Skip to main content
SpringerLink
Log in

The application and complexity analysis about a high-dimension discrete dynamical system based on heterogeneous triopoly game with multi-product

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

Abstract

In production practice, firms usually produce multi-products rather than single products to obtain cost-saving advantages, cater for the diversity of consumer tastes and provide a barrier to entry. Based on nonlinear and economics theories, this paper establishes a discrete triopoly dynamical model which considers multi-product firms with heterogeneous expectations: naïve, adaptive and bounded rationality expectations. The discrete model is described by a 6-dimensional dynamical system. Thesis explores the path of the complexity of evolution and its intrinsic regularity, studies the influence of parameter change on the sensitivity level. The stable conditions of Cournot Nash equilibrium point are analyzed. The route to complex dynamics is investigated using 2-D and creative 3-D bifurcation diagrams by numerical experiment. The results show: the adaptive parameters can modify the stability of the market, but cannot lead to chaos independently; the bounded rationality parameters can arouse chaos for the whole market; larger differentiated degree of multi-product can suppress instability which is caused by adaptive and bounded rationality adjustment parameters. These results have significant theoretical and practical value to multi-product triopoly game with heterogeneous expectations in related markets.

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
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Puu, T.: The chaotic duopolists revisited. J. Econ. Behav. Organ. 33(3–4), 385–394 (1998)

    Article  Google Scholar 

  2. Agiza, H.N., Elsadany, A.A.: Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput. 149(3), 843–860 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bischi, G.I., Naimzada, A.: Global analysis of a dynamic duopoly game with bounded rationality. Adv. Dyn. Games Appl. 5, 361–385 (2000)

    Article  MathSciNet  Google Scholar 

  4. Kopel, M.: Simple and complex adjustment dynamics in Cournot duopoly models. Chaos Solitons Fractals 7(12), 2031–2048 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, F., Ma, J.: The study of dynamic process of the triopoly games in Chinese 3G telecommunication market. Chaos Solitons Fractals 42(3), 1542–1551 (2009)

    Article  MATH  Google Scholar 

  7. Ma, J., Zhang, J.: Research on the price game and the application of delayed decision in oligopoly insurance market. Nonlinear Dyn. 70(4), 2327–2341 (2012)

    Article  Google Scholar 

  8. Elsadany, A.A.: Competion analysis of a triopoly game with bounded rationality. Chaos Solitions Fractals 45(11), 1343–1348 (2012)

    Article  Google Scholar 

  9. Elabbasy, E.M., Agiza, H.N., Elsadany, A.A.: Analysis of nonlinear triopoly game with heterogeneous players. Comput. Math. Appl. 57(3), 488–499 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tramontana, F., Elsadany, A.E.A.: Heterogeneous triopoly game with isoelastic demand function. Nonlinear Dyn. 68(1–2), 187–193 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ji, W.: Chaos and control of game model based on heterogeneous expectations in electric power triopoly. Dis. Dyn. Nat. Soc. 2009, 8 (2009). doi:10.1155/2009/469564

    Google Scholar 

  12. Ma, J., Pu, X.S.: Complex dynamics in nonlinear triopoly market with different expectations. Dis. Dyn. Nat. Soc. 2011, 12 (2011). doi:10.1155/2011/902014

    MathSciNet  Google Scholar 

  13. Zhang, J., Ma, J.: Research on the price game model for four oligarchs with different decision rules and its chaos control. Nonlinear Dyn. 70(1), 323–334 (2012)

    Article  Google Scholar 

  14. Pu, X.S., Ma, J.H.: Complex dynamics and chaos control in nonlinear four-oligopolist game with different expectations. Int. J. Bifurc. Chaos. 23 (2013). doi:10.1142/S0218127413500533.

  15. Ahmed, E., Agiza, H.N.: Dynamics of a Cournot game with n-competitors. Chaos Solitions Fractals 9(9), 1513–1517 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ma, J., Wu, K.: Complex system and influence of delayed decision on the stability of a triopoly price game model. Nonlinear Dyn. 73(3), 1741–1751 (2013)

    Article  MathSciNet  Google Scholar 

  17. Guo, Y., Ma, J.: Research on game model and complexity of retailer collecting and selling in closed-loop supply chain. Appl. Math. Model. (2012). doi:10.1016/j.apm.09.034,2012

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

    Article  Google Scholar 

  19. Mu, L.L., Liu, P.: Complexity of a real estate game model with a nonlinear demand function. Int. J. Bifurc. Chaos 21 (2011). doi:10.1142/S021812741103043X.

  20. Wang, H., Ma, J.: Complexity analysis of a cournot-bertrand duopoly game model with limited information. Dis. Dyn. Nat. Soc. 2013, 6 (2013). doi:10.1155/2013/287371

    Google Scholar 

  21. Elettreby, M.F., Mansour, M.: On Cournot dynamic multi-team game using incomplete information dynamical system. Appl. Math. Comput. 218, 10691–10696 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Xin, B.G.: Bifurcation and chaos in a price game of irrigation water in a coastal irrigation district. Dis. Dyn. Nat. Soc. 2013, 10 (2013). doi:10.1155/2013/408904

    Google Scholar 

  23. http://www.investopedia.com/terms/e/economiesofscope.asp

  24. Eaton, B.C., Schmitt, N.: Flexible manufacturing and market structure. Am. Econ. Rev. 84(4), 875–888 (1994)

    Google Scholar 

  25. Chisholma, D.C., Norman, G.: Market access and competition in product lines. Int. J. Ind. Org. 30(5), 429–435 (2012)

    Google Scholar 

  26. Gallegon, A.G., Georgantzis, N.: Multiproduct activity in an experimental differentiated oligopoly. Int. J.Ind. Org. 19(3–4), 493–518 (2001)

    Article  Google Scholar 

  27. Grossmann, V.: Firm size and diversification: multiproduct firms in asymmetric oligopoly. Int. J. Ind. Org. 25(1), 51–67 (2007)

    Article  MathSciNet  Google Scholar 

  28. Lin, P., Zhou, W.: The effects of competition on the R&D portfolios of multiproduct firms. Int. J. Ind. Org. 31(1), 83–91 (2013)

    Article  MathSciNet  Google Scholar 

  29. Kashet, L.E.: Mathematical Models in Biology. Random House, New York (1992)

    Google Scholar 

  30. Chen, Z.Q., Ip, W.H., Chan, C.Y., Yung, K.L.: Two-level chaos-based video cryptosystem on H.263 codec. Nonlinear Dyn. 62, 647–664 (2010)

    Article  Google Scholar 

  31. Chen, Z.Q., Yang, Y., Yuan, Z.Z.: A single three-wing or four-wing chaotic attractor gen-erated from a three-dimensional smooth quadratic autonomous system. Chaos Solitonsand Fractals 38, 1187–1196 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Ma, Junhai, Zhang, Qi, Gao, Qin: Stability of a three-species symbiosis model with delays. Nonlinear Dyn. 67, 567–572 (2012)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the reviewers for their careful reading and providing some pertinent suggestions. This work is supported by National Natural Science Foundation of China (Grant no. 61273231), and the Doctoral Scientific Fund Project of the Ministry of Education of China (No. 2013003211073).

Author information

Authors and Affiliations

  1. Group of Nonlinear Dynamics and Chaos, School of Management and Economics, Tianjin University, Tianjin , 300072, China

    Junhai Ma & Fang Wu

Authors
  1. Junhai Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Junhai Ma or Fang Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, J., Wu, F. The application and complexity analysis about a high-dimension discrete dynamical system based on heterogeneous triopoly game with multi-product. Nonlinear Dyn 77, 781–792 (2014). https://doi.org/10.1007/s11071-014-1340-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1340-2

Keywords

Search

Navigation