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The dynamics of Bertrand model with bounded rationality. (English) Zbl 1197.91142
Summary: The paper considers a Bertrand model with bounded rational. A duopoly game is modelled by two nonlinear difference equations. By using the theory of bifurcations of dynamical systems, the existence and stability for the equilibria of this system are obtained. Numerical simulations used to show bifurcations diagrams, phase portraits for various parameters and sensitive dependence on initial conditions. We observe that an increase of the speed of adjustment of bounded rational player may change the stability of Nash equilibrium point and cause bifurcation and chaos to occur. The analysis and results in this paper are interesting in mathematics and economics. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

91B54Special types of economies
37N40Dynamical systems in optimization and economics
91B62Growth models in economics
37A99Ergodic theory
91A26Rationality, learning (game theory)
Full Text: DOI
[1] Cournot, A.: Recherches sur LES principes mathematics de la theorie de la richesse, Hachette (1838) · Zbl 0174.51801
[2] Bertrand, J.: Revue de la thórie de la recherche sociale et des recherches sur LES principes math’ematiques de la thématiques de la th’eorie des richesses, J des savants, 499-508 (1883)
[3] Agiza, H. N.; Elsadany, A. A.: Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl math comput 149, 843-860 (2004) · Zbl 1064.91027 · doi:10.1016/S0096-3003(03)00190-5
[4] Puu, T.: Chaos in duopoly pricing, Chaos, solitons & fractals 1, 573-581 (1991) · Zbl 0754.90015
[5] Agiza, H. N.: Explicit stability zones for cournot games with 3 and 4 competitors, Chaos, solitons & fractals 9, 1955-1966 (1998) · Zbl 0952.91003
[6] Agiza, H. N.: On the stability, bifurcations, chaos and chaos control of kopel map, Chaos, solitons & fractals 11, 1909-1916 (1999) · Zbl 0955.37022 · doi:10.1016/S0960-0779(98)00210-0
[7] Kopel, M.: Simple and complex adjustment dynamics in cournot duopoly models, Chaos, solitons & fractals 12, 2031-2048 (1996) · Zbl 1080.91541 · doi:10.1016/S0960-0779(96)00070-7
[8] Zhang, Jixiang; Da, Qingli; Wang, Yanhua: Analysis of nonlinear duopoly game with heterogeneous players, Econ model 24, 138-148 (2007)
[9] Gravelle, H.; Rees, R.: Microeconomics, (1992)
[10] Bierman, H. S.; Fernandez, L.: Game theory with economic applications, (1998)
[11] Dixit, A.: Comparative statics for oligopoly, Int econ rev 27, 107-122 (1986) · Zbl 0584.90012 · doi:10.2307/2526609