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Expanding the duopoly Stackelberg game with marginal costs into a multipoly game with lowering the burden of mathematical calculations: a numerical analysis

Author

Listed:
  • Yan, Bo
  • Ahmadi, Atefeh
  • Mehrabbeik, Mahtab
  • Rajagopal, Karthikeyan
  • He, Shaobo
  • Jafari, Sajad

Abstract

In game theory, the classic Stackelberg game is a dynamic model of a leader and a follower firm contending in a market. The time evolution with the marginal cost and the dynamical traits of the duopoly and tripoly Stackelberg game can be acquired in the literature. The backward induction procedure, an analytical technique for attaining the Nash equilibrium, becomes grueling as the number of followers augments. To conquer this matter and untangle a multipoly Stackelberg game, this paper designs a numerical process based on seeking the parameter space to gain the maximum total profit. Although, due to the curse of dimensionality, a comprehensive quest in the parameter space fails as the number of followers extends. Therefore, the proposed search algorithm explores the parameter space sequentially to detect the optimum values that assure the Nash equilibrium. The significant concept of this iterative procedure is optimizing the parameters sequentially and utilizing the derived parameters for initializing the subsequent stride. The consequences of the developed numerical strategy represent that the proposed mechanism can conduce to a reasonable and even more acceptable total profit than the analytical attitude. The leader scheduled vintage is invariably greater or similar to the followers so that the leader can inevitably produce more goods and make the maximum profit in the market. The proposed algorithm can conduct a reasonable profit compared to the computational cost. In addition, solving the multipoly Stackelberg game with more than three follower firms and one leader becomes conceivable.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s0960077922008244
    DOI: 10.1016/j.chaos.2022.112645
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    1. Guo, H. & Jia, D. & Sendiña-Nadal, I. & Zhang, M. & Wang, Z. & Li, X. & Alfaro-Bittner, K. & Moreno, Y. & Boccaletti, S., 2021. "Evolutionary games on simplicial complexes," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    2. X. Henry Wang & Bill Z. Yang, 2004. "On Technology Licensing In A Stackelberg Duopoly," Australian Economic Papers, Wiley Blackwell, vol. 43(4), pages 448-458, December.
    3. Lian Shi & Yun Le & Zhaohan Sheng, 2014. "Analysis of Price Stackelberg Duopoly Game with Bounded Rationality," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-8, February.
    4. Peng, Yu & Lu, Qian & Xiao, Yue, 2016. "A dynamic Stackelberg duopoly model with different strategies," Chaos, Solitons & Fractals, Elsevier, vol. 85(C), pages 128-134.
    5. Sau‐Him Paul Lau & Felix Leung, 2010. "Estimating a Parsimonious Model of Inequality Aversion in Stackelberg Duopoly Experiments," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 72(5), pages 669-686, October.
    6. Tian, Baodan & Zhang, Yong & Li, Jiamei, 2020. "Stochastic perturbations for a duopoly Stackelberg model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    7. Xiao, Yue & Peng, Yu & Lu, Qian & Wu, Xue, 2018. "Chaotic dynamics in nonlinear duopoly Stackelberg game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 1980-1987.
    8. Ely, Jeffrey C. & Yilankaya, Okan, 2001. "Nash Equilibrium and the Evolution of Preferences," Journal of Economic Theory, Elsevier, vol. 97(2), pages 255-272, April.
    9. 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.
    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. Larry Samuelson, 2016. "Game Theory in Economics and Beyond," Journal of Economic Perspectives, American Economic Association, vol. 30(4), pages 107-130, Fall.
    12. Larry Samuelson, 2002. "Evolution and Game Theory," Journal of Economic Perspectives, American Economic Association, vol. 16(2), pages 47-66, Spring.
    13. Yu Yu & Weisheng Yu, 2019. "The Complexion of Multi-period Stackelberg Triopoly Game with Bounded Rationality," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 457-478, January.
    14. Louis Anthony (Tony) Cox, Jr., 2009. "Game Theory and Risk Analysis," Risk Analysis, John Wiley & Sons, vol. 29(8), pages 1062-1068, August.
    15. S. S. Askar, 2020. "Duopolistic Stackelberg game: investigation of complex dynamics and chaos control," Operational Research, Springer, vol. 20(3), pages 1685-1699, September.
    16. Toshihiro Matsumura, 2002. "Market Instability in a Stackelberg Duopoly," Journal of Economics, Springer, vol. 75(3), pages 199-210, April.
    17. Binmore, Ken, 2010. "Game theory and institutions," Journal of Comparative Economics, Elsevier, vol. 38(3), pages 245-252, September.
    18. Yang, Xuenan & Peng, Yu & Xiao, Yue & Wu, Xue, 2019. "Nonlinear dynamics of a duopoly Stackelberg game with marginal costs," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 185-191.
    19. Shi, Lian & Xu, Feng & Chen, Yongtai, 2021. "Quantum Cournot duopoly game with isoelastic demand function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
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