kajii & Stephen Morris, 1997. "The Robustness of Equilibria to Incomplete Information," Econometrica, Econometric Society, vol. 65(6), pages 1283-1310, November.
    5. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, vol. 82(2), pages 324-341, October.
    6. Morris, Stephen, 1995. "The Common Prior Assumption in Economic Theory," Economics and Philosophy, Cambridge University Press, vol. 11(02), pages 227-253, October.
    7. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-391, June.
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    9. Cremer, Jacques & McLean, Richard P, 1985. "Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist When Demands Are Interdependent," Econometrica, Econometric Society, vol. 53(2), pages 345-361, March.
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    18. Cremer, Jacques & McLean, Richard P, 1988. "Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions," Econometrica, Econometric Society, vol. 56(6), pages 1247-1257, November.
    19. Morris, Stephen, 1994. "Trade with Heterogeneous Prior Beliefs and Asymmetric Information," Econometrica, Econometric Society, vol. 62(6), pages 1327-1347, November.
    20. Stephen Morris & Hyun Song Shin, 2001. "Rethinking Multiple Equilibria in Macroeconomic Modeling," NBER Chapters,in: NBER Macroeconomics Annual 2000, Volume 15, pages 139-182 National Bureau of Economic Research, Inc.
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