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

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.