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

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  • Qin, Cheng-Zhong
  • Yang, Chun-Lei
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    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 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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    Bibliographic Info

    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
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    Handle: RePEc:cdl:ucsbec:qt8hq7j89k

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    Keywords: Bayesian game; consistent prior; embedding; finite-order type space; heterogeneous priors; projection type space; robustness; total variation norm; Other Applied Mathematics; Other Economics;

    References

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    1. Muhamet Yildiz & Jonathan Weinsten, 2004. "Impact of higher-order uncertainty," Econometric Society 2004 North American Winter Meetings 157, Econometric Society.
    2. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    3. Brandenburger Adam & Dekel Eddie, 1993. "Hierarchies of Beliefs and Common Knowledge," Journal of Economic Theory, Elsevier, vol. 59(1), pages 189-198, February.
    4. R. Aumann, 2010. "Correlated Equilibrium as an expression of Bayesian Rationality," Levine's Bibliography 513, UCLA Department of Economics.
    5. Aviad Heifetz & Zvika Neeman, 2004. "On the Generic (Im)possibility of Full Surplus Extraction in Mechanism Design," Discussion Paper Series dp350, The Center for the Study of Rationality, Hebrew University, Jerusalem.
    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. Neeman, Zvika, 2004. "The relevance of private information in mechanism design," Journal of Economic Theory, Elsevier, vol. 117(1), pages 55-77, July.
    8. Stephen Morris & Hyun Song Shin, 2000. "Rethinking Multiple Equilibria in Macroeconomic Modelling," Cowles Foundation Discussion Papers 1260, Cowles Foundation for Research in Economics, Yale University.
    9. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-91, June.
    10. Liu, Qingmin, 2009. "On redundant types and Bayesian formulation of incomplete information," Journal of Economic Theory, Elsevier, vol. 144(5), pages 2115-2145, September.
    11. Morris, Stephen, 1994. "Trade with Heterogeneous Prior Beliefs and Asymmetric Information," Econometrica, Econometric Society, vol. 62(6), pages 1327-47, November.
    12. Atsushi Kajii & Stephen Morris, . ""The Robustness of Equilibria to Incomplete Information*''," CARESS Working Papres 95-18, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
    13. Dirk Bergemann & Stephen Morris, 2005. "Robust Mechanism Design," Econometrica, Econometric Society, vol. 73(6), pages 1771-1813, November.
    14. Jeffrey C. Ely & Marcin Peski, . "Hierarchies Of Belief And Interim Rationalizability," Discussion Papers 1388, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    15. Faruk Gul & Hugo Sonnenschein & Robert Wilson, 2010. "Foundations of Dynamic Monopoly and the Coase Conjecture," Levine's Working Paper Archive 232, David K. Levine.
    16. Kim-Sau Chung & Jeffrey C. Ely, 2003. "Implementation with Near-Complete Information," Econometrica, Econometric Society, vol. 71(3), pages 857-871, 05.
    17. 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 345-61, March.
    18. Faruk Gul, 1998. "A Comment on Aumann's Bayesian View," Econometrica, Econometric Society, vol. 66(4), pages 923-928, July.
    19. Jonathan Weinstein & Muhamet Yildiz, 2007. "A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements," Econometrica, Econometric Society, vol. 75(2), pages 365-400, 03.
    20. Xiong, Siyang & Chen, Yi-Chun & di Tillio, Alfredo & Faingold, Eduardo, 2010. "Uniform topologies on types," Theoretical Economics, Econometric Society, vol. 5(3), September.
    21. Aviad Heifetz & Dov Samet, 1996. "Topology-Free Typology of Beliefs," Game Theory and Information 9609002, EconWPA, revised 17 Sep 1996.
    22. Robert J. Aumann, 1998. "Common Priors: A Reply to Gul," Econometrica, Econometric Society, vol. 66(4), pages 929-938, July.
    23. Cremer, Jacques & McLean, Richard P, 1988. "Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions," Econometrica, Econometric Society, vol. 56(6), pages 1247-57, November.
    24. Drew Fudenberg & David K. Levine & Jean Tirole, 1985. "Infinite-Horizon Models of Bargaining with One-Sided Incomplete Information," Levine's Working Paper Archive 1098, David K. Levine.
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