Analysis of Nonlinear Duopoly Game: A Cooperative Case

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.


Introduction
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 [1][2][3][4][5][6][7][8][9][10][11][12][13][14].
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 [15,16], theoretical and experimental studies have leaded up to several ways by which the cooperative solution can be obtained.For instance, in the well-known short game of prisoner's dilemma, the Nash equilibrium point is Pareto optimal as cooperation is obtainable.But for the repeated games, emergence of cooperation among competitors (firms) may be possible to achieve and then cooperation in iterated prisoner's dilemma can be explained [15,17].In [18], it has been shown that the conditional cooperative strategy such as the so-called "tit-for-tat" may be used to achieve cooperation among firms in repeated games.
The current paper is motivated by the work done by [15].We introduce a duopoly game based on a generalized nonlinear inverse demand function.An adjustment dynamic strategy is introduced and studied to detect the cooperation condition that may be occurred based on this strategy.Under the proposed function, the tit-for-tat is used and a two-dimensional discrete map is introduced.The complex dynamic characteristics of this map are studied and the stability of the Nash equilibrium is investigated.The qualitative study of bifurcation is studied analytically and numerically.We conclude our study with a tit-for-tat strategy with control.
The structure of the paper is as follows: In Section 2 a description of a Cournot duopoly game based on a generalized inverse demand function is presented.In Section 3, the two-dimensional map whose iteration gives the time evolution based on a proposed dynamic adjustment is defined.The steady state point of the map, which is Nash equilibrium, is computed.Then the stability of this point is investigated and its complex dynamics is detected.Section 4 introduces a tit-for-tat Cournot duopoly game using the same function.As in Section 3, the dynamic characteristics of the game are investigated.In Section 5, the system studied in Section 4 is improved by adding a control strategy in this system and some discussion is illustrated.Finally, we end the paper with some conclusions to show the significance of our results.

Cooperative Duopoly Model
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,  = 0, 1, 2, . ... The produced quantity by each firm at time  is denoted by  , ( = 1, 2).We assume that the cost of production is linear,  , =    , , where  ≥ 0 is a marginal cost.Further we assume a general inverse demand function as follows: It is well known that  indicates commodity price, while  = ∑ 2 =1   is the total quantity produced by the two firms.In addition,  and  are positive constants and  ̸ = −1.The profit of each firm is now given by In [3], the case of noncooperative duopolistic game based on incomplete information and the price function given in (1) at  = 3 has been studied.In the current paper, we study the cooperation situation under the incomplete information.We assume that both firms' profits are used in the cooperative profit that is denoted by   (cooperative profit is a maximization problem that is given by Max  =  1 +  2 ).The first-order derivative of the total profits gives the cooperative output   = (1/2)[( − )/(1 + )] 1/ and then   = (( − )/2(1+))[(−)/(1+)] 1/ .For the best of our knowledge,  = 1 gives the work done by Ding and Shi [15].

Dynamic Adjustment
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: where ] ≥ 0 is an adjusting parameter.It is assumed that the firms increase their current production in the direction (  −  , ).Now, system (3) has a unique fixed point,  1 = (  ,   ).In order to study the local stability of the fixed point  1 we need to calculate the eigenvalues of the following Jacobian: which are Therefore, the fixed point is locally stable if | 1 | < 1 that gives 0 < ] < 2. Some numerical simulations are carried out to illustrate the complex behavior of system (3).
For system (3), we start with the values  = 3,  = 2,  = 1,  = 1.7.With these values, Figure 1 shows that the equilibrium point of system (3) becomes asymptotically stable under the condition 0 < ] < 2 and then gets unstable due to bifurcation occurred.It is observed that the parameter who has affected the stability of the equilibrium point is the adjustment parameter ].In other words, for any values of the other parameters the equilibrium point will be locally asymptotically stable at the condition 0 < ] < 2. On the other hand, for any values for the other parameters, say, for example,  = 3,  = 2,  = 1,  = 1.7, and a little change in ], say ] = 2.00001, the equilibrium point becomes unstable and chaos is obtained that is illustrated in Figure 2.

Dynamic Tit-for-Tat Behavior
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  , with the cooperative profit   that is Pareto optimal.If  , >   , this means that each firm will probably reduce its output to keep the cooperation between each other.On the other hand, cooperation cannot be realized if  , <   because that indicates one of the firm is noncooperative.Based on these, map is built: where ]  > 0,  = 1, 2, is an adjusting parameter.This map can be rewritten in the following form: System (6) has a unique positive fixed point which is  1 =  2 =   .The Jacobian at this point is whose eigenvalues are where 1 is a critical condition by which we cannot detect whether system (6) is stable or not.Therefore, we perform some numerical simulation to investigate the characteristics of this system.

Dynamic Tit-for-Tat with Control
A feedback control is added in map ( 5) to improve it.This gives the following new map: where ]  > 0,  = 1, 2, is an adjusting parameter and ] > 0 and −]( , −  ),  = 1, 2, is the feedback control of the system.This feedback control shows that whether the firms cooperate or not.It is used to show that if  , >   ( = 1, 2), the firm is going to reduce its output in the next period of time since the current is over the cooperative output.On the other hand, if  , <   ( = 1, 2), the firm has to properly increase its output in the next period of time.Because of this feedback control, both firms have the intention to cooperate.Now, system (9) can be rewritten in the form: System (10) has a unique positive fixed point which is  1 =  2 =   .The Jacobian at this point is whose eigenvalues are where and therefore the fixed point is locally asymptotically stable.We perform some numerical simulation to investigate the characteristics of this system.We reconsider the unstable situation ( = 8,  = 2,  = 1,  0,1 = 2.5,  0,2 = 0.5, and  = 0.9) in the previous section.Let ] = 0.003; the equilibrium point is asymptotically stable, as shown in Figure 6.In addition, the numerical simulation has shown that as long as ] ∈ (0, 2) with the other conditions of the eigenvalues holding, the equilibrium point will be stable and the Pareto optimality holds.Figure 7 shows that Pareto optimality cannot be achieved when ] > 2.

Conclusion
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.