Essential stability for large generalized games

Correa, Sofía and Torres-Martínez, Juan Pablo (2012): Essential stability for large generalized games.

Abstract

We address the essential stability of Cournot-Nash equilibria for generalized games with a continuum of players, where only a finite number of them are atomic. Given any set of generalized games continuously parameterized by a complete metric space, we analyze the robustness of equilibria to perturbations on parameters.

• Item Type: MPRA Paper
• Language: English
• Keywords: Essential equilibria; Essential sets and components; Generalized games
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium

References

Aubin, J.P. (1982): Mathematical Methods of Games and Economic Theory, North-Holland, Amsterdam.

Aliprantis, C. and K. Border (1999): Infinite Dimensional Analysis, Springer-Verlag, Berlin, Heidelberg.

Al-Najjar, N. (1995): "Strategically stable equilibria in games with infinitely many pure strategies," Mathematical Social Sciences, volume 29, pages 151-164.

Balder, E.J. (1999): "On the existence of Cournot-Nash equilibria in continuum games," Journal of Mathematical Economics 32, pages 207-223.

Balder, E.J. (2002): "A unifying pair of Cournot-Nash equilibrium existence results," Journal of Economic Theory, volume 102, pages 437-470.

Carbonell-Nicolau, O. (2010): "Essential equilibria in normal-form games," Journal of Economic Theory, volume 145, pages 421-431.

Fort, M.K. (1949): "A unified theory of semi-continuity," Duke Mathematical Journal, volume 16, pages 237-246.

Fort, M.K. (1950): "Essential and non-essential fixed points," American Journal of Mathematics, volume 72, pages 315-322.

Hildenbrand, W. (1974): Core and Equilibria in a Large Economy, Princeton University Press, Princeton, New Jersey.

Hillas, J. (1990): "On the definition of the strategic stability of equilibria," Econometrica, volume 58, pages 1365-1390.

Jia-He, J. (1962): "Essential fixed points of the multivalued mappings," Scientia Sinica, volume XI. pages 293-298.

Jia-He, J. (1963): "Essential components of the set of fixed points of the multivalued mappings and its application to the theory of games," Scientia Sinica, volume XII. pages 951-964.

Kinoshita, S. (1952): "On essential components of the set of fixed points," Osaka Mathematical Journal, volume 4, pages 19-22.

Kohlberg, E. and J.F. Mertens (1986): "On the strategic stability of equilibrium point," Econometrica, volume 54, pages 1003-1037.

Ok, E. (2005): Real Analysis with Economic Applications, Princeton University Press, Princeton, USA.

Rath, K.P. (1992): "A direct proof of the existence of pure strategy equilibria in games with a continuum of players", Economic Theory, Volume 2, pages 427-433.

Riascos, A.J. and J.P. Torres-Martínez(2012):"On the existence of pure strategy equilibria in large generalized games with atomic players," working paper, Department of Economics, University of Chile. Available at http://www.econ.uchile.cl/cha/jutorres.

Schmeidler, D. (1973): "Equilibrium points of non-atomic games," Journal of Statistical Physics, volume 17, 295-300.

Yu, J. (1999): "Essential equilibrium of n-person noncooperative games," Journal of Mathematical Economics, volume 31, pages 361-372.

Yu, J., and H. Yang (2004): "Essential components of the set of equilibrium points for set-valued maps," Journal of Mathematical Analysis and Applications, volume 300, pages 334-342.

Yu, J., H. Yang, and S. Xiang (2005): "Unified approach to existence and stability of essential components," Nonlinear Analysis, volume 63, pages 2415-2425.

Yu, X. (2009): "Essential components of the set of equilibrium points for generalized games in the uniform topological space of best reply correspondences," International Journal of Pure and Applied Mathematics, volume 55, pages 349-357.

Zhou, Y., J.Yu, and S.Xiang (2007): "Essential stability in games with infinitely many pure strategies," International Journal of Game Theory, volume 35, pages 493-503.

Wu, W., and J. Jia-He (1962): "Essential equilibrium points of n-person noncooperative games," Scientia Sinica, volume XI, pages 1307-1322.