Abstract

A quadratic utility function is introduced in this paper to study the dynamic characteristics of Cournot duopoly game. Based on the bounded rationality mechanism, a discrete dynamical map that describes the game’s dynamic is obtained. The map possesses only one equilibrium point which is Nash point. The stability conditions for this point are analyzed. These conditions show that the point becomes unstable due to two bifurcation types that are flip and Neimark–Sacker. The synchronization property for that map is studied. Through local and global analysis, some dynamics of attracting sets are investigated. This analysis gives some insights on the basis of those sets and the shape of the critical curves. It also shows some lobes found in those attracting sets which are constructed due to the origin focal point.

1. Introduction

The Cournot duopoly game as described in literature consists of two players (or two competing firms), each wants to seek the optimal quantity of his/her production. Many scholars in literature have studied the dynamic of such games which have been described by discrete maps ([1–5]). The Nash equilibrium point of such games and its stability conditions were the core of study by scholars. Different types of bifurcations have been reported in literature as causes for the equilibrium point to be unstable. In addition, many utility functions have been adopted to model such games. Of which are Cobb–Douglas, constant elasticity of substitution (CES) and others. Interested readers are advised to see some applications of those utility functions and their properties in literature ([1–10]).

The present paper introduces a simple quadratic utility function. This function gives under Lagrangian function and its first-order condition price functions same as given by Cobb–Douglas. Our introduced utility function may be considered as a special case of Singh and Vives function [1]; however, the later does not give linear prices as reported in literature [6] if one apply Lagrangian function and its first-order conditions on it. Anyway, we highlight here some important works that have studied such kind of games. For instance, in [11], a Cournot–Bertrand duopoly game whose products are considered to be differentiated has been studied. Few useful investigations on such games and their optimality have been reported in the following studies ([12–14]). In [15], Tremblay et al. have analyzed the differentiated products in a Cournot–Bertrand duopoly game. They have focused on studying the static game and the stability/instability conditions of the game’s Nash equilibrium point. In [8], a theoretical framework of a Cournot–Bertrand game has been introduced. Based on a two-dimensional discrete linear map, Naimzada et al. [16] have analyzed a model of Cournot–Bertrand and studied its dynamic characteristics. Naimzada et al. have adopted different mechanisms such as the best response and adaptive adjustment to introduce the model which has been used to describe the dynamic of the game.

When studying such games, different types of adjustment mechanisms have been used in the modelling process. The modelling process means introducing the discrete map that describes the dynamic of game at discrete time steps. The most popular mechanism that has been used in the modelling process of such games is the bounded rationality approach. It has been considered as a gradient-based approach. Other mechanisms that has been used in few studies in literature are the naive mechanism, the tit-for-tat approach, and the approximation of local monopolistic or LMA mechanism. Information about those mechanisms and their properties can be founded in literature ([17–21]). Other interesting works on the extensions of those mechanisms and their applications have been reported ([22–26]).

The present paper belongs to the category of Cournot duopoly game on which both competing firms adopt the bounded rationality approach and want to seek the optimal quantities of production in order to achieve the profit maximization. The current game differs from those in literature [27], on which we introduce a new utility function that has not been adopted before. In this paper, local and global analyses are performed to investigate the game’s dynamics. This includes the investigation of multiple stable attractors and analyzes the attractive basins for some attracting sets. The main results in such study focus on analyzing the dynamics of the unique Nash equilibrium of the map’s game. This includes investigating the types of bifurcations by which the equilibrium point may be unstable. The obtained results show that there are two types of bifurcations by which the Nash point becomes unstable. These types are flip and Neimark–Sacker bifurcations. In addition, the form of the game’s map possesses a focal point that is the origin which gives rise to some lobes affecting on the shape of the attractive basins of some attracting sets. Furthermore, the synchronization property is studied. Moreover, the critical curves are calculated and show that the phase plane of the game’s map belongs to type.

The structure of paper is given as follows. The quadratic utility function and the discrete dynamic map describing the game are introduced in Section 2. This section includes also the stability investigation of the Nash equilibrium point and the route to chaos due to two different types of bifurcations. In addition, local and global analyses including the synchronization property are discussed in this section. Furthermore, the critical curves that divide the phase plane of the game’s map into and regions are calculated. Finally, Section 3 concludes our obtained results and suggests some future works.

2. The Model with Tax

Let us first assume the following quadratic utility function:where and denotes the quantity produced by the player (firm) . Our suggested model in this manuscript consists of two competing firms whose decision variables are the quantities produced by them. It is simple to see that the utility given in (1) is convex which economically means that the good has an increasing marginal utility. Furthermore, the marginal utility of the good does not depend on the good (as ). In addition, this utility is homogeneous. Assuming the budget constraint , then we get the following maximization problem:where represents the price of good . Solving (2), we get (see Appendix A)

Now, we assume that both firms detect the optimum according to maximizing their profits as follows:where represents the cost of the quantity and is taken as a linear cost where denotes a constant marginal cost. We assume also that the government has imposed a tax on each quantity as and . Now, (4) can be represented by

Recalling the mechanism of bounded rationality [8], both firms can update their outputs as follows:where . Using (5) in (6), we get the two-dimensional discrete dynamic map that describes the current game.

Now, we discuss the dynamic characteristics of the equilibrium point of map (7) in the next section.

2.1. The Equilibrium Point and Its Stability

The equilibrium point of the above map is a unique Nash equilibrium point and is obtained by setting and . It has the following form:

which is a positive point. We should highlight here that the equilibrium point (8) is the same as the one given in [28] but when the model has free taxes . The following propositions are used in the sequel to classify the type of the point .

Proposition 1. Let the Jacobian matrix the map (7) at any equilibrium point possesses two eigenvalues and . Then, this equilibrium point can be classified according to the following:(i)If , then it becomes an attracting node and it is stable(ii)If , then it is an unstable repelling node(iii)If and , (or and ) then it is an unstable saddle point(iv)It is a nonhyperbolic point if and (or and )In case, the eigenvalues have a complicated analytical form, we use the following proposition.

Proposition 2. Let and be the trace and determinant of the Jacobian matrix of (7), and suppose the followingThen, we have the following properties:(i)The equilibrium point is locally asymptotically stable if , and .(ii)The equilibrium point becomes unstable through a flip bifurcation if , and . In this case, the two eigenvalues are real and pass through .(iii)The equilibrium point becomes unstable through a transcritical or fold bifurcation if , and . In this case, the two eigenvalues are real and pass through 1.(iv)The equilibrium point becomes unstable through Neimark–Sacker if , and . In this case, the two eigenvalues are complex and their modulus passes through 1.The Jacobian matrix for the map (7) at the equilibrium point becomesAnd its eigenvalues are

Proposition 3. The equilibrium point is locally asymptotically stable if the following conditions are satisfied:

Proof. Simple calculations show that and for the Jacobian given in (10) becomeSubstituting (13) in (9), we get which is always positive since and are positive parameters. Simplifying the other two conditions completes the proof.

Proposition 4. The equilibrium point may be destabilized due to(i)flip bifurcation if (ii)Neimark–Sacker bifurcation

Proof. Substituting (13) in and then taking , we getSubstituting (14) in completes the proof of (i). For the part (ii), we put , and then we getThen substituting (15) in completes the proof of (ii).

2.2. The Synchronization Property

Synchronized trajectories in such game is an important property that may give more information about the dynamic of the model’s game. Indeed, such property may be occurred and there may be a transversely stable orbit on the diagonal . In case, such dynamics exist due to nonsynchronizing trajectories, and it would be important to detect the initial conditions leading to synchronization property. The possibility of such property arises when an invariant one-dimensional submanifold of exists. This means that the synchronized trajectories are described by

These trajectories given in (16) are regularized by the restriction of the map on the invariant submanifold by which synchronized dynamics occur and are described by the one-dimensional map:

For trajectories beginning outside the map given in (17) are said to be synchronized ones if as . Our map given in (7) possesses the six parameters, , and . Now, we assume the following:

Under this assumption, the map (7) becomes

The restriction conjugates the following one-dimensional map:

Studying the transverse stability of the synchronized attractors of the system (19) requires to calculate its Jacobian at the equilibrium point as follows:

Whose eigenvalues are . It is simple to see that if , and hence is an attracting stable node.

2.3. Local Analysis

In this section, we give some numerical experiments to validate the above results. As in [6], we assume the following values, , and . The Jacobian (10) at these values becomeswhose eigenvalues are real and equal and . One can see that , and hence the equilibrium point is the local stable point. Any increase in the parameters , and leads to unstable equilibrium point through flip bifurcation. In Figure 1(a), we take the parameter as the bifurcation parameter and fix the other parameters’ values to , and . As it can be seen that the point is locally stable for all the values of till this parameter reaches the point of cycle of period two and it becomes unstable. As increases further, higher periodic cycles are born, then the map enters the chaos area, and then becomes chaotic (as confirmed in the Largest Lyapunov exponent (LLE) given in Figure 1(c)). We should highlight here that the governmental taxes are different with imposed on the quantity produced by the first firm while the tax imposed on the second firm is . Numerical experiments show that higher governmental taxes imposed lead to shrinking of the stability region as will be discussed in the global analysis later on. Figure 1(b) shows that the impact on the parameter is slightly different on the quantities produced by both firms. This may be due to the cost of productions and taxes imposed that are slightly different. The same discussions are for the impact of the parameter when it is taken as the bifurcation diagram. Figures 1(d)–1(f) show the bifurcation diagram with respect to and the corresponding LLE at the parameters’ values, , and . On the other hand, we give in Figure 1(g)–1(l) the taxation impact on the equilibrium point and the corresponding LLE. Figure 1(g) shows that at the parameters’ values , and , the equilibrium point is stable at taxation rate ; then, it becomes unstable through Neimark–Sacker bifurcation. As increases further, a closed ring is born that is followed by a cycle of period 7 and then the map enters the chaotic region. The same discussion and observations are for the other taxation parameter on where its effect is given in Figures 1(j) and 1(k). The Figures 1(j) and 1(k) show the impact of on the quantity . Simulation shows that in the interval , the quantity bifurcates into two chaotic bands not period 2-cycle, and this is observed in the corresponding Lyapunov exponent given in Figure 1(l).

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Figure 1 

(a–l) Different types of bifurcation diagrams with respect to the parameters , and when and their corresponding Lyapunov exponents. Each figure is plotted at some parameters values given in the text and within the figure itself.

Now, we assume the set of parameters’ values , , and . The Jacobian [10] at these values becomes

Whose eigenvalues are complex and equal . One can see that , and hence the equilibrium point is local stable point. Any increase in the parameters , and leads to unstable equilibrium point through Neimark–Sacker (N-S) bifurcation. In Figures 2(a) and 2(b), we take the parameter as the bifurcation parameter and fix the other parameters’ values to , and . It is clear that the equilibrium point loses its stability through N-S bifurcation. The same observations are obtained when we take the parameters , and as the bifurcation’s parameter. The Figures 2(c)–2(h) give the N-S bifurcations for those parameters.

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(a–h) The N-S bifurcation diagrams with respect to the parameters , and when . Each figure is plotted at some parameters values given in the text and within the figure itself.

2.4. Global Analysis

The above local analysis regarding the bifurcation diagrams does not permit further understanding about the future evolution of the map (7). For this reason, some complex behaviors and their basins of attraction are given here. As said before, the map (7) depends on the parameters , and . Let us first investigate the influences of the parameters and by assuming the following parameters values, . This set of parameters values gives the two-dimensional bifurcation diagram in the presented by Figure 3(a). It is clear that this plane is divided into different periods by different colors besides the stability region. The figure shows that the system behaves chaotically when the speed of adjustment parameters and are all high. Any changing in those two parameters while keeping the other parameters fixed makes the system enter the chaotic region through Flip and Neimark Sacker bifurcation. The system’s attractors and multistability have been investigated by many scholars in literature ([4, 6, 16]). Such investigations have shown one of the important conditions of chaos which is the sensitivity to the initial conditions. In economic market, different strategies by decision makers may lead to different directions to the development of the firm’s direction. Consequently, adopting a good way of choosing initial conditions is very important which in turn gives an indication of whether the firm will behave well or not in the future. This includes the coexistence of multiple attractors and their influences on the stability of the equilibrium point. The basins of attraction can help in investigating such attractors. It can be used to analyze the convergence of system after a series of iterations (games) based on certain initial conditions. It helps the firm’s decision makers to choose a range of initial values by which the firm can develop better. For instance, let us assume the following initial set: , and . This set gives rise to a period of 3-cycle with its basins of attraction plotted in Figure 3(b). In Figure 3(b), the gray color characterizes the divergence (or the unfeasible trajectories), while the other colors represent the attractive basin of the equilibrium point and the basin of attraction of period 3-cycle coexisting with . The peculiar shape of the basin of attraction given in Figure 3(b) will be because of the origin point as will be discussed in the next section. Figure 3(c) gives a chaotic situation of the system around the equilibrium point as increases to 1.26, and the other parameters values are fixed to . Increasing the taxes and the speed of adjustment parameters gives rise to instability situation of the equilibrium point due to Neimark–Sacker bifurcation. For instance, the parameter values , and present a quasiperiodic dynamic possessing an attracting invariant closed ring around the equilibrium point. Other closed invariant curve is given when increasing the speed of adjustment parameter to . As shown previously, the stability conditions depend on the tax parameters and any changes in those parameters give rise to instability of the equilibrium point. For this reason, the two-dimensional bifurcation diagram in the is depicted in Figure 3(f). We finish this section by giving in Figure 4, an example of period 2-cycle at fixing the parameters values: , and .

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Figure 3 

(a) 2D-bifurcation diagram at . (b) The period of 3-cycle at , and . (c) Chaotic dynamic behavior at . (d) Closed invariant curve at . (e) Closed ring at . (f) 2D-bifurcation diagram at .

Figure 4 

The period of 2-cycle at , and .

2.5. Critical Curves and Preimage Zones

The phase plane of the map (7) at any sets of parameters’ values possesses some important characteristics. Looking at this map, one can observe that at or , one gets or , respectively. This means that the origin point traps the map (7), and therefore, it is important to start with this point in order to calculate the boundaries of the attractive basins for any attracting set for that map. To do that, we set and in (7) where ‘indicates time evolution. This means that the map’s time evolution is attained by the iteration of as follows: