The Existence of Equilibria in Discontinuous and Nonconvex Games
This paper investigates the existence of pure strategy, dominant-strategy, and mixed strategy Nash equilibria in discontinuous and nonconvex games. We introduce a new notion of very weak continuity, called weak transfer continuity, which holds in a large class of discontinuous economic games and is easy to check. We show that it, together with the compactness of strategy space and the quasiconcavity of payoff functions, permits the existence of pure strategy Nash equilibria. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in Baye et al.  and Reny . We provide sufficient conditions for weak transfer continuity by introducing notions of weak transfer upper continuity and weak transfer lower continuity. These conditions are satisfied in many economic games and are often quite simple to check. We also introduce the notion of weak dominant transfer upper continuity, and use it to study the existence of dominant strategy equilibria. We then generalize these results and those in Baye et al.  and Reny  without assuming any form of quasi-concavity of payoff functions or convexity of strategy spaces.
|Date of creation:||Nov 2008|
|Date of revision:||Mar 2010|
|Contact details of provider:|| Postal: 3, rue de la Digue, FR-59000 Lille|
Web page: http://www.ieseg.fr/
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Robson, A.J., 1990.
"An "Informationally Robust Equilibrium" For Two-Person Nonzero-Sum Games,"
9039, Tilburg - Center for Economic Research.
- Robson~ Arthur J., 1994. "An Informationally Robust Equilibrium for Two-Person Nonzero-Sum Games," Games and Economic Behavior, Elsevier, vol. 7(2), pages 233-245, September.
- Robson, A.J., 1990. "An "informationally robust equilibrium" for two-person nonzero-sum games," Discussion Paper 1990-39, Tilburg University, Center for Economic Research.
- Nishimura, Kazuo & Friedman, James, 1981. "Existence of Nash Equilibrium in n Person Games without Quasi-Concavity," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 22(3), pages 637-48, October.
- Guilherme Carmona, 2005.
"On the existence of equilibria in discontinuous games: three counterexamples,"
International Journal of Game Theory,
Springer, vol. 33(2), pages 181-187, 06.
- Guilherme Carmona, 2004. "On the Existence of Equilibria in Discontinuous Games: Three Counterexamples," Game Theory and Information 0402001, EconWPA.
- Carmona, Guilherme, 2003. "On the Existence of Equilibria in Discontinuous Games: Three Counterexamples," FEUNL Working Paper Series wp438, Universidade Nova de Lisboa, Faculdade de Economia.
- Morgan, Jacqueline & Scalzo, Vincenzo, 2007. "Pseudocontinuous functions and existence of Nash equilibria," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 174-183, February.
- Guoqiang Tian, 2009. "The Existence of Equilibria in Games with Arbitrary Strategy Spaces and Payoffs: A Full Characterization," Levine's Working Paper Archive 814577000000000160, David K. Levine.
- Simon, Leo K. & Zame, William R., 1987.
"Discontinous Games and Endogenous Sharing Rules,"
Department of Economics, Working Paper Series
qt8n46v2wv, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
- Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
- Monteiro, Paulo Klinger & Page Jr, Frank H., 2007. "Uniform payoff security and Nash equilibrium in compact games," Journal of Economic Theory, Elsevier, vol. 134(1), pages 566-575, May.
- 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.
- Guoqiang Tian, 1993. "Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations," Review of Economic Studies, Oxford University Press, vol. 60(4), pages 949-958.
- Paul Rothstein, 2007. "Discontinuous Payoffs, Shared Resources, and Games of Fiscal Competition: Existence of Pure Strategy Nash Equilibrium," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 9(2), pages 335-368, 04.
- Vives, X., 1988.
"Nash Equilibrium With Strategic Complementarities,"
UFAE and IAE Working Papers
107-88, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
- 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.
- Michael R. Baye & Guoqiang Tian & Jianxin Zhou, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," Review of Economic Studies, Oxford University Press, vol. 60(4), pages 935-948.
- Leo K. Simon, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Oxford University Press, vol. 54(4), pages 569-597.
When requesting a correction, please mention this item's handle: RePEc:ies:wpaper:e200814. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Monika Marin)
If references are entirely missing, you can add them using this form.