Nessah, Rabia and Tian, Guoqiang (2008): Existence of Equilibria in Discontinuous Games.

PDF
MPRA_paper_41206.pdf Download (307kB)  Preview 
Abstract
This paper investigates the existence of pure strategy, dominant strategy, and mixed strategy Nash equilibria in discontinuous games. We introduce a new notion of weak continuity, called weak transfer quasicontinuity, which is weaker than most known weak notions of continuity, including diagonal transfer continuity in Baye et al (1993) and betterreply security in Reny (1999), and holds in a large class of discontinuous games. We show that it, together with strong diagonal transfer quasiconcavity introduced in the paper, is enough to guarantee the existence of Nash equilibria in compact and convex normal form games. We provide sufficient conditions for weak transfer quasicontinuity by introducing notions of weak transfer continuity, weak transfer upper continuity, and weak transfer lower continuity. Moreover, an analogous analysis is applied to show the existence of dominant strategy and mixed strategy Nash equilibria in discontinuous games.
Item Type:  MPRA Paper 

Original Title:  Existence of Equilibria in Discontinuous Games 
Language:  English 
Keywords:  Discontinuous games, weak transfer quasicontinuity, pure strategy, mixed strategy, dominant strategy, Nash equilibrium, existence 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C70  General 
Item ID:  41206 
Depositing User:  Guoqiang Tian 
Date Deposited:  12. Sep 2012 12:49 
Last Modified:  12. Feb 2013 12:09 
References:  Aliprantis, C.B., Border, K.C. (1994): Infinite Dimensional Analysis. SpringerVerlag, New York. Athey, S. (2001): Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information. Econometrica, 69, 861–889. Bagh, A., Jofre, A. (2006): Reciprocal Upper Semicontinuity and Better Reply Secure Games: A Comment. Econometrica, 74, 1715–1721. Barelli, P., Soza, I. (2009): On the Existence of Nash Equilibria in Discontinuous and Qualitative Games, mimeo. Baye, M.R., Tian, G., Zhou, J. (1993): Characterizations of the Existence of Equilibria in Games with Discontinuous and NonQuasiconcave Payoffs. The Review of Economic Studies, 60, 935– 948. Carmona, G. (2005): On the Existence of Equilibria in Discontinuous Games: Three Counterexamples. International Journal of Game Theory, 33, 181–187. Carmona, G. (2008): An Existence Result of Equilibrium in Discontinuous Economic Games, mimeo. Dasgupta, P., Maskin, E. (1986): The Existence of Equilibrium in Discontinuous Economic Games, I: Theory. The Review of Economic Studies, 53, 1–26. Debreu, G. (1952): A Social Equilibrium Existence Theorem. Proceedings of the National Academy of Sciences of the U. S. A., 38. Deguire, P., Lassonde, M. (1995): Familles S´electantes. Topological Methods in Nonlinear Analysis, 5, 261–269. Gatti, R.J. (2005): A Note on the Existence of Nash Equilibrium in Games with Discontinuous Payoffs. Cambridge Economics Working Paper No. CWPE 0510. Available at SSRN: http://ssrn.com/abstract=678701. Glicksberg, I.L. (1952): A Further Generalization of the Kakutani Fixed Point Theorem. Proceedings of the American Mathematical Society, 3, 170–174. Jackson, M. O. (2005): Nonexistence of Equilibrium in Vickrey, Secondprice, and English Auctions, working paper, Stanford University. Karlin, S. (1959): Mathematical Methods and Theory in Games, Programming and Economics, Vol. II (London: Pregamon Press). McLennan, A., Monteiro, P. K., Tourky, R. (2009): Games with Discontinuous Payoffs: a Strengthening of Reny’ s Existence Theorem, mimeo. Milgrom, P., and Roberts, H. (1990): Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities, Econometrica 58, 12551277. Milgrom, P., Weber, R. (1985): Distributional Strategies for Games with Incomplete Information, Mathematics of Operations Research 10, 619632. Monteiro, P.K., Page, F.H.Jr. (2007): Uniform Payoff Security and Nash Equilibrium in Compact Games. Journal of Economic Theory, 134, 566–575. Morgan, J., Scalzo, V. (2007): Pseudocontinuous Functions and Existence of Nash Equilibria. Journal of Mathematical Economics, 43, 174–183. Nash, J. (1950): Equilibrium Points in nPerson Games. Proceedings of the National Academy of Sciences, 36, 48–49. Nash, J.F. (1951): Noncooperative Games. Annals of Maths, 54, 286–295. Nishimura, K., Friedman, J. (1981): Existence of Nash Equilibrium in nPerson Games without QuasiConcavity. International Economic Review, 22, 637–648. Roberts, J. and Sonnenschein, H. (1977): On the Existence of Cournot Equilibrium Without Concave Profit Functions, Econometrica, 45, 101–113. Reny, P. J. (1999): On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games. Econometrica, 67, 1029–1056. Reny, P. J. (2009): Further Results on the Existence of Nash Equilibria in Discontinuous Games, mimeo. Robson, A. J. (1994): An Informationally Robust Equilibrium in TwoPerson NonzeroSum Games. Games and Economic Behavior, 2, 233245. Rosen, J.B. (1965): Existence and Uniqueness of Equilibrium Point for Concave nPerson Games. Econometrica, 33, 520–534. Rothstein, P. (2007): Discontinuous Payoffs, Shared Resources, and Games of Fiscal Competition: Existence of Pure Strategy Nash Equilibrium. Journal of Public Economic Theory, 9, 335–368. Simon, L. (1987): Games with Discontinuous Payoffs. Review of Economic Studies, 54, 569–597. Simon, L., Zame. W. (1990): Discontinuous Games and Endogenous Sharing Rules. Econometrica, 58, 861–872. Tian, G. (1992a): Generalizations of the KKM Theorem and Ky Fan Minimax Inequality, with Applications to Maximal Elements, Price Equilibrium, and Complementarity, Journal of Mathematical Analysis and Applications, 170, 457–471. Tian, G. (1992b): “Existence of Equilibrium in Abstract Economies with Discontinuous Payoffs and NonCompact Choice Spaces,” Journal of Mathematical Economics, 21, pp. 379388. Tian, G. (1992c) “On the Existence of Equilibria in Generalized Games,” International Journal of Game Theory, 20, pp.247254. Tian, G. (1993): Necessary and Sufficent Conditions for Maximization of a Class of Preference Relations. Review of Economic Studies, 60, 949–958. Tian, G. (2009): Existence of Equilibria in Games with Arbitrary Strategy Spaces and Payoffs: A Full Characterization, mimeo. Tian, G., Zhou, Z. (1995): Transfer Continuities, Generalizations of the Weierstrass Theorem and Maximum Theorem: A Full Characterization. Journal of Mathematical Economics, 24, 281– 303. Topkis, D. M.(1979): Equilibrium Points in NonzeroSum nPerson Submodular Games, SIAM Journal on Control and Optimization 17(6), 773–787. Vives, X. (1990): Nash Equilibrium with Strategic Complementarities. Journal of Mathematical Economics, 19, 305–321. Yao, J.C. (1992): Nash Equilibria in nPerson Games without Convexity. Applied Mathematics Letters, 5, 67–69. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/41206 