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Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game

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  • Sameh Askar

    (Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

Abstract

This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.

Suggested Citation

  • Sameh Askar, 2021. "Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game," Mathematics, MDPI, vol. 9(20), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2558-:d:654825
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    References listed on IDEAS

    as
    1. Puu, Tonu, 2017. "A New Approach to Modeling Bertrand Duopoly," Review of Behavioral Economics, now publishers, vol. 4(1), pages 51-67, April.
    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., 2021. "On complex dynamics of Cournot-Bertrand game with asymmetric market information," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    4. 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.
    5. Li, Yan & Wang, Lidong, 2019. "Chaos in a duopoly model of technological innovation with bounded rationality based on constant conjectural variation," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 116-126.
    6. 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.
    7. 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.
    8. Radi, Davide & Gardini, Laura & Avrutin, Viktor, 2014. "The role of constraints in a segregation model: The symmetric case," Chaos, Solitons & Fractals, Elsevier, vol. 66(C), pages 103-119.
    9. 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.
    10. 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.
    11. Commendatore, Pasquale & Kubin, Ingrid & Petraglia, Carmelo & Sushko, Iryna, 2014. "Regional integration, international liberalisation and the dynamics of industrial agglomeration," Journal of Economic Dynamics and Control, Elsevier, vol. 48(C), pages 265-287.
    12. Hommes,Cars, 2015. "Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems," Cambridge Books, Cambridge University Press, number 9781107564978, January.
    13. S. S. Askar & A. Al-khedhairi, 2017. "Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-13, June.
    14. Askar, S.S. & Elettreby, M.F., 2017. "The impact of cost uncertainty on Cournot oligopoly games," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 169-176.
    15. 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.
    16. Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
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