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Existence of Nash equilibria in large games

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  • Noguchi, Mitsunori
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    Abstract

    Podczeck [Podczeck, K., 1997. Markets with infinitely many commodities and a continuum of agents with non-convex preferences. Economic Theory 9, 385-426] provided a mathematical formulation of the notion of "many economic agents of almost every type" and utilized this formulation as a sufficient condition for the existence of Walras equilibria in an exchange economy with a continuum of agents and an infinite dimensional commodity space. The primary objective of this article is to demonstrate that a variant of Podczeck's condition provides a sufficient condition for the existence of pure-strategy Nash equilibria in a large non-anonymous game G when defined on an atomless probability space not necessary rich, and equipped with a common uncountable compact metric space of actions A. We also investigate to see whether the condition can be applied as well to the broader context of Bayesian equilibria and prove an analogue of Yannelis's results [Yannelis, N.C., in press. Debreu's social equilibrium theorem with asymmetric information and a continuum of agents. Economic Theory] on Debreu's social equilibrium theorem with asymmetric information and a continuum of agents.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Mathematical Economics.

    Volume (Year): 45 (2009)
    Issue (Month): 1-2 (January)
    Pages: 168-184

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    Handle: RePEc:eee:mateco:v:45:y:2009:i:1-2:p:168-184

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    Web page: http://www.elsevier.com/locate/jmateco

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    Keywords: Large non-anonymous game Pure-strategy Nash equilibrium Large anonymous game Cornot-Nash equilibrium distribution Loeb space Rich probability space Bayesian games;

    References

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    1. Bernard Cornet & Mihaela Topuzu, 2005. "Existence Of Equilibria For Economies With Externalities And A Measure Space Of Consumers," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 200505, University of Kansas, Department of Economics, revised Feb 2005.
    2. Khan, M. Ali & Sun, Yeneng, 1999. "Non-cooperative games on hyperfinite Loeb spaces1," Journal of Mathematical Economics, Elsevier, vol. 31(4), pages 455-492, May.
    3. HART, Sergiu & HILDENBRAND, Werner & KOHLBERG, Elon, . "On equilibrium allocations as distributions on the commodity space," CORE Discussion Papers RP -183, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Sun, Yeneng & Yannelis, Nicholas C., 2008. "Saturation and the integration of Banach valued correspondences," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 861-865, July.
    5. Mas-Colell, Andreu, 1984. "On a theorem of Schmeidler," Journal of Mathematical Economics, Elsevier, vol. 13(3), pages 201-206, December.
    6. M Ali Khan & Yeneng Sun, 1996. "Non-Cooperative Games with Many Players," Economics Working Paper Archive, The Johns Hopkins University,Department of Economics 382, The Johns Hopkins University,Department of Economics.
    7. Podczeck, Konrad, 2008. "On the convexity and compactness of the integral of a Banach space valued correspondence," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 836-852, July.
    8. Balder, Erik J & Yannelis, Nicholas C, 1993. "On the Continuity of Expected Utility," Economic Theory, Springer, Springer, vol. 3(4), pages 625-43, October.
    9. Khan, M. Ali & Yeneng, Sun, 1995. "Pure strategies in games with private information," Journal of Mathematical Economics, Elsevier, vol. 24(7), pages 633-653.
    10. Zame, William R. & Noguchi, Mitsunori, 2006. "Competitive markets with externalities," Theoretical Economics, Econometric Society, Econometric Society, vol. 1(2), pages 143-166, June.
    11. Balder, Erik J., 2008. "More on equilibria in competitive markets with externalities and a continuum of agents," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 575-602, July.
    12. Konard Podczeck, 1997. "Markets with infinitely many commodities and a continuum of agents with non-convex preferences (*)," Economic Theory, Springer, Springer, vol. 9(3), pages 385-426.
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    Cited by:
    1. M. Ali Khan & Kali P. Rath & Haomiao Yu & Yongchao Zhang, 2012. "Large Distributional Games with Traits," Working Papers, Ryerson University, Department of Economics 037, Ryerson University, Department of Economics.
    2. Wang, Jianwei & Zhang, Yongchao, 2010. "Purification, Saturation and the Exact Law of Large Numbers," MPRA Paper 22119, University Library of Munich, Germany.
    3. Haomiao Yu, 2014. "Rationalizability in large games," Economic Theory, Springer, Springer, vol. 55(2), pages 457-479, February.
    4. Noguchi, Mitsunori, 2010. "Large but finite games with asymmetric information," Journal of Mathematical Economics, Elsevier, vol. 46(2), pages 191-213, March.

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