IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v155y2022ics0960077921010651.html
   My bibliography  Save this article

Nonlinear dynamic investigations and global analysis of a Cournot duopoly game with two different objectives

Author

Listed:
  • Askar, S.S.

Abstract

Considering a nonlinear price function a duopoly game with quantities setting is introduced. The two competitors in this game seek maximization of two different objectives. The first competitor want to detect the optimum of his/her production by maximizing an average of social welfare and profit while the second competitor wants to maximize his/her profit only. Due to the lack of market information, each competitor behaves rationally and so the bounded rationality mechanism is adopted in order to build the model describing the game. Studying the evolution of the game requires to investigate the model in discrete time periods. So a two-dimensional map is introduced to analyze the game’s evolution. For this map, we calculate its equilibrium points and study their stability. Through local and global dynamic analysis we prove that the Nash equilibrium point loses its stability because of flip bifurcation only. Other dynamic characteristics for the map such as contact bifurcation and multi-stability are analyzed. The obtained results show that the manifold of game’s map can be investigated based on a one-dimensional map whose analytical form looks like the famous logistic map. Through the critical curves analysis we prove that the phase plane of game’s map is divided into three zones that are Zi,i=0,2,4 and hence the map is noninvertible. Furthermore, an analysis of two types of contact bifurcation are discussed through simulation.

Suggested Citation

  • Askar, S.S., 2022. "Nonlinear dynamic investigations and global analysis of a Cournot duopoly game with two different objectives," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    • Handle: RePEc:eee:chsofr:v:155:y:2022:i:c:s0960077921010651
    DOI: 10.1016/j.chaos.2021.111711
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077921010651
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2021.111711?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. William Novshek, 1985. "On the Existence of Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 52(1), pages 85-98.
    2. Zhou, Jie & Zhou, Wei & Chu, Tong & Chang, Ying-xiang & Huang, Meng-jia, 2019. "Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 358-378.
    3. Ravi P. Agarwal & Sadek Gala & Maria Alessandra Ragusa, 2020. "A Regularity Criterion in Weak Spaces to Boussinesq Equations," Mathematics, MDPI, vol. 8(6), pages 1-11, June.
    4. Luciano Fanti & Domenico Buccella, 2017. "Corporate social responsibility, profits and welfare with managerial firms," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 64(4), pages 341-356, December.
    5. Ahmed, E. & Hegazi, A.S. & Elettreby, M.F. & Askar, S.S., 2006. "On multi-team games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(2), pages 809-816.
    6. Seung-Leul Kim & Sang-Ho Lee & Toshihiro Matsumura, 2019. "Corporate social responsibility and privatization policy in a mixed oligopoly," Journal of Economics, Springer, vol. 128(1), pages 67-89, September.
    7. Matsumura, Toshihiro & Matsushima, Noriaki & Cato, Susumu, 2013. "Competitiveness and R&D competition revisited," Economic Modelling, Elsevier, vol. 31(C), pages 541-547.
    8. Peng, Yu & Lu, Qian, 2015. "Complex dynamics analysis for a duopoly Stackelberg game model with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 259-268.
    9. Xavier Vives, 2001. "Oligopoly Pricing: Old Ideas and New Tools," MIT Press Books, The MIT Press, edition 1, volume 1, number 026272040x, December.
    10. Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
    11. Naimzada, A.K. & Tramontana, F., 2012. "Dynamic properties of a Cournot–Bertrand duopoly game with differentiated products," Economic Modelling, Elsevier, vol. 29(4), pages 1436-1439.
    12. Ahmed, E. & Elettreby, M.F., 2014. "Controls of the complex dynamics of a multi-market Cournot model," Economic Modelling, Elsevier, vol. 37(C), pages 251-254.
    13. Sameh S. Askar, 2020. "A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit," Mathematics, MDPI, vol. 8(10), pages 1-13, October.
    14. Elsadany, A.A., 2017. "Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 253-263.
    15. Nirvikar Singh & Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, The RAND Corporation, vol. 15(4), pages 546-554, Winter.
    16. Rand, David, 1978. "Exotic phenomena in games and duopoly models," Journal of Mathematical Economics, Elsevier, vol. 5(2), pages 173-184, September.
    17. Tremblay, Carol Horton & Tremblay, Victor J., 2011. "The Cournot-Bertrand model and the degree of product differentiation," Economics Letters, Elsevier, vol. 111(3), pages 233-235, June.
    18. 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.
    19. Arya, Anil & Mittendorf, Brian & Sappington, David E.M., 2008. "Outsourcing, vertical integration, and price vs. quantity competition," International Journal of Industrial Organization, Elsevier, vol. 26(1), pages 1-16, January.
    20. Arghya Ghosh & Manipushpak Mitra, 2014. "Reversal of Bertrand-Cournot Rankings in the Presence of Welfare Concerns," Journal of Institutional and Theoretical Economics (JITE), Mohr Siebeck, Tübingen, vol. 170(3), pages 496-519, September.
    21. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
    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. Caravaggio, Andrea & Gori, Luca & Sodini, Mauro, 2022. "Endogenous preferences in a dynamic Cournot duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. 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).

    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. Sameh Askar & Abdulaziz Foul & Tarek Mahrous & Saleh Djemele & Emad Ibrahim, 2021. "Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions," Mathematics, MDPI, vol. 9(23), pages 1-14, December.
    2. 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).
    3. Askar, S.S., 2021. "On complex dynamics of Cournot-Bertrand game with asymmetric market information," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    4. Askar, S.S., 2022. "On the dynamics of Cournot duopoly game with private firms: Investigations and analysis," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    5. Sameh S. Askar, 2020. "The Influences of Asymmetric Market Information on the Dynamics of Duopoly Game," Mathematics, MDPI, vol. 8(7), pages 1-12, July.
    6. Sameh S. Askar, 2020. "A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit," Mathematics, MDPI, vol. 8(10), pages 1-13, October.
    7. S. Askar, 2014. "On Cournot–Bertrand competition with differentiated products," Annals of Operations Research, Springer, vol. 223(1), pages 81-93, December.
    8. Joaquín Andaluz & Gloria Jarne, 2016. "Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational," Annals of Operations Research, Springer, vol. 238(1), pages 1-25, March.
    9. 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.
    10. Alexandrov, Alexei & Bedre-Defolie, Özlem, 2017. "LeChatelier–Samuelson principle in games and pass-through of shocks," Journal of Economic Theory, Elsevier, vol. 168(C), pages 44-54.
    11. Bian, Junsong & Lai, Kin Keung & Hua, Zhongsheng & Zhao, Xuan & Zhou, Guanghui, 2018. "Bertrand vs. Cournot competition in distribution channels with upstream collusion," International Journal of Production Economics, Elsevier, vol. 204(C), pages 278-289.
    12. José Méndez‐Naya & José A. Novo‐Peteiro, 2023. "Partial privatization with endogenous choice of strategic variable," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 44(2), pages 1215-1227, March.
    13. Anne-Christine Barthel & Eric Hoffmann, 2020. "On the existence and stability of equilibria in N-firm Cournot–Bertrand oligopolies," Theory and Decision, Springer, vol. 88(4), pages 471-491, May.
    14. 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).
    15. Elias Asproudis & Eleftherios Filippiadis, 2021. "Environmental Technological Choice in a Cournot-Bertrand Model," Journal of Industry, Competition and Trade, Springer, vol. 21(1), pages 43-58, March.
    16. Joaquín Andaluz & Gloria Jarne, 2019. "On price stability and the nature of product differentiation," Journal of Evolutionary Economics, Springer, vol. 29(2), pages 741-762, April.
    17. Rodney Beard, 2015. "N-Firm Oligopoly With General Iso-Elastic Demand," Bulletin of Economic Research, Wiley Blackwell, vol. 67(4), pages 336-345, October.
    18. Claude d'Aspremont & Rodolphe Dos Santos Ferreira & Louis-André Gérard-Varet, 2007. "Competition For Market Share Or For Market Size: Oligopolistic Equilibria With Varying Competitive Toughness," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 48(3), pages 761-784, August.
    19. Xiaoliang Li & Bo Li, 2023. "A Bertrand duopoly game with differentiated products reconsidered," Papers 2301.01007, arXiv.org.
    20. Andaluz, J. & Elsadany, A.A. & Jarne, G., 2017. "Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 86-99.

    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:eee:chsofr:v:155:y:2022:i:c:s0960077921010651. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.