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Tripoly Stackelberg game model: One leader versus two followers

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  • Askar, S.S.

Abstract

This paper is devoted to introduce and study a Stackelberg game consisting of three competed firms. The three firms are classified as a leader which is the first firm and the other two firms are called the followers. A linear inverse demand function is used. In addition a quadratic cost based on an actual and announced quantities is adopted. Based on bounded rationality, a three-dimensional discrete dynamical system is constructed. For the system, the backward induction is used to solve the system and to get Nash equilibrium. The obtained results are shown that Nash equilibrium is unique and its stability is affected by the system’s parameters by which the system behaves chaotically due to bifurcation and chaos appeared. Some numerical experiments are performed to portrays such chaotic behavior. A control scheme is used to return the system back to its stability state and is supported by some simulations.

Suggested Citation

  • Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
    • Handle: RePEc:eee:apmaco:v:328:y:2018:i:c:p:301-311
    DOI: 10.1016/j.amc.2018.01.041
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    References listed on IDEAS

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    1. 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.
    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. Askar, S.S., 2013. "On complex dynamics of monopoly market," Economic Modelling, Elsevier, vol. 31(C), pages 586-589.
    4. 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.
    5. S. Askar, 2014. "On Cournot–Bertrand competition with differentiated products," Annals of Operations Research, Springer, vol. 223(1), pages 81-93, December.
    6. 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.
    7. 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.
    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. 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.
    10. Ding, Zhanwen & Li, Qiang & Jiang, Shumin & Wang, Xuedi, 2015. "Dynamics in a Cournot investment game with heterogeneous players," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 939-950.
    11. Hołyst, Janusz A & Urbanowicz, Krzysztof, 2000. "Chaos control in economical model by time-delayed feedback method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 587-598.
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    Citations

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    Cited by:

    1. Askar, S.S., 2021. "On complex dynamics of Cournot-Bertrand game with asymmetric market information," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    2. Yan, Bo & Ahmadi, Atefeh & Mehrabbeik, Mahtab & Rajagopal, Karthikeyan & He, Shaobo & Jafari, Sajad, 2022. "Expanding the duopoly Stackelberg game with marginal costs into a multipoly game with lowering the burden of mathematical calculations: a numerical analysis," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Long, Jianjun & Wang, Fenglian, 2024. "Complexity of a two-stage R&D game within a cluster supply chain considering vertical R&D spillovers, effective information, and government subsidies," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 606-621.
    4. S. S. Askar & A. Al-khedhairi, 2019. "Analysis of a Four-Firm Competition Based on a Generalized Bounded Rationality and Different Mechanisms," Complexity, Hindawi, vol. 2019, pages 1-12, May.
    5. 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).
    6. 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).
    7. Fang Wu & Junhai Ma, 2023. "Research Trend, Logical Structure and Outlook on Complex Economic Game," Mathematics, MDPI, vol. 11(5), pages 1-16, February.
    8. 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.
    9. Nan Xu & Yaoqun Xu & Haiyan Zhong, 2023. "Pricing Decisions for Power Battery Closed-Loop Supply Chains with Low-Carbon Input by Echelon Utilization Enterprises," Sustainability, MDPI, vol. 15(23), pages 1-30, December.
    10. Zhang, Ming & Wang, Guanghui & Xu, Jin & Qu, Cunquan, 2020. "Dynamic contest model with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    11. Sameh Askar, 2021. "Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game," Mathematics, MDPI, vol. 9(20), pages 1-13, October.
    12. Jie, Yingmo & Liu, Charles Zhechao & Li, Mingchu & Choo, Kim-Kwang Raymond & Chen, Ling & Guo, Cheng, 2020. "Game theoretic resource allocation model for designing effective traffic safety solution against drunk driving," Applied Mathematics and Computation, Elsevier, vol. 376(C).
    13. Askar, S.S. & Al-khedhairi, A., 2020. "On complex dynamic investigations of a piecewise smooth nonlinear duopoly game," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    14. 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.
    15. Zhao, Liuwei & Jin, Shuai & Jiang, Hongyun, 2022. "Investigation of complex dynamics and chaos control of the duopoly supply chain under the mixed carbon policy," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    16. Peng, Yu & Lu, Qian & Wu, Xue & Zhao, Yueru & Xiao, Yue, 2020. "Dynamics of Hotelling triopoly model with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    17. Liuwei, Zhao & Shuai, Jin & Hongyun, Jiang, 2024. "Complex dynamics of dual oligopoly demand uncertainty under carbon emission trading mechanism," Chaos, Solitons & Fractals, Elsevier, vol. 179(C).

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