We study the dynamics of a nonlinear discrete-time duopoly game, where the players have homogenous knowledge on the market demand and decide their outputs based on adaptive expectation. The Nash equilibrium and its local stability are investigated. The numerical simulation results show that the model may exhibit chaotic phenomena. Quasiperiodicity is also found by setting the parameters at specific values. The system can be stabilized to a stable state by using delayed feedback control method. The discussion of control strategy shows that the effect of both firms taking control method is better than that of single firm taking control method.

An oligopoly is a market form in which a market or industry is dominated by a small number of sellers. Thus, they have power to decide the market price and the yields. The behavior of one player inevitable influences the other players’ behavior. To maximize their profits, each oligopolist is likely to be aware of the actions of the others.

Cournot, in 1838, introduced the first formal theory of oligopoly [

Expectations play an important role in modeling economic phenomena. A firm can choose its expectation rules to adjust its production. There exist three different firms’ expectations: naive, bounded rational, and adaptive [

Recently, the dynamics of duopoly game has been studied by many researchers. Chaotic phenomena in duopoly game were first found by Puu [

Since chaotic phenomena were found in duopoly game, there are also many researchers focusing on the chaos control in duopoly game [

The paper is organized as follows. In Section

We assume that the duopoly players produce homogenous goods which are perfect substitutes and offer goods at discrete-time periods. Therefore, the duopoly players face the same market demands. They both choose adaptive expectation rule to decide the amounts of the goods in the next period as their response strategy. Assume that the total demand is reciprocal to price

By setting

The Jacobian matrix of the two-dimensional map (

In this section, numerical simulation results corresponding to model (

Figure

Bifurcation diagram with respect to parameter

Now, set

Bifurcation diagram with respect to parameter

Figure

(a) Phase portrait of map (

Figure

(a) Phase portrait of map (

Any firm does not want to adjust its outputs too frequently and largely. So it is important for firms to control their outputs to a stable process. Chaos control has been studied by many researchers since chaos has been found in economy [

To examine the effects of the control function, we compare the output process of firm 1 before and after adding the control function. Set

Bifurcation diagram with respect to parameter

(a) Time series of output of firm 1 before using control function. (b) Time series of output of firm 1 after only firm 1 uses control function. (c) Time series of output of firm 1 after both firms use control function.

Now, consider the case that both firms realize that they will generate chaos in the future if they do not take any measure to control it. Therefore, both firms use control function to stabilize their outputs. Then, system (

In this paper, we have investigated a Cournot duopoly model where both the players decide their outputs weighting on their own previous outputs and the optimal outputs on the condition that their rival produces as their previous step. The Nash equilibrium and its local stability were analyzed. The numerical simulations show that the changes of marginal cost

In view of the fact that firms may not produce any goods if they suffer losses, it is needed to take firms’ profits into consideration in modeling the outputs of firms. This problem will be investigated in our future research.

The authors declare that they have no competing interests.

This work was supported by the National Natural Science Foundation of China (Grant no. 71571007), the State Key Program of National Natural Science Foundation of China (Grant no. 71333014), and the Aeronautical Science Foundation of China (Grant no. 2012ZG51079).