IDEAS home Printed from https://ideas.repec.org/a/psc/journl/v2y2010i4p95-116.html
   My bibliography  Save this article

Complex Dynamics in a Bertrand Duopoly Game with Heterogeneous Players

Author

Listed:
  • Tomasz Dubiel-Teleszyński

    (Warsaw School of Economics)

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.

Suggested Citation

  • Tomasz Dubiel-Teleszyński, 2010. "Complex Dynamics in a Bertrand Duopoly Game with Heterogeneous Players," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 2(2), pages 95-116, March.
    • Handle: RePEc:psc:journl:v:2:y:2010:i:4:p:95-116
    as

    Download full text from publisher

    File URL: http://www.cejeme.pl/publishedarticles/2011-59-06-634455827759843750-2922.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Den Haan, Wouter J., 2001. "The importance of the number of different agents in a heterogeneous asset-pricing model," Journal of Economic Dynamics and Control, Elsevier, vol. 25(5), pages 721-746, May.
    2. Agiza, H.N. & Elsadany, A.A., 2003. "Nonlinear dynamics in the Cournot duopoly game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 512-524.
    3. 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.
    4. Puu, T., 1998. "The chaotic duopolists revisited," Journal of Economic Behavior & Organization, Elsevier, vol. 33(3-4), pages 385-394, January.
    5. Medio,Alfredo & Lines,Marji, 2001. "Nonlinear Dynamics," Cambridge Books, Cambridge University Press, number 9780521558747, October.
    6. 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.
    7. Medio,Alfredo & Gallo,Giampaolo, 1995. "Chaotic Dynamics," Cambridge Books, Cambridge University Press, number 9780521484619, October.
    8. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2009. "The dynamics of Bertrand model with bounded rationality," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2048-2055.
    9. Rothman, Philip, 1995. "Chaotic dynamics. Theory and applications to economics : Alfredo Medio, (Cambridge University Press, Cambridge 1992) pp. xv + 344, $54.95," Journal of Economic Behavior & Organization, Elsevier, vol. 26(2), pages 308-310, March.
    10. Dana, Rose-Anne & Montrucchio, Luigi, 1986. "Dynamic complexity in duopoly games," Journal of Economic Theory, Elsevier, vol. 40(1), pages 40-56, October.
    11. 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.
    12. G.‐I. Bischi & M. Gallegati & A. Naimzada, 1999. "Symmetry‐breaking bifurcations and representativefirm in dynamic duopoly games," Annals of Operations Research, Springer, vol. 89(0), pages 252-271, January.
    13. 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 107-122, February.
    14. Medio,Alfredo & Lines,Marji, 2001. "Nonlinear Dynamics," Cambridge Books, Cambridge University Press, number 9780521551861, October.
    15. 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.
    16. Onozaki, Tamotsu & Sieg, Gernot & Yokoo, Masanori, 2003. "Stability, chaos and multiple attractors: a single agent makes a difference," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1917-1938, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ciprian Rusescu & Mihai Daniel - Roman, 2021. "Dynamic Behaviour In A Bertrand Model With Bounded Rational Players," Annals - Economy Series, Constantin Brancusi University, Faculty of Economics, vol. 3, pages 45-55, June.
    2. Puu, Tonu & Tramontana, Fabio, 2019. "Can Bertrand and Cournot oligopolies be combined?," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 97-107.
    3. Zhang, Xinli & Sun, Deshan & Ma, Sijia & Zhang, Shuning, 2020. "The dynamics of a quantum Bertrand duopoly with differentiated products and heterogeneous expectations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luciano Fanti & Luca Gori, 2013. "Stability Analysis in a Bertrand Duopoly with Different Product Quality and Heterogeneous Expectations," Journal of Industry, Competition and Trade, Springer, vol. 13(4), pages 481-501, December.
    2. Georges SARAFOPOULOS & Kosmas PAPADOPOULOS, 2017. "On A Cournot Duopoly Game With Differentiated Goods, Heterogeneous Expectations And A Cost Function Including Emission Costs," Scientific Bulletin - Economic Sciences, University of Pitesti, vol. 16(1), pages 11-22.
    3. Fanti, Luciano & Gori, Luca, 2012. "The dynamics of a differentiated duopoly with quantity competition," Economic Modelling, Elsevier, vol. 29(2), pages 421-427.
    4. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
    5. Xin, Baogui & Chen, Tong, 2011. "On a master-slave Bertrand game model," Economic Modelling, Elsevier, vol. 28(4), pages 1864-1870, July.
    6. Fanti, Luciano & Gori, Luca & Sodini, Mauro, 2012. "Nonlinear dynamics in a Cournot duopoly with relative profit delegation," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1469-1478.
    7. Fanti, Luciano & Gori, Luca, 2011. "The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited," MPRA Paper 33268, University Library of Munich, Germany.
    8. Luciano Fanti, 2015. "Environmental Standards and Cournot Duopoly: A Stability Analysis," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 61(4), pages 577-593, August.
    9. Shi, Lian & Sheng, Zhaohan & Xu, Feng, 2015. "The dynamics of competition in remanufacturing: A stability analysis," Economic Modelling, Elsevier, vol. 50(C), pages 245-253.
    10. Ding, Zhanwen & Wang, Qiao & Jiang, Shumin, 2014. "Analysis on the dynamics of a Cournot investment game with bounded rationality," Economic Modelling, Elsevier, vol. 39(C), pages 204-212.
    11. Yu, Weisheng & Yu, Yu, 2014. "A dynamic duopoly model with bounded rationality based on constant conjectural variation," Economic Modelling, Elsevier, vol. 37(C), pages 103-112.
    12. PAPADOPOULOS Kosmas & SARAFOPOULOS Georges, 2019. "Dynamics of a Cournot Game with Differentiated Goods and Asymmetric Cost Functions based on Relative Profit Maximization," European Journal of Interdisciplinary Studies, Bucharest Economic Academy, issue 02, June.
    13. Fanti, Luciano, 2014. "The dynamics of a banking duopoly with capital regulations," Economic Modelling, Elsevier, vol. 37(C), pages 340-349.
    14. Fanti, Luciano & Gori, Luca, 2013. "Efficient bargaining versus right to manage: A stability analysis in a Cournot duopoly with trade unions," Economic Modelling, Elsevier, vol. 30(C), pages 205-211.
    15. Agiza, H.N. & Elsadany, A.A., 2003. "Nonlinear dynamics in the Cournot duopoly game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 512-524.
    16. Zhu, Xiaolong & Zhu, Weidong & Yu, Lei, 2014. "Analysis of a nonlinear mixed Cournot game with boundedly rational players," Chaos, Solitons & Fractals, Elsevier, vol. 59(C), pages 82-88.
    17. Fanti, Luciano & Gori, Luca, 2011. "Stability analysis in a Cournot duopoly with managerial sales delegation and bounded rationality," MPRA Paper 33828, University Library of Munich, Germany.
    18. S. S. Askar & Mona F. EL-Wakeel & M. A. Alrodaini, 2018. "Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products," Complexity, Hindawi, vol. 2018, pages 1-13, February.
    19. 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.
    20. Li, Wen-na & Elsadany, A.A. & Zhou, Wei & Zhu, Yan-lan, 2021. "Global Analysis, Multi-stability and Synchronization in a Competition Model of Public Enterprises with Consumer Surplus," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

    More about this item

    Keywords

    Bertrand duopoly; heterogeneous expectations; nonlinear dynamics; chaos;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D84 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Expectations; Speculations

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:psc:journl:v:2:y:2010:i:4:p:95-116. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Damian Jelito (email available below). General contact details of provider: http://cejeme.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.