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Refinements of Nash equilibrium in potential games

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  • Oriol Carbonell-Nicolau

    ()
    (Rutgers University)

  • Richard McLean

    ()
    (Rutgers University)

Abstract

We prove the existence of strategically stable sets of pure-strategy Nash equilibria (and hence the existence of pure-strategy trembling-hand perfect equilibria) in potential games that admit an upper semicontinuous potential, and we show that generic potential games possess pure-strategy strictly perfect and essential equilibria. In addition, we provide a link between upper semicontinuity of a potential and conditions defined directly on the payo ff functions of a potential game. Finally, we show that stable sets and (strictly) perfect equilibria are related to the set of maximizers of a potential, which refines the set of Nash equilibria. Specifically, the set of maximizers of a potential contains a strategically stable set of pure-strategy Nash equilibria (and hence a pure-strategy trembling-hand perfect equilibrium) and, for generic games, any maximizer of a potential is a pure-strategy strictly perfect and essential equilibrium.

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Bibliographic Info

Paper provided by Rutgers University, Department of Economics in its series Departmental Working Papers with number 201125.

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Length: 20 pages
Date of creation: 06 Jun 2011
Date of revision:
Handle: RePEc:rut:rutres:201125

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Keywords: discontinuous game; potential game; trembling-hand perfect equilibrium; stable set; essential equilibrium;

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  1. E. Kohlberg & J.-F. Mertens, 1998. "On the Strategic Stability of Equilibria," Levine's Working Paper Archive 445, David K. Levine.
  2. Carbonell-Nicolau, Oriol, 2011. "On the existence of pure-strategy perfect equilibrium in discontinuous games," Games and Economic Behavior, Elsevier, vol. 71(1), pages 23-48, January.
  3. Gerhard SORGER, 1998. "Perfect Foresight and Equilibrium Selection in Symmetric Potential Games," Vienna Economics Papers 9802, University of Vienna, Department of Economics.
  4. Ui, Takashi, 2001. "Robust Equilibria of Potential Games," Econometrica, Econometric Society, vol. 69(5), pages 1373-80, September.
  5. Oriol Carbonell-Nicolau, 2011. "The Existence of Perfect Equilibrium in Discontinuous Games," Games, MDPI, Open Access Journal, vol. 2(3), pages 235-256, July.
  6. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
  7. DavidP. Myatt & Chris Wallace, 2009. "Evolution, Teamwork and Collective Action: Production Targets in the Private Provision of Public Goods," Economic Journal, Royal Economic Society, vol. 119(534), pages 61-90, 01.
  8. Carbonell-Nicolau, Oriol, 2011. "Perfect and limit admissible perfect equilibria in discontinuous games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 531-540.
  9. Carbonell-Nicolau, Oriol, 2011. "On strategic stability in discontinuous games," Economics Letters, Elsevier, vol. 113(2), pages 120-123.
  10. Michael Ostrovsky & Michael Schwarz, 2005. "Adoption of Standards Under Uncertainty," RAND Journal of Economics, The RAND Corporation, vol. 36(4), pages 816-832, Winter.
  11. Carbonell-Nicolau, Oriol, 2010. "Essential equilibria in normal-form games," Journal of Economic Theory, Elsevier, vol. 145(1), pages 421-431, January.
  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. Mclennan, A., 1989. "Selected Topics In The Theory Of Fixed Points," Papers 251, Minnesota - Center for Economic Research.
  14. Armstrong, Mark & Vickers, John, 2001. "Competitive Price Discrimination," RAND Journal of Economics, The RAND Corporation, vol. 32(4), pages 579-605, Winter.
  15. Armstrong, Mark, 2006. "Price discrimination," MPRA Paper 4693, University Library of Munich, Germany.
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