Nonlinear Dynamics

, Volume 78, Issue 4, pp 2759–2768 | Cite as

Complexity analysis of a Cournot–Bertrand duopoly game with different expectations

Original Paper


In this paper, we consider a Cournot–Bertrand mixed duopoly model with different expectations, where the market has linear demand and the firms have fixed marginal cost functions. Two firms choose output and price as decision variables, respectively, under the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market is occupied by the one that applies a lower price. The two players are considered to have bounded rational and static expectations. The existence and local stability condition of Nash equilibrium is investigated. We find the stability region of Cournot–Bertrand system is bigger than that of Cournot or Bertrand system under the same conditions. Furthermore, there are two different kinds of bifurcations when the parameters pass through the different boundary curves of the stability region, which is different from the Cournot or Bertrand model. Numerical simulation method is used to display the dynamic behaviors of the dynamical system, such as periodic cycles, bifurcation diagrams and strange attractors of the systems. The economic explanations of the complex dynamic behaviors are also given.


Cournot–Bertrand model Stability Dynamics Bifurcation Oligopoly theory 



The authors thank the reviewers for their careful reading and providing some pertinent suggestions. The research is supported by the National Natural Science Foundation of China (No: 61273231) and Tianjin City High School Science and Technology Fund Planning Project (No: 20130823).

Supplementary material

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Supplementary material 1 (eps 155 KB)
11071_2014_1623_MOESM2_ESM.eps (142 kb)
Supplementary material 2 (eps 142 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.College of ScienceTianjin University of Science and TechnologyTianjinChina
  2. 2.School of ManagementTianjin UniversityTianjinChina

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