×

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.

MSC:

91B54 Special types of economic markets (including Cournot, Bertrand)
37N40 Dynamical systems in optimization and economics
91B62 Economic growth models
37A99 Ergodic theory
91A26 Rationality and learning in game theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
[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
[7] Kopel, M., Simple and complex adjustment dynamics in Cournot duopoly models, Chaos, solitons & fractals, 12, 2031-2048, (1996) · Zbl 1080.91541
[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), Longman Harlow
[10] Bierman, H.S.; Fernandez, L., Game theory with economic applications, (1998), Addison-Wesley Reading, MA
[11] Dixit, A., Comparative statics for oligopoly, Int econ rev, 27, 107-122, (1986) · Zbl 0584.90012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.