IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i11p1079-d285190.html
   My bibliography  Save this article

Cournot Duopoly Games: Models and Investigations

Author

Listed:
  • S. S. Askar

    (Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    Department of Mathematics, College of Science, Mansoura University, Mansoura 35516, Egypt
    These authors contributed equally to this work.)

  • A. Al-khedhairi

    (Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

Abstract

This paper analyzes Cournot duopoly games that are constructed based on Cobb–Douglas preferences. We introduce here two models whose dynamic adjustments depend on bounded rationality, dynamic adjustment, and tit-for-tat mechanism. In the first model, we have two firms with limited information and due to that they adopt the bounded rationality mechanism. They update their productions based on the changing occurred in the marginal profit. For this model, its fixed point is obtained and its stability condition is calculated. In addition, we provide conditions by which this fixed point loses its stability due to flip and Neimark–Sacker bifurcations. Furthermore, numerical simulation shows that this model possesses some chaotic behaviors which are recovered due to corridor stability. In the second model, we handle two different mechanisms of cooperation. These mechanisms are dynamic adjustment process and tit-for-tat strategy. The players who use the dynamic adjustment increase their productions based on the cooperative output while, in tit-for-tat mechanism, they increase the productions based on the cooperative profit. The local stability analysis shows that adopting tit-for-tat makes the model unstable and then the system becomes chaotic for any values of the system’s parameters. The obtained results show that the dynamic adjustment makes the system’s fixed point stable for a certain interval of the adjustment parameter.

Suggested Citation

  • S. S. Askar & A. Al-khedhairi, 2019. "Cournot Duopoly Games: Models and Investigations," Mathematics, MDPI, vol. 7(11), pages 1-15, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1079-:d:285190
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/11/1079/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/11/1079/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Giulio Codognato & Ludovic A. Julien, 2013. "Noncooperative Oligopoly in Markets with a Cobb-Douglas Continuum of Traders," Recherches économiques de Louvain, De Boeck Université, vol. 79(4), pages 75-88.
    3. Anna Agliari & Laura Gardini & Tonu Puu, 2006. "Global Bifurcations In Duopoly When The Cournot Point Is Destabilized Via A Subcritical Neimark Bifurcation," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 1-20.
    4. Ding, Zhanwen & Shi, Guiping, 2009. "Cooperation in a dynamical adjustment of duopoly game with incomplete information," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 989-993.
    5. S. S. Askar & Ahmad M. Alshamrani & K. Alnowibet, 2015. "Analysis of Nonlinear Duopoly Game: A Cooperative Case," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-5, January.
    6. Rotemberg, Julio J & Saloner, Garth, 1986. "A Supergame-Theoretic Model of Price Wars during Booms," American Economic Review, American Economic Association, vol. 76(3), pages 390-407, June.
    7. Cafagna, Vittorio & Coccorese, Paolo, 2005. "Dynamical systems and the arising of cooperation in a Cournot duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 655-664.
    8. Naimzada, Ahmad K. & Sbragia, Lucia, 2006. "Oligopoly games with nonlinear demand and cost functions: Two boundedly rational adjustment processes," Chaos, Solitons & Fractals, Elsevier, vol. 29(3), pages 707-722.
    9. S. Askar, 2014. "On Cournot–Bertrand competition with differentiated products," Annals of Operations Research, Springer, vol. 223(1), pages 81-93, December.
    10. Peng, Yu & Lu, Qian & Xiao, Yue & Wu, Xue, 2019. "Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 658-670.
    11. Matsumoto, Akio & Szidarovszky, Ferenc, 2014. "Complex dynamics of monopolies with gradient adjustment," Economic Modelling, Elsevier, vol. 42(C), pages 220-229.
    12. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2015. "Dynamic Cournot duopoly games with nonlinear demand function," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 427-437.
    13. Rand, David, 1978. "Exotic phenomena in games and duopoly models," Journal of Mathematical Economics, Elsevier, vol. 5(2), pages 173-184, September.
    14. Askar, S.S. & Al-khedhairi, A., 2020. "The dynamics of a business game: A 2D-piecewise smooth nonlinear map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    Full references (including those not matched with items on IDEAS)

    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. Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
    2. Sameh S Askar & Abdulrahman Al-Khedhairi, 2020. "Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game," Mathematics, MDPI, vol. 8(9), pages 1-16, August.
    3. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2016. "The arising of cooperation in Cournot duopoly games," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 535-542.
    4. S. S. Askar, 2020. "Duopolistic Stackelberg game: investigation of complex dynamics and chaos control," Operational Research, Springer, vol. 20(3), pages 1685-1699, September.
    5. Askar, S.S. & Alnowibet, K., 2016. "Cooperation versus noncooperation: Cournot duopolistic game based on delay and time-dependent parameters," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 580-584.
    6. Gian Italo Bischi & Fabio Lamantia & Davide Radi, 2018. "Evolutionary oligopoly games with heterogeneous adaptive players," Chapters, in: Luis C. Corchón & Marco A. Marini (ed.), Handbook of Game Theory and Industrial Organization, Volume I, chapter 12, pages 343-370, Edward Elgar Publishing.
    7. Sameh Askar, 2021. "Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game," Mathematics, MDPI, vol. 9(20), pages 1-13, October.
    8. Askar, S.S. & Alnowibet, K., 2016. "Nonlinear oligopolistic game with isoelastic demand function: Rationality and local monopolistic approximation," Chaos, Solitons & Fractals, Elsevier, vol. 84(C), pages 15-22.
    9. Ding, Zhanwen & Shi, Guiping, 2009. "Cooperation in a dynamical adjustment of duopoly game with incomplete information," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 989-993.
    10. Villena, Marcelo J. & Araneda, Axel A., 2017. "Dynamics and stability in retail competition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 134(C), pages 37-53.
    11. Askar, S.S. & Al-khedhairi, A., 2020. "The dynamics of a business game: A 2D-piecewise smooth nonlinear map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    12. S. S. Askar & A. Al-khedhairi, 2019. "Investigations of Nonlinear Triopoly Models with Different Mechanisms," Complexity, Hindawi, vol. 2019, pages 1-15, December.
    13. 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).
    14. Liu, Cui & He, Rui-chun & Zhou, Wei & Li, Hui, 2021. "Dynamic analysis of airline bidding game based on nonlinear cost," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    15. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2015. "Dynamic Cournot duopoly games with nonlinear demand function," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 427-437.
    16. 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.
    17. Zhang, Ming & Wang, Guanghui & Xu, Jin & Qu, Cunquan, 2020. "Dynamic contest model with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    18. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
    19. Arranz Sombría, M. Rosa, 2011. "Cooperación en modelos de Cournot con información incompleta/Cooperation in Cournot’s Models with Incomplete Information," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 29, pages 397(18á)-39, Abril.
    20. Zhu, Yan-lan & Zhou, Wei & Chu, Tong, 2022. "Analysis of complex dynamical behavior in a mixed duopoly model with heterogeneous goods," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).

    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:gam:jmathe:v:7:y:2019:i:11:p:1079-:d:285190. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.