# Nonlinear Dynamics of Cournot Duopoly Game: When One Firm Considers Social Welfare

Authors: S. S. Askar; A. A. Elsadany
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, we study the competition between two firms whose outputs are quantities. The first firm considers maximization of its profit while the second firm considers maximization of its social welfare. Adopting a gradient-based mechanism, we introduce a nonlinear discrete dynamic map which is used to describe the dynamics of this game. For this map, the fixed points are calculated and their stability conditions are analyzed. This includes investigating some attracting set and chaotic behaviors for the complex dynamics of the map. We have also investigated the types of the preimages that characterize the phase plane of the map and conclude that the game’s map is noninvertible of typeZ4−Z2.


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## Body

## 1. Introduction

Because of the appearance of the wealth theory in [1, 2], many studies on Cournot and Bertrand games have been raised. For instance, Singh and Vives introduced a quadratic utility function that has been used to model and study a Cournot duopoly game in [3]. This utility function has been also adopted in different games by many authors such as Askar [4–7], Elsadany [8], Naimzada and Tramontana [9], and Ma and Pu [10]. Modelling such games in a discrete time periods requires some mechanisms such as bounded rationality, Puu’s approach, naive expectation, and other adaptive methods.Several studies have been adopted both bounded rationality and Puu’s incomplete information. They are two different mechanisms. For the bounded rationality, the game’s players (or firms) are updating their output production depending on discrete time steps and by using a local estimate of their marginal profits. Furthermore, the players do not have to possess a complete knowledge of both demand and cost functions. However, they instead want to discover whether the market responses to small changes in production using an estimation of their marginal profits. For more applications on this mechanism which is sometimes called a myopic mechanism, one can see [11, 12]. The so-called Puu’s mechanism has been introduced in [13]. It is characterized by its realistic feature that is the firms do not have to know the profit function in order to get estimation of the commodity produced at the next period of time. The firms only require the commodity and profit at the past two periods of time. There are some other updating mechanisms that have been reported in the literature. For instance, Long and Huang [14], Agiza and Elsadany [15], Kopel [16], Elabbasy et al. [17, 18], Askar and Abouhawwash [19], Hommes [20], Tremblay et al. [21], Ahmed et al. [22], Baiardi and Naimzada [23], Fanti et al. [24], Elsadany and Awad [25], Tremblay and Tremblay [26], Askar and Al-Khedhairi [27], and Gao and Du [28].The majority of the literature has been analyzed for models of mixed oligopoly on the basis that the game has been in a static case. Liu et al. [29] investigated the static game of endogenous horizontal product differentiation in a mixed duopoly. The relationship between privatization and corporate tax policies has been studied in [30]. Nie [31] has analyzed the effects of capacity constraints on the mixed duopoly game. The strategy for cost-reduction innovation in the mixed economy has been explored in [32]. All of earlier papers have not discussed the case of a dynamic mixed oligopoly model.The current paper introduces a dynamic game of Cournot duopoly on which two firms are competing but they are different in optimization process. The first firm focuses on maximizing its profit that depends on a quadratic production cost. The second firm is different and wants to maximize its social welfare using the same quadratic cost form. The main contribution considers introducing such optimization problem and dynamic characteristics emanated from the game’s map. Furthermore, the monopolistic case is studied and is shown that each firm behaves monopolistically such as the standard logistic process. In the duopolistic case, the equilibrium points are calculated and their stability conditions are analyzed. Indeed, this includes local and global analysis of the routes by which these equilibrium points can be destabilized. The numerical simulation shows that some attracting sets are born due to both period-doubling and Neimark–Sacker bifurcations. In addition, the nonlinearity and noninvertibility of the game’s map give rise to such complex behaviors.The outline of current paper is divided into the following sections. After the introduction, we give in Section2 the description of the game represented by a two-dimensional discrete dynamic map. In Section 3, we calculate the game’s fixed points and study their stability. Furthermore, we give in this section a detailed discussion on the monopolistic case. In addition, we study by numerical simulation the attracting sets and chaotic behaviors arise due to the dynamics of the duopoly case. Moreover, we investigate the critical curves of the map and categorize the phase plane regions. Finally, we give our conclusion on the obtained results within this paper.


## 2. Cournot Duopoly Game

The game is constructed based on two competing firms with quantity-based strategies and differentiated products. The quantities produced by the two firms are denoted byq1 and q2. Their demand functions are obtained by recalling the following utility function:(1)Uq1,q2=aq1+q2−12q12+2dqq2+q22.More information on this utility is given in [3]. Both a and d are constants. Supposing the budget constraint p1q1+p2q2=M, we get the following maximization problem:(2)MaxUq1,q2,s.t.p1q1+p2q2=M.Solving (2) gives(3)p1=a−q1−dq2,p2=a−q2−dq1,where p1 and p2 denote the retail prices of the two firms’ products, respectively. The constants a>0 represent the maximum price while d∈−0.5,1 represents the product differentiation. d=1 indicates homogeneous products while d=0 means we have two monopolistic firms. Consider the following cost function:(4)Ciqi=c2qi2,i=1,2,where c>0. Now, the profits of the two firms are given by(5)π1=p1q1−c2q12=1−q1−dq2q1−c2q12,π2=1−q2−dq1q2−c2q22.The consumer surplus is assumed to beCS=1/2q12+q22+2dq1q2. The social welfare is defined as the sum of consumer surplus and profits as follows: W=CS+π1+π2. Now, both firms want to maximize the following payoffs:(6)π1=a−q1−dq2q1−c2q12,Π2=ωπ2+1−ωW,where ω∈0,1. We should highlight that lack of market demand and consumer information bring difficulty for producers. Therefore, producers estimate the market demand by adopting the gradient mechanism defined below:(7)q1t+1=q1t+v1q1∂π1∂q1,q2t+1=q2t+v2q2∂Π2∂q2,where vi represents the adjustment speed for the ith firm. Using (6) in (7), we get the map’s game:(8)Tq1,q2:q1t+1=q1t+ν1q1a−2+cq1−dq2,q2t+1=q2t+ν2q2a−1+ω+cq2−dq1.This map is a quadratic discrete dynamic map, and it is converted into Fanti’s map [24] at ω=1.


## 3. Equilibrium Points and Their Stability

Settingqit+1=qit=qi,i=1,2 in (8), one obtains(9)q1a−2+cq1−dq2=0,q2a−1+ω+cq2−dq1=0.Solving algebraically system (9), we get four fixed points as follows:(10)E0=0,0,E1=ac+2,0,E2=0,ac+ω+1,E∗=q1∗,q2∗=a+aω+ac−ad3c+2ω+cω+c2−d2+2,2a+ac−ad3c+2ω+cω+c2−d2+2,where E0,E1,E2 are called boundary fixed points while E∗ is called a Nash equilibrium point. It should be noted that the equilibrium points E∗ become the same as of Fanti’s equilibrium point when ω=1. Studying the stability of these points requires to calculate the Jacobian matrix of the map:(11)J=1+ν1a−22+cq1−dq2−ν1dq1−ν2dq21+ν2a−21+ω+cq2−dq1.It has the following characteristic polynomial:(12)fλ=λ2−trJEλ+detJE,where trJ and detJ represent trace and determinant of (11). They are used in Jury conditions [4]. These conditions are given by(13)f1=1−trJE+detJE>0,f−1=1+trJE+detJE>0,detJ<1.The above conditions characterize different types of bifurcations by which the equilibrium points may be unstable. These types are summarized in the following:(1
)Period-doubling bifurcation is raised whenf−1=0(2)
Transcritical or fold bifurcation is raised whenf1=0(3)
Neimark–Sacker bifurcation is raised when detJ<1Now, we study the stability of the fixed points.Theorem 1.
The fixed pointE0 is unstable point.Proof.
The Jacobian matrix given in (11) at this point becomes(14)JE0=1+ν1a001+ν2a.
It is clear that (15) is a diagonal matrix and hence its eigenvalues become λ1=1+ν1a and λ2=1+ν2a. Because of the positivity of the parameters a,v1, and v2, we have λ1,2>1 and so E0 is unstable repelling node.Theorem 2.
The fixed pointE1 is saddle point.Proof.
The Jacobian matrix given in (11) at this point becomes(15)JE1=1−ν1aν1dc+201+ν22ac+2.
It is clear that (15) is a triangular matrix and hence its eigenvalues become λ1=1−ν1a and λ2=1+ν22a/c+2. It is simple to see λ1<1 and λ2>1. Therefore, the fixed point E1 is saddle point.Theorem 3.
The fixed pointE2 is saddle point.Proof.
The Jacobian matrix given in (11) at this point becomes(16)JE2=1+ν1ac+ω+1−dc+ω+10−ν2adc+ω+11−aν2.
It is clear that (16) is a triangular matrix and hence its eigenvalues become λ1=1+ν1ac+ω+1−d/c+ω+1 and λ2=1−aν2. Because of the positivity of the parameters a,v1,v2, and d<1, we have λ1>1 and λ2<1. Therefore, the point E2 is saddle point.
For Nash equilibrium point, the Jacobian becomes(17)JE∗=1−a2+c1+ω+c−d2+2ω+3c−d2+cω+c2ν1−ad1+ω+c−d2+2ω+3c−d2+cω+c2ν1−ad2+c−d2+2ω+3c−d2+cω+c2ν21−a2+c−d1+ω+c2+2ω+3c−d2+cω+c2ν2,whose trace and determinant are given by(18)trJE∗=2−a2+c1+ω+c−dν1+a1+ω+c2+c−dν22+2ω+3c−d2+cω+c2,DetJE∗=1+a22+c−d1+ω+c−dν1ν2−a1+ω+c2+c−dν2−2a2+c1+ω+c−dν12+2ω+3c−d2+cω+c2.
The eigenvalues ofJE∗ have a long analytical form, and instead we discuss the stability of the Nash equilibrium point by using Jury conditions:(19)1−trJE∗+DetJE∗>0,1+trJE∗+DetJE∗>0,1−DetJE∗>0,which can be rewritten in the form(20)a22+c−d1+ω+c−dν1ν22+2ω+3c−d2+cω+c2>0,4−Φ>0,Φ>0,Φ=a1+ω+c2+c−dν2+2a2+c1+ω+c−dν1−a22+c−d1+ω+c−dν1ν22+2ω+3c−d2+cω+c2.
It is easy to see that the first condition of (20) is always fulfilled. If the other two conditions are fulfilled, then E∗ is locally asymptotically stable provided that 0<Φ<4. On the other hand, if Φ≥4, this means E∗ gets unstable due to the coexistence of period-doubling bifurcation. In addition, it becomes unstable due to Neimark–Sacker bifurcation provided that Φ≤0. The next section gives some insights about the above analytical analysis.


### 3.1. Discussion and Numerical Simulation

Let us now discuss the monopoly case of map (8). It is easy to see that this map is trapped to the point 0,0 which means at q1t=0 or q2t=0, it gives q1t+1=0 or q2t+1=0. Setting q1t=0 or q2t=0 in (8), one gets the following:(21)q1t+1=q1t+ν1q1ta−2+cq1t,q2t+1=q2t+ν2q2ta−1+ω+cq2t,which can be simplified to(22)q1t+1=1+aν1q1t1−ν12+c1+aν1q1t,q2t+1=1+aν2q2t1−ν21+ω+c1+aν2q2t.Separately, each part of (22) conjugates the standard logistic map, yjt+1=μjyjt1−yjt,j=1,2. Then, we have the following linear transformations for (22):(23)q1t=1+aν1ν12+cy1t,q2t=1+aν2ν21+ω+cy2t,μj=1+aνj;j=1,2.This implies that the dynamics of (22) are the same as the logistic map. Each part in (22) is separately a unimodel map. Both have unique critical points Cq1−1 and Cq2−1 with coordinates given by(24)q⌣1=1+aν12ν12+c,q⌣2=1+aν22ν21+ω+c.These coordinates conjugate the critical pointsy1=1/2 and y2=1/2. In addition, system (22) has the following fixed points:(25)q1O=0,q1O1=a2+c,q2O=0,q2O2=a1+ω+c,which conjugate y1=0, y1=1−1/μ1, y2=0, and y2=1−1/μ2 of the logistic map. It is easy to see that dqit+1/dqitqiO=0=1+aνi>1,i=1,2 and then qiO,i=1,2 is an unstable repelling point. In addition, we can see that both q1O1 and q2O2 are stable attracting points under the condition 0<aνi<2,i=1,2. At aνi>2,i=1,2, the dynamics of each part in (22) may be a period cycle or a cyclic chaotic attractor. These behaviors are characterized by basins of attraction that are bounded by the repelling points q1O or q2O and their preimages. These preimages are obtained by setting qit+1=0,i=1,2 in (22) as follows:(26)q1O−1=1+aν1ν12+c,q2O−1=1+aν2ν21+ω+c,which conjugate y1=1 and y2=1 in the logistic map. Therefore, for the first part of (22), any trajectories starting out of the interval 0,q1O−1 will be divergent to −∞. The same observation is for the second part of (22). In order to validate the obtained results in the monopoly case, we use numerical simulation by assuming the following parameters’ values: a=0.5,c=0.2. Figure 1(a) shows that q1O1 is stable for all the values of ν1 until the parameter reaches the value ν1=2/a on where the period-2 cycle arises. At ν1=3.88, Figure 1(b) presents the basins of attraction of the stable point q1O1. It is also obvious that the basins are bounded by the box defined by 0,q1O−1×0,q1O−1. At k=5.9, we give a situation of unstable q1O1 due to a chaotic attractor behavior. As shown in Figure 1(c), the basins of this chaotic attractor lie within the box 0,q1O−1×0,q1O−1. For the second part of (22), we have the same discussions. Figure 1(d) shows that q2O2 is stable for the values of ν2 until the parameter reaches 2/a. Reaching this value gives rise to periodic cycle and chaotic attractor. For instance, at the parameters set a=0.5,ω=0.9,c=0.2, and ν2=3.88, the basins of attraction of the stable point q2O2 are given in Figure 1(e). As ν2 increases to 5.9, the point q2O2 gets unstable and chaotic attractor appears. The basins for this chaotic attractor are given in Figure 1(f). We can conclude that as ν2 increases further, any dynamic behavior will be bounded by the box 0,q2O−1×0,q2O−1.Figure 1
(a) Bifurcation diagram inν1,q1−plane. (b) The basins of attraction of the stable point q1O1 at k=3.88. (c) The basins of attraction of the stable point q1O1 at k=5.9. (d) Bifurcation diagram in ν2,q2−plane. (e) The basins of attraction of the stable point q2O2 at k=3.88. (f) The basins of attraction of the stable point q2O2 at k=5.9. The other values of the parameters are a=0.5,c=0.2,andω=0.9.
(a)(b)(c)(d)(e)(f)Now, we carry out some numerical simulations in order to investigate and analyze the influences of the parametersν1 and ν2 on the map given in (8). All numerical simulations in this section are performed at the initial datum q10,q20=0.11,0.12. Assuming the parameters set, a=0.5,c=0.2,d=0.35, and ω=0.45. This gives E∗=0.1853171775,0.2637205988. Assuming ν1=3.5 and ν2=4.2, then (18) becomes(27)JE∗≈−0.42694−0.22701−0.38767−0.82758,which has two real eigenvalues, λ1≈−0.26930 and λ2≈−0.98522. One can see that λ1,2<1 and hence E∗ is a local stable point. Keeping the parameter set including ν2 fixed and increasing ν1 to 3.65, the Jacobian JE∗ gets(28)JE∗≈−0.48810−0.23674−0.38767−0.82758,and then the real eigenvalues become λ1≈−0.31058 and λ2≈−1.0051. This means that E∗ is changed into an unstable saddle point. Now, we perform some numerical simulation experiments in order to get more insights on the dynamic of map (8) around the equilibrium point E∗. We start our analysis by investigating the effects of the adjustment parameters ν1 and ν2 on the map. In Figures 2(a) and 2(b), we present the bifurcation diagram for the influences of the parameters ν1 and ν2 on the quantities q1 and q2 at the parameters values, a=0.5,c=0.2,d=0.35, and ω=0.45. They show that the equilibrium point may be destabilized due to period-doubling bifurcation. Figure 2(c) confirms the chaotic behavior of the map by presenting the largest Lyapunov exponent. In Figures 2(d)–2(h), we give some different dynamic situations of the map due to varying the parameter ν1 and keeping the parameter ν2=4.2. They present the attractive basins of periodic cycles 2, 4, and 8. Besides that, we show in Figures 2(g) and 2(h) two different chaotic attractors for the map around the equilibrium point. We have two disconnected attractors around the equilibrium point given in Figure 2(g) which gather together to form a one chaotic attractor as given in Figure 2(h).Figure 2
(a),(b) Bifurcation diagram with respect toq1 and q2 on varying ν1 and ν2. (c) Largest Lyapunov exponents with respect to ν1 and ν2. Basins of attraction for (d) period-2 cycle at ν1=4 and ν2=4.2. (e) Period-4 cycle at ν1=5.16 and ν2=4.2. (f) Period-8 cycle at ν1=5.398 and ν2=4.2. (g) Two unconnected chaotic attractors at ν1=5.592 and ν2=4.2. (h) One chaotic attractor at ν1=6 and ν2=4.2. Other parameters are a=0.5,c=0.2,d=0.35, and ω=0.45.
(a)(b)(c)(d)(e)(f)(g)(h)Now, we study another situation when the parameterν2 is varied and ν1 becomes constant at the value 5. Figure 3(a) shows the 1D bifurcation diagram taking ν1 as the bifurcation parameter and the other parameters are selected to be a=0.5,c=0.2,d=0.35,ω=0.45, and ν2=5 while Figure 3(b)) depicts the bifurcation diagram with respect to the parameter ν2 and the other parameters’ values are a=0.5,c=0.2,d=0.35,ω=0.45, and ν1=3. It is clear that the equilibrium point becomes locally asymptotically stable till it reaches the point on where it can be destabilized due to period-doubling bifurcation. Due to a series of period-doubling bifurcated points, the map becomes chaotic and enters the chaos region and this is confirmed in Figure 3(c) which shows the Lyapunov exponent with respect to the variables ν1 and ν2. Now, we use some numerical experiments to investigate more the dynamics of the map. Setting the parameters’ values to a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5, we get in Figure 3(d)) four closed invariant sets around the equilibrium point. Further increase in ν2 to 5.2 makes these four sets convert into four disconnected chaotic attractors as shown in Figure 3(e) which turn into two chaotic attractors as ν2 increases to 5.3 as given in Figure 3(f). At ν2=5.4, a period-6 cycle is emerged and plotted in Figure 3(g) with its attractive basins. Increasing ν2 to 5.55, the dynamics of the map become chaotic as depicted in Figure 3(h).Figure 3
(a),(b) Bifurcation diagram with respect toq1 and q2 on varying ν1 and ν2. (c) Largest Lyapunov exponents with respect to ν1 and ν2. Phase plane for (d) four closed invariant sets at ν1=5 and ν2=5. (e) Four chaotic areas at ν1=5 and ν2=5.2. (f) Two chaotic areas at ν1=5 and ν2=5.3. (g) Basins of attraction of period-6 cycle at ν1=5 and ν2=5.4. (h) Phase plane for one chaotic attractor at ν1=5 and ν2=5.55. Other parameters are a=0.5,c=0.2,d=0.35, and ω=0.45.
(a)(b)(c)(d)(e)(f)(g)(h)


### 3.2. Noninvertible Map

We have previously discussed that the mapT is being trapped in the point 0,0. This means that at q1t=0 or q2t=0, we get q1t+1=0 or q2t+1=0, respectively. So, the point 0,0 is used to calculate basins’ boundaries for any attracting set of the map. Doing that requires setting q1t+1=q´1 and q2t+1=q´2 in (8) as follows:(29)T:q´1=q1t+ν1q1a−2+cq1−dq2,q´2=q2t+ν2q2a−1+ω+cq2−dq1,where ′ indicates evolution of time. For map (29), if T−1:q´1,q´2⟶q1,q2 gets unique value for each point in the range, then we call T an invertible map and then the point q´1,q´2∈ℝ2 is a rank-1 image while q1,q2 is a rank-1 preimage. If there exist at least two rank-1 preimages for an image q´1,q´2, then the map T is called a noninvertible map. Now, we calculate the real rank-1 preimages for the point 0,0.Proposition 4.
The pointO=0,0 possesses four real rank-1 preimages,Proof.
Settingq´1=0 and q´2=0 in (29) and solving algebraically, we get(30)O−10=0,0,O−11=1+aν1ν12+c,0,O−12=0,1+aν2ν21+c+ω,O−13=aν1ν21+c+ω−d+1+c+ων2−dν1ν1ν22+2ω+3c+cω+c2−d2,aν1ν22+c−d+2+cν1−dν2ν1ν22+2ω+3c+cω+c2−d2.
This completes the proof.The above proposition indicates that any attracting set for map (27) has an attractive basin that is bounded by a quadrilateral shape whose boundaries are defined by the line segments ξ1=OO−11,ξ2=OO−12 and their preimages ξ1−1 and ξ2−1, respectively. These preimages are given by(31)ξ1−1:q2=1+aν2−dν2q1ν21+c+ω,ξ2−1:q2=1+aν1−ν12+cq1dν1.Figure4 displays those line segments and their preimages at the parameters’ values: a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5.55.Figure 4
The boundaries of the chaotic attractor given in Figure3(h).In Figure5, we plot the critical curves LC and LC−1 at the parameters set, a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5.55. It is clear that LC divides the phase plane into the two regions Z4 and Z2 as shown in Figures 5(a) and 5(b). Therefore, the map is noninvertible. In addition, the points 0,0 and the equilibrium point O are belonging to the region Z4.Figure 5
(a), (b) The critical curvesLC, LC−1 and the region, Zi, i=2,4 at the parameters’ values: a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5.55.
(a)(b)


## 3.1. Discussion and Numerical Simulation

Let us now discuss the monopoly case of map (8). It is easy to see that this map is trapped to the point 0,0 which means at q1t=0 or q2t=0, it gives q1t+1=0 or q2t+1=0. Setting q1t=0 or q2t=0 in (8), one gets the following:(21)q1t+1=q1t+ν1q1ta−2+cq1t,q2t+1=q2t+ν2q2ta−1+ω+cq2t,which can be simplified to(22)q1t+1=1+aν1q1t1−ν12+c1+aν1q1t,q2t+1=1+aν2q2t1−ν21+ω+c1+aν2q2t.Separately, each part of (22) conjugates the standard logistic map, yjt+1=μjyjt1−yjt,j=1,2. Then, we have the following linear transformations for (22):(23)q1t=1+aν1ν12+cy1t,q2t=1+aν2ν21+ω+cy2t,μj=1+aνj;j=1,2.This implies that the dynamics of (22) are the same as the logistic map. Each part in (22) is separately a unimodel map. Both have unique critical points Cq1−1 and Cq2−1 with coordinates given by(24)q⌣1=1+aν12ν12+c,q⌣2=1+aν22ν21+ω+c.These coordinates conjugate the critical pointsy1=1/2 and y2=1/2. In addition, system (22) has the following fixed points:(25)q1O=0,q1O1=a2+c,q2O=0,q2O2=a1+ω+c,which conjugate y1=0, y1=1−1/μ1, y2=0, and y2=1−1/μ2 of the logistic map. It is easy to see that dqit+1/dqitqiO=0=1+aνi>1,i=1,2 and then qiO,i=1,2 is an unstable repelling point. In addition, we can see that both q1O1 and q2O2 are stable attracting points under the condition 0<aνi<2,i=1,2. At aνi>2,i=1,2, the dynamics of each part in (22) may be a period cycle or a cyclic chaotic attractor. These behaviors are characterized by basins of attraction that are bounded by the repelling points q1O or q2O and their preimages. These preimages are obtained by setting qit+1=0,i=1,2 in (22) as follows:(26)q1O−1=1+aν1ν12+c,q2O−1=1+aν2ν21+ω+c,which conjugate y1=1 and y2=1 in the logistic map. Therefore, for the first part of (22), any trajectories starting out of the interval 0,q1O−1 will be divergent to −∞. The same observation is for the second part of (22). In order to validate the obtained results in the monopoly case, we use numerical simulation by assuming the following parameters’ values: a=0.5,c=0.2. Figure 1(a) shows that q1O1 is stable for all the values of ν1 until the parameter reaches the value ν1=2/a on where the period-2 cycle arises. At ν1=3.88, Figure 1(b) presents the basins of attraction of the stable point q1O1. It is also obvious that the basins are bounded by the box defined by 0,q1O−1×0,q1O−1. At k=5.9, we give a situation of unstable q1O1 due to a chaotic attractor behavior. As shown in Figure 1(c), the basins of this chaotic attractor lie within the box 0,q1O−1×0,q1O−1. For the second part of (22), we have the same discussions. Figure 1(d) shows that q2O2 is stable for the values of ν2 until the parameter reaches 2/a. Reaching this value gives rise to periodic cycle and chaotic attractor. For instance, at the parameters set a=0.5,ω=0.9,c=0.2, and ν2=3.88, the basins of attraction of the stable point q2O2 are given in Figure 1(e). As ν2 increases to 5.9, the point q2O2 gets unstable and chaotic attractor appears. The basins for this chaotic attractor are given in Figure 1(f). We can conclude that as ν2 increases further, any dynamic behavior will be bounded by the box 0,q2O−1×0,q2O−1.Figure 1
(a) Bifurcation diagram inν1,q1−plane. (b) The basins of attraction of the stable point q1O1 at k=3.88. (c) The basins of attraction of the stable point q1O1 at k=5.9. (d) Bifurcation diagram in ν2,q2−plane. (e) The basins of attraction of the stable point q2O2 at k=3.88. (f) The basins of attraction of the stable point q2O2 at k=5.9. The other values of the parameters are a=0.5,c=0.2,andω=0.9.
(a)(b)(c)(d)(e)(f)Now, we carry out some numerical simulations in order to investigate and analyze the influences of the parametersν1 and ν2 on the map given in (8). All numerical simulations in this section are performed at the initial datum q10,q20=0.11,0.12. Assuming the parameters set, a=0.5,c=0.2,d=0.35, and ω=0.45. This gives E∗=0.1853171775,0.2637205988. Assuming ν1=3.5 and ν2=4.2, then (18) becomes(27)JE∗≈−0.42694−0.22701−0.38767−0.82758,which has two real eigenvalues, λ1≈−0.26930 and λ2≈−0.98522. One can see that λ1,2<1 and hence E∗ is a local stable point. Keeping the parameter set including ν2 fixed and increasing ν1 to 3.65, the Jacobian JE∗ gets(28)JE∗≈−0.48810−0.23674−0.38767−0.82758,and then the real eigenvalues become λ1≈−0.31058 and λ2≈−1.0051. This means that E∗ is changed into an unstable saddle point. Now, we perform some numerical simulation experiments in order to get more insights on the dynamic of map (8) around the equilibrium point E∗. We start our analysis by investigating the effects of the adjustment parameters ν1 and ν2 on the map. In Figures 2(a) and 2(b), we present the bifurcation diagram for the influences of the parameters ν1 and ν2 on the quantities q1 and q2 at the parameters values, a=0.5,c=0.2,d=0.35, and ω=0.45. They show that the equilibrium point may be destabilized due to period-doubling bifurcation. Figure 2(c) confirms the chaotic behavior of the map by presenting the largest Lyapunov exponent. In Figures 2(d)–2(h), we give some different dynamic situations of the map due to varying the parameter ν1 and keeping the parameter ν2=4.2. They present the attractive basins of periodic cycles 2, 4, and 8. Besides that, we show in Figures 2(g) and 2(h) two different chaotic attractors for the map around the equilibrium point. We have two disconnected attractors around the equilibrium point given in Figure 2(g) which gather together to form a one chaotic attractor as given in Figure 2(h).Figure 2
(a),(b) Bifurcation diagram with respect toq1 and q2 on varying ν1 and ν2. (c) Largest Lyapunov exponents with respect to ν1 and ν2. Basins of attraction for (d) period-2 cycle at ν1=4 and ν2=4.2. (e) Period-4 cycle at ν1=5.16 and ν2=4.2. (f) Period-8 cycle at ν1=5.398 and ν2=4.2. (g) Two unconnected chaotic attractors at ν1=5.592 and ν2=4.2. (h) One chaotic attractor at ν1=6 and ν2=4.2. Other parameters are a=0.5,c=0.2,d=0.35, and ω=0.45.
(a)(b)(c)(d)(e)(f)(g)(h)Now, we study another situation when the parameterν2 is varied and ν1 becomes constant at the value 5. Figure 3(a) shows the 1D bifurcation diagram taking ν1 as the bifurcation parameter and the other parameters are selected to be a=0.5,c=0.2,d=0.35,ω=0.45, and ν2=5 while Figure 3(b)) depicts the bifurcation diagram with respect to the parameter ν2 and the other parameters’ values are a=0.5,c=0.2,d=0.35,ω=0.45, and ν1=3. It is clear that the equilibrium point becomes locally asymptotically stable till it reaches the point on where it can be destabilized due to period-doubling bifurcation. Due to a series of period-doubling bifurcated points, the map becomes chaotic and enters the chaos region and this is confirmed in Figure 3(c) which shows the Lyapunov exponent with respect to the variables ν1 and ν2. Now, we use some numerical experiments to investigate more the dynamics of the map. Setting the parameters’ values to a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5, we get in Figure 3(d)) four closed invariant sets around the equilibrium point. Further increase in ν2 to 5.2 makes these four sets convert into four disconnected chaotic attractors as shown in Figure 3(e) which turn into two chaotic attractors as ν2 increases to 5.3 as given in Figure 3(f). At ν2=5.4, a period-6 cycle is emerged and plotted in Figure 3(g) with its attractive basins. Increasing ν2 to 5.55, the dynamics of the map become chaotic as depicted in Figure 3(h).Figure 3
(a),(b) Bifurcation diagram with respect toq1 and q2 on varying ν1 and ν2. (c) Largest Lyapunov exponents with respect to ν1 and ν2. Phase plane for (d) four closed invariant sets at ν1=5 and ν2=5. (e) Four chaotic areas at ν1=5 and ν2=5.2. (f) Two chaotic areas at ν1=5 and ν2=5.3. (g) Basins of attraction of period-6 cycle at ν1=5 and ν2=5.4. (h) Phase plane for one chaotic attractor at ν1=5 and ν2=5.55. Other parameters are a=0.5,c=0.2,d=0.35, and ω=0.45.
(a)(b)(c)(d)(e)(f)(g)(h)


## 3.2. Noninvertible Map

We have previously discussed that the mapT is being trapped in the point 0,0. This means that at q1t=0 or q2t=0, we get q1t+1=0 or q2t+1=0, respectively. So, the point 0,0 is used to calculate basins’ boundaries for any attracting set of the map. Doing that requires setting q1t+1=q´1 and q2t+1=q´2 in (8) as follows:(29)T:q´1=q1t+ν1q1a−2+cq1−dq2,q´2=q2t+ν2q2a−1+ω+cq2−dq1,where ′ indicates evolution of time. For map (29), if T−1:q´1,q´2⟶q1,q2 gets unique value for each point in the range, then we call T an invertible map and then the point q´1,q´2∈ℝ2 is a rank-1 image while q1,q2 is a rank-1 preimage. If there exist at least two rank-1 preimages for an image q´1,q´2, then the map T is called a noninvertible map. Now, we calculate the real rank-1 preimages for the point 0,0.Proposition 4.
The pointO=0,0 possesses four real rank-1 preimages,Proof.
Settingq´1=0 and q´2=0 in (29) and solving algebraically, we get(30)O−10=0,0,O−11=1+aν1ν12+c,0,O−12=0,1+aν2ν21+c+ω,O−13=aν1ν21+c+ω−d+1+c+ων2−dν1ν1ν22+2ω+3c+cω+c2−d2,aν1ν22+c−d+2+cν1−dν2ν1ν22+2ω+3c+cω+c2−d2.
This completes the proof.The above proposition indicates that any attracting set for map (27) has an attractive basin that is bounded by a quadrilateral shape whose boundaries are defined by the line segments ξ1=OO−11,ξ2=OO−12 and their preimages ξ1−1 and ξ2−1, respectively. These preimages are given by(31)ξ1−1:q2=1+aν2−dν2q1ν21+c+ω,ξ2−1:q2=1+aν1−ν12+cq1dν1.Figure4 displays those line segments and their preimages at the parameters’ values: a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5.55.Figure 4
The boundaries of the chaotic attractor given in Figure3(h).In Figure5, we plot the critical curves LC and LC−1 at the parameters set, a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5.55. It is clear that LC divides the phase plane into the two regions Z4 and Z2 as shown in Figures 5(a) and 5(b). Therefore, the map is noninvertible. In addition, the points 0,0 and the equilibrium point O are belonging to the region Z4.Figure 5
(a), (b) The critical curvesLC, LC−1 and the region, Zi, i=2,4 at the parameters’ values: a=0.5,c=0.2,d=0.35,ω=0.45,ν1=5, and ν2=5.55.
(a)(b)


## 4. Conclusion

The current paper has studied a two-dimensional map that has described the competition between two Cournot firms one of which has considered the maximization of social welfare instead of profit. Based on a gradient-based mechanism, the map has been modelled in discrete time steps. It has been analyzed that when the two firms have become monopolistic, their dynamics have turned into just like the standard coupled logistic map. While in the duopolistic case, the map’s fixed points have been calculated and their stability conditions have been investigated showing that the Nash equilibrium point may be unstable due to two types of bifurcations. Through some numerical analysis, we have shown some attracting sets with their basins of attraction and other chaotic behaviors of the map around the equilibrium point have been detected. The critical curves of the game’s map have been calculated, and the corresponding preimages regions have been identified. We have shown that the map is noninvertible and has belonged toZ4−Z2 type. For future studies, we plan to investigate more such adoption of maximization of social welfare on heterogeneous players (or firms). We have shown that the model parameters, in particular the speed of adjustment and the degree of privatization of the second firm, have an effect on the long-term dynamic response of the game, which is important for understanding the functioning of the mixed duopoly game. This result allows players to gain a specific understanding of the mixed oligopoly market and to recognize that the choice of decision criteria would have a certain effect on the system’s actions.



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*Source: 6697341-2021-01-27.xml*