Skip to main content
SpringerLink
Log in

Complex dynamics investigations of a mixed Bertrand duopoly game: synchronization and global analysis

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, an economic competition between two firms that want to maximize the weighted-average social welfare and own profits is proposed. This kind of competition is eligible in the complex economic market. The competition is described by a nonlinear discrete dynamic map whose variables are prices of quantities produced by firms. Complex dynamic characteristics such as global dynamic behavior, multi-stability and synchronization are investigated for the competition’s map. The map’s fixed points are calculated, and their stability conditions are discussed. The obtained results show that the map’s Nash point can be destabilized through flip bifurcation. The global bifurcation of the game is analyzed using a 2D map, which corresponds to the model via critical curves. The results show that the map’s synchronization is equivalent to the standard logistic map. Through numerical simulation, some attracting sets of the synchronized map are given. Finally, we calculate the critical curves of the map and prove that it belongs to \(Z_{4}-Z_{2}-Z_{0}\) type, and hence, it is noninvertible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Availability of data and material

All the data required are within the paper.

References

  1. Xin, B., Chen, T.: On a master-slave Bertrand game model. Econ. Model. 28, 1864–1870 (2011)

    Article  Google Scholar 

  2. Garmani, H., Omar, D., El Amrani, M., Baslam, M., Jourhmane, M.: Analysis of a dynamics duopoly game with two content providers. Chaos Solitons Fractals 131, 109466 (2020)

    Article  MathSciNet  Google Scholar 

  3. Xin, B., Cao, F., Peng, W., Elsadany, A.A.: A Bertrand Duopoly Game with Long-Memory Effects. Hindawi Complexity Volume 2020, Article ID 2924169, 7 pages

  4. Yi, Q., Zeng, X.J.: Complex dynamics and chaos control of duopoly Bertrand model in Chinese air-conditioning market. Chaos Solitons Fractals 76, 231–237 (2015)

    Article  Google Scholar 

  5. Peng, J., Miao, Z., Peng, F.: Study on a 3-dimensional game model with delayed bounded rationality. Appl. Math. Comput. 218, 1568–1576 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Sun, Z., Ma., J.: Complexity of triopoly price game in Chinese cold rolled steel market. Nonlinear Dyn. 67, 2001–2008 (2012)

  7. Bischi, G., Kopel, M.: Equilibrium selection in a nonlinear duopoly game with adaptive expectations. J. Econ. Behav. Organ. 46, 73–100 (2001)

    Article  Google Scholar 

  8. Fanti, L., Gori, L., Mammana, C., Michetti, E.: The dynamics of a Bertrand duopoly with differentiated products: synchronization, intermittency and global dynamics. Chaos Solitons Fractals 52, 73–86 (2013)

    Article  MathSciNet  Google Scholar 

  9. Andaluz, J., Jarne, G.: Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational. Ann. Oper. Res. 238, 1–25 (2016)

    Article  MathSciNet  Google Scholar 

  10. Andaluz, J., Jarne, G.: On the dynamics of economic games based on product differentiation. Math. Comput. Simul. 113, 16–27 (2015)

    Article  MathSciNet  Google Scholar 

  11. Bischi, G., Naimzada, A.: Global analysis of a dynamic duopoly game with bounded rationality. In: Filar, J.A., et al. (eds.) Advances in Dynamic Games and Applications. Birkhauser, Boston (2000)

  12. De Fraja, G., Delbono, F.: Alternative strategies of a public enterprise in oligopoly. Oxf. Econ. Pap. 41, 302–311 (1989)

    Article  Google Scholar 

  13. De Fraja, G., Delbono, F.: Game-theoretic models of mixed oligopoly. J. Econ. Surv. 4, 1–17 (1990)

    Article  Google Scholar 

  14. White, M.D.: Mixed oligopoly, privatization and subsidization. Econ. Lett. 53(2), 189–195 (1996)

    Article  Google Scholar 

  15. Bennett, J., Maw, J.: Privatization, partial state ownership and competition. J. Comp. Econ. 31(1), 58–74 (2003)

    Article  Google Scholar 

  16. Matsumura, T., Shimizu, D.: Privatization waves. Manch. Sch. 78, 609–625 (2010)

    Article  Google Scholar 

  17. Zhu, X., Zhu, W., Yu, L.: Analysis of a nonlinear mixed Cournot game with boundedly rational players. Chaos Solitons Fractals 59, 82–88 (2014)

    Article  MathSciNet  Google Scholar 

  18. Ahmed, E., Ashry, G.A., Askar, S.S.: On multi-objective optimization and game theory in production management. Int. J. Nonlinear Sci. 24, 29–33 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Mert, M.: What does a firm maximize? A simple explanation with regard to economic growth. Int. J. Eng. Bus. Manag. 10, 1847979018815296 (2018)

    Google Scholar 

  20. Elsadany, A.A., Awad, A.M.: Nonlinear dynamics of Cournot duopoly game with social welfare. Electron. J. Math. Anal. Appl. 4(2), 173–191 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Fanti, L., Buccella, D.: Corporate social responsibility, profits and welfare with managerial firms. Int. Rev. Econ. 64, 341–56 (2017)

    Article  Google Scholar 

  22. Cao, Y., Zhou, W., Chu, T., Chang, T.: Global dynamics and synchronization in a duopoly game with bounded rationality and consumer surplus. Int. J. Bifurc. Chaos 29, 1930031 (2019)

    Article  MathSciNet  Google Scholar 

  23. Puu, T., Tramontana, F.: Can Bertrand and Cournot oligopolies be combined? Chaos Solitons Fractals 125, 97–107 (2019)

    Article  MathSciNet  Google Scholar 

  24. Zhou, J., Zhou, W., Chu, T., Chang. Y., Huang, M.: Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation. Appl. Math. Comput. 341, 358–78 (2019)

  25. Zhou, W., Zhao, N., Chu, T., Chang, Y.X.: Stability and multistability of a bounded rational mixed duopoly model with homogeneous product. Complexity 2020, 8 (2020)

  26. Li, W.N., Elsadany, A.A., Zhou, W., Zhu, Y.L.: Global analysis, multi-stability and synchronization in a competition model of public enterprises with consumer surplus. Chaos Solitons Fractals 143, 110604 (2021)

  27. Yu, Y., Yu, W.: The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference. Appl. Math. Comput. 395, 125852 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Askar, S.S., Abouhawwash, M.: Quantity and price competition in a differentiated triopoly: static and dynamic investigations. Nonlinear Dyn. 91(3), 1963–1975 (2018)

    Article  Google Scholar 

  29. Askar, S.S.: The impact of cost uncertainty on Cournot duopoly game with concave demand function. J. Appl. Math. 2013, Article ID 809795 (2013)

  30. Andaluz, J., Elsadany, A.A., Jarne, G.: Dynamic Cournot oligopoly game based on general isoelastic demand. Nonlinear Dyn. 99(2), 1053–1063 (2020)

    Article  Google Scholar 

  31. Zhou, W., Wang, X.X.: On the stability and multistability in a duopoly game with R&D spillover and price comeptition. Discret. Dyn. Nat. Soc. 2019 (2019)

  32. Zhou, J., Zhou, W., Chu, T., Chang, Y.X., Huang, M.J.: Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation. Appl. Math. Comput. 341, 358–378 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Askar, S.S.: On complex dynamics of Cournot-Bertrand game with asymmetric market information. Appl. Math. Comput. 393, 125823 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Elaydi, S.N.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2005)

    MATH  Google Scholar 

Download references

Funding

This research has not received any types of funding.

Author information

Authors and Affiliations

  1. Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt

    A. M. Awad & A. A. Elsadany

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

    S. S. Askar

  3. Department of Mathematics, College of Sciences and Humanities Studies, Prince Sattam Bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia

    A. A. Elsadany

Authors
  1. A. M. Awad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. S. Askar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Elsadany
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors are equally contributed in this paper.

Corresponding author

Correspondence to S. S. Askar.

Ethics declarations

Conflicts of interest

The authors declare that this article content has no conflict of interest.

Code availability

The software used in the paper is free and is called E&F Chaos (https://cendef.uva.nl/software/ef-chaos/ef-chaos.html).

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Awad, A.M., Askar, S.S. & Elsadany, A.A. Complex dynamics investigations of a mixed Bertrand duopoly game: synchronization and global analysis. Nonlinear Dyn 107, 3983–3999 (2022). https://doi.org/10.1007/s11071-021-07143-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07143-2

Keywords

Search

Navigation