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Topologies on types

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  • Dekel, Eddie

    ()
    (Northwestern University and Tel Aviv University)

  • Fudenberg, Drew

    ()
    (Harvard University)

  • Morris, Stephen

    ()
    (Princeton University)

Abstract

We define and analyze a "strategic topology'' on types in the Harsanyi-Mertens-Zamir universal type space, where two types are close if their strategic behavior is similar in all strategic situations. For a fixed game and action define the distance between a pair of types as the difference between the smallest epsilon for which the action is epsilon interim correlated rationalizable. We define a strategic topology in which a sequence of types converges if and only if this distance tends to zero for any action and game. Thus a sequence of types converges in the strategic topology if that smallest epsilon does not jump either up or down in the limit. As applied to sequences, the upper-semicontinuity property is equivalent to convergence in the product topology, but the lower-semicontinuity property is a strictly stronger requirement, as shown by the electronic mail game. In the strategic topology, the set of "finite types'' (types describable by finite type spaces) is dense but the set of finite common-prior types is not.

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Bibliographic Info

Article provided by Econometric Society in its journal Theoretical Economics.

Volume (Year): 1 (2006)
Issue (Month): 3 (September)
Pages: 275-309

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Handle: RePEc:the:publsh:141

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Web page: http://econtheory.org

Related research

Keywords: Rationalizability; incomplete information; common knowledge; universal type space; strategic topology;

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References

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  1. Harsanyi, John C., 1994. "Games with Incomplete Information," Nobel Prize in Economics documents 1994-1, Nobel Prize Committee.
  2. Philippe Jehiel & Benny Moldovanu, 1998. "Efficient Design with Interdependent Valuations," Discussion Papers 1244, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  3. Jonathan Weinstein & Muhamet Yildiz, 2004. "Finite-Order Implications of Any Equilibrium," Levine's Working Paper Archive 122247000000000065, David K. Levine.
  4. Dirk Bergemann & Stephen Morris, 2005. "Robust Mechanism Design," NajEcon Working Paper Reviews 666156000000000593, www.najecon.org.
  5. Brandenburger, Adam & Dekel, Eddie, 1987. "Rationalizability and Correlated Equilibria," Econometrica, Econometric Society, vol. 55(6), pages 1391-1402, November.
  6. Mertens, J.-F., 1986. "Repeated games," CORE Discussion Papers 1986024, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  7. Eddie Dekel & Drew Fudenberg & David K Levine, 2002. "Learning to Play Bayesian Games," Levine's Working Paper Archive 625018000000000151, David K. Levine.
  8. 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.
  9. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, vol. 82(2), pages 324-341, October.
  10. Fudenberg, Drew & Levine, David, 1986. "Limit Games and Limit Equilibria," Scholarly Articles 3350443, Harvard University Department of Economics.
  11. P. Battigalli & M. Siniscalchi, 2002. "Rationalization and Incomplete Information," Princeton Economic Theory Working Papers 9817a118e65062903de7c3577, David K. Levine.
  12. Barton L. Lipman, 2003. "Finite Order Implications of Common Priors," Econometrica, Econometric Society, vol. 71(4), pages 1255-1267, 07.
  13. MERTENS , Jean-François & SORIN , Sylvain & ZAMIR , Shmuel, 1994. "Repeated Games. Part A : Background Material," CORE Discussion Papers 1994020, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  14. 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.
  15. 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.
  16. Atsushi Kajii & Stephen Morris, 1997. "Payoff Continuity in Incomplete Information Games," Discussion Papers 1193R, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  17. Eddie Dekel & Drew Fudenberg & Stephen Morris, 2005. "Interim Rationalizability," Harvard Institute of Economic Research Working Papers 2064, Harvard - Institute of Economic Research.
  18. Neeman, Z., 1998. "The Relevance of Private Infromation in Mechanism Design," Papers 93, Boston University - Department of Economics.
  19. 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.
  20. Brandenburger Adam & Dekel Eddie, 1993. "Hierarchies of Beliefs and Common Knowledge," Journal of Economic Theory, Elsevier, vol. 59(1), pages 189-198, February.
  21. McAfee, R Preston & Reny, Philip J, 1992. "Correlated Information and Mechanism Design," Econometrica, Econometric Society, vol. 60(2), pages 395-421, March.
  22. Geanakoplos, John D. & Polemarchakis, Heraklis M., 1982. "We can't disagree forever," Journal of Economic Theory, Elsevier, vol. 28(1), pages 192-200, October.
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