Analysis and Control of the Complex Dynamics of a Multimarket Cournot Investment Game with Bounded Rationality

A dynamic multimarket Cournot model is introduced based on a specific inverse demand function. Puu’s incomplete information approach, as a realistic method, is used to contract the corresponding dynamical model under this function. Therefore, some stability analysis is carried out on the model to detect the stability and instability conditions of the system’s Nash equilibrium. Based on the analysis, some dynamic phenomena such as bifurcation and chaos are found. Numerical simulations are used to provide experimental evidence for the complicated behaviors of the system evolution. It is observed that the equilibrium of the system can lose stability via flip bifurcation or Neimark-Sacker bifurcation and time-delayed feedback control is used to stabilize the chaotic behaviors of the system.


Introduction
The dynamical behaviors of oligopoly games are complex because every oligopolistic producer in each period must consider not only its own decision but also the reactions of all other competitors.Cournot competition is an economic model used to describe the competition between some companies on the amount of output they will produce [1].Thus a generalization of this game to the case of two markets is done.It is shown that the resulting dynamics is quite variable.In the classic model, each participant uses a naïve expectation to suppose that the opponents' output keeps the same level as the previous period and adopts an output strategy to maximize the expected profit.Many researchers have analyzed the system stability and the complex phenomena in Cournot oligopoly games with this kind of expectation [2][3][4][5][6][7][8][9].In an early work [10], a kind of bounded rationality is assumed for the dynamical Cournot game, where each producer does not have complete knowledge of the market and updates its production by the local profit maximization method.
In recent years, a great amount of work has been done on the dynamical Cournot games with homogeneous or heterogeneous expectations.Bounded rationality in the marginal profit method is assumed to all producers in the models considering homogeneous expectation [10][11][12][13].The models with heterogeneous expectations (naïve, boundedly rational, or adaptive) have been discussed in many other works [14][15][16][17][18][19][20].
In this study, a dynamic multimarket Cournot model is introduced based on a specific inverse demand function.The main purpose of our work is to formulate a novel model, which puts investment decision as a substitute for output adjustment into the dynamical Cournot game.In the model, all producers are also assumed to have bounded rationality and make their investment decisions in line with the marginal profit in the previous period.Meanwhile each firm will increase its investment if it perceives a growing marginal profit and decreases its investment if the perceived marginal profit is decreased.During a local adjustment process, this novel dynamical Cournot game aims to develop the equilibrium or demonstrate complex dynamic behaviors.
This paper is organized as follows.In Section 2, we model the dynamical game played by players with bounded rationality.In Section 3, we discuss the existence and local stability of the equilibrium points for the system.In Section 4, we show the dynamic features of this system with numerical simulations, including bifurcation diagram, phase portrait, and sensitive dependence on initial conditions.In Section 5, time-delayed feedback control is used to stabilize the chaotic behaviors of the system.

The Multimarket Cournot Model
In this model, the products are near substitutes but different in the quality levels.So, firms can charge different prices for different markets.Suppose we have  firms ( > 2) that compete in two interrelated markets  and .For the price in the market, we consider Bowley [21] who has introduced the demand function where 0 ≤  ≤ 1.We also suppose that the production cost function of each firm takes a specific inverse demand function; [22][23][24] used this form in their work.Billand et al. [25] used the same demand function with  = 1.Let the demand in market  for firm  be where   is the firm 's price in markets ,   is the firm 's quantity and the constant   > 0,   > 0 for market .The demand in market  for firm  is where   is the firm 's price in markets ,   is the firm 's quantity and the constant   > 0,   > 0 for market .
The cost function of the firm is Hence the profit of firm  is given by A standard approach to generalize static games to dynamic ones is called the bounded rationality approach [3,5].The corresponding dynamical system for the static multimarket Cournot model is Here, we study the monopoly case.In this case there is only one firm ( = 1) and two markets.We use the speeds (rates) of adjustment as linear functions of the quantities.For instance, we take   (  ()) =   (  ()) and   (  ()) =   (  ()) as an example, where   and   are constants,  1 = ,  1 = ,  1 = , and  1 = .With all the assumptions above, we get firm profit in period  as follows: By differentiating (  ,   ), we obtain firm marginal profit with respect to its investment in period  at the markets  and , respectively, as follows: According to the theory of Ding [26], hence, we get the dynamic system as follows: where 0 ≤  ≤ 1 is a weight coefficient assigned to the nondelayed period  and 1 −  is assigned to the delayed period  − 1.
Let  1 () =   ( − 1),  2 () =   ( − 1); we obtain a fourdimensional discrete dynamic system: System (10) describes a duopoly game played by boundedly rational players making decision in a process of dynamical investment in the two markets.In the following sections, we are to investigate the dynamical properties of this model.

Analysis of the Equilibrium Points and Stability
Let   ( + 1) =   () and   ( + 1) =   () ( = , ) in system (10); then we get Solving equations in (11), we obtain four equilibrium states of dynamics (see (10)), which are listed as follows: In order to make these equilibrium points have economic meaning, we only consider the nonnegative cases.Since , , , and  are positive parameters,  1 ,  2 , and  * are all positive provided that  +  −  > 0, (13a) In the following, all the nonnegativity conditions (13a) and (13b) are assumed. 0 ,  1 , and  2 are all boundary equilibriums and  * is a unique interior equilibrium.Next, we will analyze the stability of the equilibrium.
Proposition 1.The boundary equilibrium  0 is an unstable equilibrium.
Proof.Taking the expression of equilibrium  0 into (14), we get the Jacobian matrix at  0 as follows: which has four eigenvalues: Proof.At the boundary equilibrium point  1 , the Jacobian matrix (see ( 14)) is given by By simple calculation, we get four eigenvalues of the matrix ( 1 ): Obviously  2 > 1, so equilibrium  1 is unstable.A similar approach shows that  2 is unstable too. where If () denotes the characteristic polynomial of the Jacobian matrix ( * ), then By calculation, we get where  = ++,  = +−, and  = −+.

Numerical Simulation
In this section, numerical simulations show how the system evolves under different levels of parameters, especially with adjustment speeds   and   .In all the numerical simulations, the other parameters are fixed:  = 1,  = 0.6,  = 0.6,  = 0.4,  = 5, and  = 4. So, we will use some numerical simulations to show the complicated behavior of the model (stability, period-doubling bifurcation, and chaos).
Figure 1 shows the bifurcation diagram of quantities   and   with the adjustment speed   while the other parameters are constant and have taken the value   = 0.6.This figure shows that the equilibrium points   and   are locally stable for   < 0.0755.
Figure 2 shows the bifurcation diagram of the quantities   and   with the adjustment speed   while the other parameters are constant and have taken the value   = 0.5.This figure shows that the equilibrium points   and   are locally stable for   < 0.443, and such a complicated process (period-doubling bifurcation, Neimark-Sacker bifurcation, periodic window, etc.) continues to lead the system to chaos.
The observations from Figures 1 and 2 show that when   and   are increasing, it shows more complex dynamic phenomena by appearing in the process of the evolution of Neimark-Sacker bifurcation and cycle window.
In Figure 3, the nonlinear dynamic system (10) can be showed from three dimensions of strange attractor, and strange attractor shows in the system space which is comprised by three arbitrary state variables.
In Figures 5(a), 5(b), and 5(c), the bifurcation diagrams are plotted for the weight coefficient assigned  when   is fixed as 0.6.Figure 5(a) is for   = 0.105, Figure 5(b) is for   = 0.515, and Figure 5(c) is for   = 0.615.Figure 5(a) shows a reverse period-doubling bifurcation, while Figures 5(b) and 5(c) combine reverse period-doubling bifurcation and reverse Neimark-Sacker bifurcation.The three figures show that the system tends toward stability with the increased weight coefficient, which also shows that a high residual rate has a positive effect on the system stability.

Chaos Control
From the numerical simulations, the adjustment rate and the weight coefficient have great influence on the stability of system (10).If the model parameters fail to locate into the stable region required, the behaviors of the dynamics will be much complicated.In a real economic system, chaos is not desirable and will be not expected, and it is needed to be avoided or controlled so that the dynamic system would work better.In this section, we use the time-delayed feedback control (e.g., [18,28]) to control system chaos.We modify the first equation of system (10) by intercalating a controller ( 1 () −  1 ( + 1)) as a small perturbation, where  > 0 is a controlling coefficient.Then the controlled system is given by It is easy to see that the new system (25) has the same equilibriums as system (10) and it takes the following equivalent form: The Jacobian matrix of the controlled system ( 26) is given by where As has been shown in Figures 1 and 2, chaotic behavior of system (10) occurs when all the model parameters take their values as (, , , , ,   ,   , ) = (1, 0.6, 0.4, 5, 4, 0.55, 0.6, 0.55).
Using this group of parameters values, we obtain the Jacobian matrix (see (26)) at the interior equilibrium as follows.
As shown in Figure 1, the controlled system near the adjustment coefficient   = 0.081 is entered into a state of chaos.When the model parameter selection, the initial conditions ( 1 (0),  1 (0),  1 (0),  2 (0)) = (1, 2, 1, 2), and system (10) show the messy chaos phenomenon, the external variable parameter   is in a stable area.Analysis under the controlled system (26) in this parameter values of the chaos represents the stability of the new system (26).The parameter values in the system use (26) the Jacobian matrix as follows: Corresponding characteristic polynomial calculates the matrix () =  4 with the coefficient of theory, as follows: According to Schur-Cohn stability criterion, when the Jacobi matrix of the characteristic polynomial () of the coefficients satisfy the following conditions: the entire characteristic root of the matrix is less than 1, which shows that dynamic system ( 26) is to take the set of parameter values under stable, so the controlled system (10) of chaos control tends to be a stable orbit.From the stability conditions, we get that all the eigenvalues of the matrix will lie inside the unit disk by providing  > 0.0917.When  > 0.0917, the controlled system (26) will be asymptotically locally stable.
In Figure 6, it is obviously observed that, with the control coefficient  increasing, the system gradually gets out of chaos and periodic windows and achieves stability when  > 0.0917.When  = 1.5 and  = 3.5, Figures 7 and 8 show the stable behaviors of the orbits of the controlled system beginning from the initial state ( 1 (0),  2 (0),  1 (0),  2 (0)) = (1, 2, 1, 2).These two graphs can be found in the stable region, the feedback gains strength value, and the more chaotic behavior can quickly control the stable orbit.

Conclusion
In this study we have taken into consideration a dynamic Cournot game based on quadratic cost function which is built with the same assumption of bounded rationality.We discuss the stability of each equilibrium solution by using the nonlinear system.Numerical simulations are used to provide experimental evidence for the complicated evolution behaviors of the system.It proves that the chaotic behavior of the system can be controlled.It proves that parameters play an important role in the stability of the economy systems based on analysis of dynamical behaviors of the established game models.It shows that the weight coefficient assigned can enhance the stability of systems.If each firm renews its investment strategy too fast or too slow, the systems will postpone converging to equilibrium and may respond intricately, including bifurcation, chaos, and initial state of sensitivity.Meanwhile, time-delayed feedback control could be used to stabilize the chaotic behaviors of the dynamic systems.

Figure 1 :Figure 2 :
Figure 1: Bifurcation diagram for the quantities   and   with the adjustment speed   .

Figure 5 :
Figure 5: Bifurcation diagrams of dynamic system (10) with the weight coefficient assigned .

Figure 7 :
Figure 7: Evolution for the dynamic system (26) with the controlling factor  = 1.5.

Figure 8 :
Figure 8: Evolution for the dynamic system (26) with the controlling factor  = 3.5.
0, and  4 = 0.Because ,   , and   are positive parameters, it can be seen that  1 ,  2 > 1 does not satisfy  0 point stability condition.So equilibrium  0 is unstable.
Proposition 2. The boundary equilibriums  1 and  2 are both unstable.