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We make further attempts to investigate equilibrium stability of a nonlinear Cournot duopoly game. Our studies in this paper focus on the cooperation that may be obtained among duopolistic firms. Discrete time scales under the assumption of unknown inverse demand function and linear cost are used to build our models in the proposed games. We introduce and study here an adjustment dynamic strategy beside the so-called tit-for-tat strategy. For each model, the stability analysis of the fixed point is analyzed. Numerical simulations are carried out to show the complex behavior of the proposed models and to point out the impact of the models’ parameters on the cooperation.

There are often several duopolistic firms in economic market where competition among them is controlled by the amount of commodities they produce, the demand scheme they adopt, and the profit each firm wants to maximize. In the competition, firms produce the same or homogenous goods and they must focus not only on the market size, but also on the actions their competitors do. Game theory is one of the most important theories that is used to describe and study such competition statically and dynamically. Game theory is characterized by its ability to consider interactions among firms. The dynamic case in which the equilibrium point (Nash equilibrium) is sought and its complex dynamic characteristics are of main interest has been studied in literature [

In this paper, we argue that there is a cooperation between firms in repeated Cournot duopoly games with a generalized price function. In Cournot duopoly games, Nash equilibrium or Cournot equilibrium is the basic solution in such games and reflects the rationality of the firms within games. Since firm rationality contradicts with Pareto optimality (in cooperation case), then Nash equilibrium in duopoly game is not Pareto optimal. In other words, Pareto optimality in such games cannot be achieved by firm interest’s maximization. As reported in [

The current paper is motivated by the work done by [

The structure of the paper is as follows: In Section

Suppose a market with two firms producing the same product or homogenous product. The decisions in this market are the quantities both firms sell in the market and are taken at discrete time scale,

It is well known that

In [

To achieve the cooperation between the two firms in the repeated game, we assume that the firms revise their beliefs by using the following dynamic adjustment:

For system (

Behavior of

Behavior of

Another strategy for achieving the cooperation between the two firms is the tit-for-tat strategy. With this strategy, every firm is doing what its opponent has done in the previous move. This is an incomplete information scenario; however the only things each firm knows are the output and the profit. In this situation each firm compares its profit

System (

Behavior of

Behavior of

Behavior of

A feedback control is added in map (

System (

We reconsider the unstable situation (

Behavior of

Behavior of

In this paper we have studied the cooperation that may be obtained among duopolistic firms in the economic market. Based on a general nonlinear price function, three duopolistic Cournot models have been investigated. For each model, the fixed point has been computed and complete analytical and numerical studies of the stability conditions for the fixed point have been obtained. The analyses show that under the dynamic adjustment strategy and the tit-for-tat strategy, the cooperation may be achieved, but the stability in both systems is sensitive to the parameters, and the Pareto optimality cannot be assured; by improving the model—adding the feedback control to consider the cooperation intention of the firms—the firms’ cooperation can be achieved, and the Pareto optimality is stable within the parameters’ certain field. So the cooperation can be the result of such strategy under certain condition.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Research group no. RG-1435-054.