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Complex Dynamics in a Bertrand Duopoly Game with Heterogeneous Players

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  • Tomasz Dubiel-Teleszyński

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    (Warsaw School of Economics)

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    Abstract

    A heterogeneous Bertrand duopoly game with bounded rational and adaptive players manufacturing differentiated products is subject of investigation. The main goal is to demonstrate that participation of one bounded rational player in the game suffices to destabilize the duopoly. The game is modelled with a system of two difference equations. Evolution of prices over time is obtained by iteration of a two dimensional nonlinear map. Equilibria are found and local stability properties thereof are analyzed. Complex behavior of the system is examined by means of numerical simulations. Region of stability of the Nash equilibrium is demonstrated in the plane of the speeds of adjustment. Period doubling route to chaos is presented on the bifurcation diagrams and on the largest Lyapunov characteristic exponent graph. Lyapunov time is calculated. Chaotic attractors are depicted and their fractal dimensions are computed. Sensitive dependence on initial conditions is evidenced.

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    Bibliographic Info

    Article provided by CEJEME in its journal Central European Journal of Economic Modelling and Econometrics.

    Volume (Year): 2 (2010)
    Issue (Month): 2 (March)
    Pages: 95-116

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    Handle: RePEc:psc:journl:v:2:y:2010:i:4:p:95-116

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    Related research

    Keywords: Bertrand duopoly; heterogeneous expectations; nonlinear dynamics; chaos;

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    1. Wouter J. Denhaan, 2000. "The Importance Of The Number Of Different Agents In A Heterogeneous Asset-Pricing Model," Computing in Economics and Finance 2000 349, Society for Computational Economics.
    2. Rassenti, Stephen & Reynolds, Stanley S. & Smith, Vernon L. & Szidarovszky, Ferenc, 2000. "Adaptation and convergence of behavior in repeated experimental Cournot games," Journal of Economic Behavior & Organization, Elsevier, vol. 41(2), pages 117-146, February.
    3. Agiza, H.N. & Hegazi, A.S. & Elsadany, A.A., 2002. "Complex dynamics and synchronization of a duopoly game with bounded rationality," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(2), pages 133-146.
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    7. Bischi, Gian Italo & Kopel, Michael, 2001. "Equilibrium selection in a nonlinear duopoly game with adaptive expectations," Journal of Economic Behavior & Organization, Elsevier, vol. 46(1), pages 73-100, September.
    8. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2007. "Analysis of nonlinear duopoly game with heterogeneous players," Economic Modelling, Elsevier, vol. 24(1), pages 138-148, January.
    9. Dixit, Avinash K, 1986. "Comparative Statics for Oligopoly," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 27(1), pages 107-22, February.
    10. Medio,Alfredo & Gallo,Giampaolo, 1995. "Chaotic Dynamics," Cambridge Books, Cambridge University Press, number 9780521484619, April.
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