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An existence result for discontinuous games

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  • Carmona, Guilherme

Abstract

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 144 (2009)
Issue (Month): 3 (May)
Pages: 1333-1340

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Handle: RePEc:eee:jetheo:v:144:y:2009:i:3:p:1333-1340

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Web page: http://www.elsevier.com/locate/inca/622869

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Keywords: Discontinuous games Nash equilibrium Upper semicontinuity Lower semicontinuity;

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  1. Baye, Michael R & Tian, Guoqiang & Zhou, Jianxin, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," Review of Economic Studies, Wiley Blackwell, vol. 60(4), pages 935-48, October.
  2. Adib Bagh & Alejandro Jofre, 2006. "Reciprocal Upper Semicontinuity and Better Reply Secure Games: A Comment," Econometrica, Econometric Society, vol. 74(6), pages 1715-1721, November.
  3. 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.
  4. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Wiley Blackwell, vol. 53(1), pages 1-26, January.
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Citations

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Cited by:
  1. Guilherme Carmona, 2011. "Understanding some recent existence results for discontinuous games," Economic Theory, Springer, vol. 48(1), pages 31-45, September.
  2. Guilherme Carmona, 2011. "Symposium on: Existence of Nash equilibria in discontinuous games," Economic Theory, Springer, vol. 48(1), pages 1-4, September.
  3. Oriol Carbonell-Nicolau & Richard McLean, 2013. "Approximation results for discontinuous games with an application to equilibrium refinement," Economic Theory, Springer, vol. 54(1), pages 1-26, September.
  4. repec:hal:journl:halshs-00717135 is not listed on IDEAS
  5. Vincenzo Scalzo, 2013. "Essential equilibria of discontinuous games," Economic Theory, Springer, vol. 54(1), pages 27-44, September.
  6. Bich Philippe, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," International Journal of Game Theory, Springer, vol. 38(3), pages 395-410, November.
  7. Carmona, Guilherme & Podczeck, Konrad, 2013. "Existence of Nash Equilibrium in games with a measure space of players and discontinuous payoff functions," MPRA Paper 44263, University Library of Munich, Germany.
  8. Nessah, Rabia, 2011. "Generalized weak transfer continuity and the Nash equilibrium," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 659-662.
  9. Pavlo Prokopovych, 2011. "On equilibrium existence in payoff secure games," Economic Theory, Springer, vol. 48(1), pages 5-16, September.
  10. Alejandro Saporiti, 2013. "Power Sharing and Electoral Equilibrium," The School of Economics Discussion Paper Series 1301, Economics, The University of Manchester.
  11. Bagh, Adib, 2010. "Variational convergence: Approximation and existence of equilibria in discontinuous games," Journal of Economic Theory, Elsevier, vol. 145(3), pages 1244-1268, May.
  12. Philip J. Reny, 2013. "Nash Equilibrium in Discontinuous Games," Working Papers 2013-004, Becker Friedman Institute for Research In Economics.
  13. Scalzo, Vincenzo, 2012. "Discontinuous stable games and efficient Nash equilibria," Economics Letters, Elsevier, vol. 115(3), pages 387-389.
  14. Idione Soza & Paulo Barelli, 2012. "A note on the equilibrium existence problem in discontinuous games," Discussion Papers Series 467, School of Economics, University of Queensland, Australia.
  15. repec:hal:wpaper:hal-00846143 is not listed on IDEAS
  16. Philippe Bich & Rida Laraki, 2013. "On the Existence of Approximated Equilibria and Sharing-Rule Equilibria in Discontinuous Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00846143, HAL.
  17. repec:hal:journl:halshs-00426402 is not listed on IDEAS
  18. Rabia Nessah, 2013. "Weakly Continuous Security in Discontinuous and Nonquasiconcave Games: Existence and Characterization," Working Papers 2013-ECO-20, IESEG School of Management.
  19. Tadeusz Radzik, 2014. "Poor convexity and Nash equilibria in games," International Journal of Game Theory, Springer, vol. 43(1), pages 169-192, February.
  20. Philippe Bich & Rida Laraki, 2012. "A Unified Approach to Equilibrium Existence in Discontinuous Strategic Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00717135, HAL.
  21. Philippe Bich & Rida Laraki, 2012. "A Unified Approach to Equilibrium Existence in Discontinuous Strategic Games," Documents de travail du Centre d'Economie de la Sorbonne 12040, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.

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