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Two different time delay structures for the dynamical Cournot game with two heterogeneous players are considered in this paper, in which a player is assumed to make decision via his marginal profit with time delay and another is assumed to adjust strategy according to the delayed price. The dynamics of both players output adjustments are analyzed and simulated. The time delay for the marginal profit has more influence on the dynamical behaviors of the system while the market price delay has less effect, and an intermediate level of the delay weight for the marginal profit can expand the stability region and thus promote the system stability. It is also shown that the system may lose stability due to either a period-doubling bifurcation or a Neimark-Sacker bifurcation. Numerical simulations show that the chaotic behaviors can be stabilized by the time-delayed feedback control, and the two different delays play different roles on the system controllability: the delay of the marginal profit has more influence on the system control than the delay of the market price.

A monopoly market is the case where a trade is completely controlled by a small number of firms. The few firms produce the same or homogeneous products and they must take into account all the information in the market and the actions of the competitors. Cournot [

Rather than the naïve expectation, a so-called boundedly rationality based on players’ marginal profits has received great attention in recent years. Bischi and Naimzada [

Besides the work on the models with homogeneous players, there is another branch of literature that is interested in the games with heterogeneous players. The dynamic duopoly model with one boundedly rational player and one naïve player has been studied by Agiza and Elsadany [

In the work on the dynamic Cournot model, time delay has also attracted the attention of many scholars. Many researches examine the effect of delay on dynamics. One-time delay, two-time delay, continuously distributed time delay, and geometric delay are systematically reviewed and studied in [

In this work, we reconsider the duopoly model in Ding et al. [

We assume that there are two firms producing homogeneous goods for sale and the total supply

Suppose that firm 1 adjusts its output with the same time delay structure as discussed in [

Next, we consider firm 2 that is supposed to have a time delay structure built for the price history. In a real market, the market price is usually a common knowledge to all players so that the information of the price history

It is supposed that in period

For a duopoly game with two homogeneous players, Section

To give a specific form for dynamics (

Taking (

To study dynamics (

Letting

At an equilibrium point

We know that the local stability of an equilibrium point is determined by the eigenvalues of the above matrix. That is, an equilibrium

Taking the expression of the boundary point

Next we study the local stability of the interior equilibrium

The local stability of

The determinants of the

In our model,

If the inequalities in (S1) and (S2) are all satisfied, then the equilibrium

In this section, the constants

In Figure

Bifurcation diagrams with respect to the adjustment speed

To study the influence of the delay weight

Stability regions in the

Figure

Bifurcation diagrams with respect to the adjustment speed

The stability regions of the system are shown in Figure

Stability regions in the

Comparing Figures

As is shown in Figures

Phase portrait for Figure

We know that chaos is not desirable in a real economic system and it is often hoped to be controlled so that the dynamic system could run in a stable status. In this section, we show that the usually used method of time-delayed feedback control (e.g., [

Using the time-delayed feedback control method, we adjust (

It is obvious that controlled system (

By a similar approach in Section

To show the availability of the control method, we consider the case

Bifurcation diagram with respect to

Below we numerically check the

Stability regions in the

Stability regions in the

In this paper, we reconsider the duopoly game with time delay and formulate a model with two different delay structures. The main consideration is that the price history may be observed by all players in the market and there may be players making decision according to the historic information of the market price. In this work, we modify the model in [

The dynamics of the players’ output adjustments has been established, the equilibrium solutions are obtained, and the stability of the equilibriums is mathematically analyzed. The boundary equilibrium is proved to be unstable and the stability conditions for the interior equilibrium are obtained. Numerical simulations are done to show the influence of the main model parameters on the system stability and on the complicated behaviors of the system when losing stability. It is observed that there is difference in the roles played by the two delay structures. The time delay for the marginal profit has more influence on the dynamical behaviors of the system but the market price delay has less effect. An intermediate level of the delay weight for the marginal profit can promote the system stability and thus reduce the occurrence of complex behavior such as bifurcation and chaos. It is shown that the system stability losing may be due to either period-doubling bifurcations or Neimark-Sacker bifurcations. Numerical simulations show that the chaotic behaviors can be stabilized by the time-delayed feedback control, and the two different delays play also different roles in the system controllability: the delay for the marginal profit has more influence on the system control than the delay for the market price, and an intermediate level of delay for the marginal profit has a positive effect on the chaos control.

The authors declare that they have no conflicts of interest.

Financial support by the National Natural Science Foundation of China (nos. 71171098, 51306072, 71690242, and 71273120) is gratefully acknowledged.