Karafyllis, Iasson and Jiang, ZhongPing and Athanasiou, George (2010): Nash Equilibrium and Robust Stability in Dynamic Games: A SmallGain Perspective. Published in: Computers and Mathematics with Applications , Vol. 60, No. 11 (1. December 2010): pp. 29362952.

PDF
MPRA_paper_26890.pdf Download (357kB)  Preview 
Abstract
This paper develops a novel methodology to study robust stability properties of Nash equilibrium points in dynamic games. Smallgain techniques in modern mathematical control theory are used for the first time to derive conditions guaranteeing uniqueness and global asymptotic stability of Nash equilibrium point for economic models described by functional difference equations. Specification to a Cournot oligopoly game is studied in detail to demonstrate the power of the proposed methodology.
Item Type:  MPRA Paper 

Original Title:  Nash Equilibrium and Robust Stability in Dynamic Games: A SmallGain Perspective 
English Title:  Nash Equilibrium and Robust Stability in Dynamic Games: A SmallGain Perspective 
Language:  English 
Keywords:  Dynamic game; Cournot oligopoly; Nash equilibrium; Robust stability; Small gain 
Subjects:  C  Mathematical and Quantitative Methods > C0  General C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C70  General C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62  Existence and Stability Conditions of Equilibrium 
Item ID:  26890 
Depositing User:  George Athanasiou 
Date Deposited:  29. Dec 2010 20:33 
Last Modified:  16. Feb 2013 00:23 
References:  Agiza, H. N., G. I. Bischi and M. Kopel (1999). Multistability in a Dynamic Cournot Game with Three Oligopolists. Mathematics and Computers in Simulation, 51, 6390. Athanasiou, G., I. Karafyllis and S. Kotsios (2008). Price Stabilization Using Buffer Stocks. Journal of Economic Dynamics and Control, 32, 12121235. Bischi, G. and M. Kopel (2001). Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior & Organization, 46, 73100. Chiarella, C. and F. Szidarovszky (2004). Dynamic oligopolies without full information and with continuously distributed time lags. Journal of Economic Behavior & Organization, 54, 495511. Droste, E., C. Hommes, and J. Tuinstra (2002), Endogenous fluctuations under evolutionary pressure in Cournot competition. Games and Economic Behavior, 40, 232–269. Forgo, F., J. Szep and F. Szidarovszky (1999). Introduction to the Theory of Games. Concepts, Methods, Applications. Kluwer Academic Publishers. Grune, L.(2002). InputtoState Dynamical Stability and its Lyapunov Function Characterization, IEEE Transactions on Automatic Control, 47, 14991504. Guo, Q. and P. Niu (2009). Some Theorems on Existence and Uniqueness of Fixed Points for Decreasing Operators. Computers and Mathematics with Applications, 57, 15151521. Hale, J. K. and S. M. V. Lunel (1993). Introduction to Functional Differential Equations, SpringerVerlag, New York. Hommes, C. H. (1998). On the consistency of backwardlooking expectations: The case of the cobweb. Journal of Economic Behavior & Organization, 33, 333362. Huang, W. (2008). Information lag and dynamic stability. Journal of Mathematical Economics, 44, 513529. Ito, H. and Z.P. Jiang (2005). Nonlinear SmallGain Condition Covering iISS Systems: Necessity and Sufficiency from a Lyapunov Perspective. Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, 355360. Ito, H. and Z.P. Jiang (2009). Necessary and Sufficient SmallGain Conditions for Integral InputtoState Stable Systems: A Lyapunov Perspective. IEEE Transactions on Automatic Control, 54, 23892404. Jiang, Z.P., A. Teel and L. Praly (1994). SmallGain Theorem for ISS Systems and Applications. Mathematics of Control, Signals and Systems, 7, 95120. Karafyllis, I. (2007). A SystemTheoretic Framework for a Wide Class of Systems I: Applications to Numerical Analysis. Journal of Mathematical Analysis and Applications, 328, 876899. Karafyllis, I. (2007). A SystemTheoretic Framework for a Wide Class of Systems II: InputtoOutput Stability. Journal of Mathematical Analysis and Applications, 328, 466486. Karafyllis, I. and Z.P. Jiang (2007). A SmallGain Theorem for a Wide Class of Feedback Systems with Control Applications. SIAM Journal on Control and Optimization, 46, 14831517. Karafyllis, I., P. Pepe and Z.P. Jiang (2009). Stability Results for Systems Described by Coupled Retarded Functional Differential Equations and Functional Difference Equations. Nonlinear Analysis, Theory, Methods and Applications, 71, 33393362. Karafyllis, I. and Z.P. Jiang (2009). A Vector SmallGain Theorem for General Nonlinear Control Systems. Proceedings of the 48th IEEE Conference on Decision and Control 2009, Shanghai, China, 79968001. Also available in http://arxiv.org/abs/0904.0755. Lin, G., and Y. Hong (2007). Delay induced oscillation in predatorprey system with BeddingtonDeAngelis functional response. Applied Mathematics & Computation, 190, 12971311. Luenberger, D. G. (1978). Complete Stability of Noncooperative Games. Journal of Optimization Theory and Applications, 25, 485505. Mazenc, F. and P.A. Bliman (2006). Backstepping Design for TimeDelay Nonlinear Systems. IEEE Transactions on Automatic Control, 51, 149154. Mazenc, F., M. Malisoff and Z. Lin (2008). Further Results on InputtoState Stability for Nonlinear Systems with Delayed Feedbacks. Automatica, 44, 24152421. Nesic, D. and A. R. Teel (2004). Input output stability properties of networked control systems. IEEE Transactions on Automatic Control, 49, 16501667. Nesic, D. and A. R. Teel (2004). Input to state stability of networked control systems. Automatica, 40,21212128. Sontag, E.D. (1989). Smooth Stabilization Implies Coprime Factorization. IEEE Transactions on Automatic Control, 34, 435443. Sontag, E.D. and Y. Wang (1999).Notions of Input to Output Stability. Systems and Control Letters, 38, 235248. Sontag, E.D. and Y. Wang (2001). Lyapunov Characterizations of InputtoOutput Stability. SIAM Journal on Control and Optimization, 39, 226249. Panchuk, A. and T. Puu (2009). Stability in a NonAutonomous Iterative System: An Application to Oligopoly. Computers and Mathematics with Applications, 58, 20222034. Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. Okuguchi, K., and T. Yamazaki (2008). Global stability of unique Nash equilibrium in Cournot oligopoly and rentseeking game. Journal of Economic Dynamics and Control, 32, 12041211. Pepe, P. (2003). The Liapunov’s second method for continuous time difference equations. International Journal of Robust and Nonlinear Control, 13, 13891405. Szidarovszky, F. and W. Li (2000). A Note on the Stability of a CournotNash Equilibrium: the Multiproduct Case with Adaptive Expectations. Journal of Mathematical Economics, 33, 101107. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/26890 