# On Comparing between Two Nonlinear Cournot Duopoly Models

Authors: S. S. Askar
Journal: Complexity
 (2021)

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## Abstract

The comparison between two nonlinear duopoly models constructed based on symmetric utility function that is derived from Cobb–Douglas is investigated in this paper. The first model consists of two firms which update their outputs using gradient-based mechanism called bounded rationality. The second model contains a bounded rational firm that is competing with a firm whose outputs depend on a trade-off between market share maximization and profit maximization. For the two models, the fixed points are calculated and their conditions of stability are analyzed. The obtained results show that the second model is more stabilizing provided that the second firm adopts low weights of trade-offs. We show that the two models can be destabilized via flip bifurcation only. Furthermore, the noninvertibility of the two models that can give rise to several stable attractors is discussed.


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## Body

## 1. Introduction

Literature has reported several works that have studied the complex dynamic characteristics of Cournot duopoly games. Such games have given rise to interesting results about the models describing them. These results have included investigations of the types of bifurcation responsible for instability of games’ equilibrium points and chaos routes. Some of those games have been formed based on popular utility function such as Cobb–Douglas [1], constant elasticity of substitution (CES) [2], and Singh and Vives utility function [3]. There are other utility production functions that have been adopted to model Cournot games like, for example, the isoelastic utility derived from Cobb–Douglas (see [4]). The current paper adopts a symmetric utility function that is derived from the well-known Cobb–Douglas one. Although the adopted symmetric utility in the current paper is simple, the two models formed based on it are complicated and provide interesting results about their dynamics as will be shown later.In order to make the current paper self-contained, we report in the introduction some important works that have been cited in this direction of research. For instance, we cite from literature the reported works ([5–10]) which have discussed and investigated the routes of chaos in Cournot games whose models have been constructed based on naive expectations. Such works have analyzed the chaotic behaviors on those models via detecting the types of bifurcations appeared on them. In [11], a gradient-based mechanism called bounded rationality has been adopted to globally study the competition of Cournot game. In this study, a rich analysis has shown that the game’s equilibrium points can be destabilized due to coexistence of periodic cycles and chaotic behaviors. The adoption of bounded rationality approach has opened the gate to several studies on its influences on duopoly games and it has been applied to other games whose players may be homogeneous or heterogeneous. In [12–15], the authors have applied the bounded rationality and other approaches such as Puu’s approach in order to analyze Cournot games and the cooperation that might be occurred between their players. These studies have confirmed what has been reported in literature on the destabilization routes of chaos that affected the stability of the equilibrium points in those games. Other useful works have been reported in [16, 17], in which Agiza et al. have investigated oligopoly games with both linear and nonlinear inverse demand function that have been used to study those games under limited rationality. In [18], chaos has been controlled in a duopoly game of master and slave players after the game’s dynamics have been analyzed and routes to chaos have been identified. The complex dynamic characteristics of both triopoly Cournot and Bertrand games whose players produce differentiated commodities have been investigated in [19]. Here, we cannot ignore to cite two important works of Bischi et al. ([20, 21]) which introduced an intensive analysis of bifurcation and chaos occurred in the dynamics of oligopoly games. In [22], a Cournot duopoly game whose players are homogeneous and their inverse demand functions are isoelastic has been introduced and its dynamic characteristics have been discussed. For more works in this direction of research, interesting readers are advised to see the literature [23–30].The current paper belongs to both homogeneous and heterogeneous duopoly games in which games’ players adopt similar and different decisional mechanisms. The main results of this paper concern with the comparison between two nonlinear duopoly games which are formed based on a symmetric utility function derived from Cobb–Douglas well-known production utility. The comparison includes investigation of routes where the equilibrium points can be destabilized, analyzing the global behaviors of the two models describing the games that include investigation of the structure of attractive basins. Our obtained results show that the equilibrium points of both models cannot preserve their stabilities due to flip bifurcation only. Moreover, they show that the model with heterogeneous players seems to be more stable if the second firm uses low trade-off weights between market share and profit maximization. The structure of basins for both models may be quite complicated as we deal with noninvertible maps.After this introduction, we can summarize the parts of the paper as follows. Section2 introduces the two maps describing the duopoly games in this paper. In Section 3, we analyze the properties of the first map and this includes discussing the stability conditions of its equilibrium points and proves it is a noninvertible map. As in Section 3, Section 4 discusses the second map and analyzes its complex characteristics. Some comparisons between the two maps are given in Section 5. In Section 6 ,we end the paper with some conclusions and future works.


## 2. The Game of Competition

We consider a game of two firms (or players) whose decision variables in the market are quantities. We takeq1 and q2 as the quantities produced by each firm. We recall the Cobb–Douglas function of two variables appearing in many economic models as follows:(1)U=q1aq2b,0<a,b<1.This function is popular as Cobb–Douglas (this function is a production function used to organize production possibilities for firms from different and various branches [31]. In the application to production, as a firm’s output is something measurable, the sum a+b> or <1 makes sense as watershed between increasing and decreasing returns to scale. In our assumption in this paper, maximizing utility U=q1q2 returns p1/p2=q2/q1 or p1q1=p2q2 that means the total revenue for both firms is equal and this is due to the assumed symmetry as we take a=b.) function which should properly be called Wicksell function since the Swedish economist Knut Wicksell was the first to introduce this production function [32]. It represents the agent’s preferences. Both a and b are constants. It has the following properties:(i)
It is a monotonic function under the transformationu=lnU. This means that ∂u/∂q1=a/q1 and ∂u/∂q2=b/q2 and hence the marginal utility of each quantity is positive.(ii)
It is a concave utility function. This means that∂2u/∂q12=−a/q12<0 and ∂2u/∂q22=−b/q22<0 and hence the marginal utility of each quantity is decreasing.(iii)
It is strongly additive. This means that∂2u/∂q1∂q2=0 and hence the marginal of the quantity q1 is independent of the quantity q2. So, the quantities cannot represent substitutes or complements.(iv)
It is a homothetic function. This means that ifx=q1,q2,y=q⌣1,q⌣2, ux=uy, and θ>0, then uθx=uθy.We assume the symmetric case,a=b=1/2, with the budget constraint p1q1+p2q2=1 where pi is the price of qi,i=1,2. Now, we have the following maximization problem:(2)maxU=q1q2,s.t.p1q1+p2q2=1.Solving (2) gives the following prices:(3)p1=12q2q1,p2=12q1q2.At the economic market, each competitor wants to maximize its profit given by(4)π1=12q1q2−c1q12,π2=12q1q2−c2q22,where Cqi=ciqi2 is the cost function of each quantity. It is a quadratic function and is often met in applications. For example, in the modeling of renewable resources exploitation, such as fisheries [33], in [34], it has been considered in order to investigate the role of convexity. So, the marginal cost is not constant, ∂Cqi/∂qi=2ciqi, and ci,i=1,2 is a constant parameter. The marginal profits take the following form:(5)∂π1∂q1=14q2q1−2c1q1,∂π2∂q2=14q1q2−2c2q2.Here, we discuss two situations. The first situation assumes that both firms use a gradient-based mechanism such as the bounded rationality defined in the literature [4, 35–37]. Using this mechanism, we get the following discrete dynamical system:(6)Tq1,q2:q1t+1=q1t+k1q1t14q2tq1t−2c1q1t,q2t+1=q2t+k2q2t14q1tq2t−2c2q2t,where ki,i=1,2 is a speed of adjustment parameter. The second situation is a heterogeneous situation. We assume that the second firm shares the market with certain profit. It trades off between market share maximization and profit maximization. The market share means it seeks the optimum output by putting π2=0 and then we get the optimum, q⌣2=q1/4c223, while the profit maximization gives q^2=1/4q1/c223. Since it is traded off between those, we get the following:(7)q˜2=ωq⌣2+1−ωq^2,where ω∈0,1. When the second firm shares market maximization only, we take ω=1, while ω=0 means it seeks profit maximization only. Now, the second firm updates its output according to the following:(8)q2t+1=1−sq2t+sω43+1−ω4q1tc223,where s∈0,1 is a constant weight. So, we have another model with heterogeneous players given by(9)T1q1,q2:q1t+1=q1t+kq1t14q2tq1t−2c1q1t,q2t+1=1−sq2t+sω43+1−ω4q1tc223.It is obvious that ifs=0, then the second firm in (9) is naive, while at s=1 means it updates its production based on market share maximization. Economically, profit maximization only is mainly a short-term goal and is primarily restricted to the accounting analysis of the financial situation. Furthermore, it is primarily concerned as to how the company will survive and grow in the existing competitive business environment [38]. On the other hand, market share can allow firms to improve profitability either by lowering prices, using advertising, or introducing new or different products. It can also grow the size of its market share by appealing to other audiences or demographics. In our paper, we introduce a weighted average decision-making mechanism between those two mechanisms discussed above. This includes the main contribution in this paper that is to compare between the maps (6) and (9) in order to see which model is more stable and gives large region of stability [39].


## 3. Properties of Map (6)

### 3.1. Analytical Results

The map (6) can make negative or unbounded trajectories if the initial condition q0,1,q0,2 is chosen far from O=0,0. It is obvious that if qi0>1/2ciki,i=1,2, negative values for qit+1 can be obtained.Proposition 1.
The map (6) has two fixed points that are O1=0,0 and O2=1/8c1c1/c24,1/8c1c1/c234.Proof.
Settingq1t+1=q1t and q2t+1=q2t in (6) and solving algebraically, we get the two fixed points.Proposition 2.
The fixed pointO1 is an unstable point.Proof: The Jacobian matrix of map (6) at O1 becomes(10)00000000,which contains indefinite form and then the point is unstable and the proof is completed.Proposition 3.
The fixed pointO2 is locally asymptotically stable provided that(11)k1>163c2c14−c2c1k2.Proof.
AtO2, the Jacobian matrix becomes(12)1−3k18c1c24k18c2c14k28c1c241−3k28c2c14,whose trace and determinant are given by(13)τ=2−3k18c1c24−3k28c2c14,δ=1−3k18c1c24−3k28c2c14+18k1k2.
Substituting (13) in Jury conditions [40] gives(14)Δ1≔1−τ+δ=k1k28,Δ2≔1+τ+δ=4−3k14c1c24−3k24c2c14+k1k28,Δ3≔1−δ=3k18c1c24+3k28c2c14−k1k28.
Sincek1 and k2 are positive speeds of adjustment parameters then Δ1>0 is always satisfied. Δ2>0 and Δ3>0 give(15)4−34c1c24k1−34c2c14k2+18k1k2>0,38c1c24k1+38c2c14k2−18k1k2>0.
Combining the above inequalities completes the proof.Proposition 4.
The fixed pointO2 can be destabilized due to flip bifurcation only.Proof.
The Jacobian (12) has the following eigenvalues:(16)λ±=1−3k116c1c24−3k216c2c14±9c1k1−c2k22+4c1c2k1k216c1c24.
It is clear that9c1k1−c2k22+4c1c2k1k2 is always positive, and therefore, we have only two real eigenvalues. So the point becomes unstable because of flip bifurcation.


### 3.2. Numerical Results

The following numerical simulation experiments validate the above theoretical results. We start our numerical experiments by assuming the following parameters’ setc1>c2, c1=0.7,c2=0.5. At this set, O2=0.1942423762,0.2298306788 and the Jacobian becomes(17)−0.223730.344750.54388−0.37898,where we assume k1=3 and k2=4. The eigenvalues then become λ±=0.138563,−0.741273 with λ±<1 and then O2 is locally asymptotically stable. Keeping the costing shift parameters c1 and c2 as previously, we now study the influence of k1 or k2 on the stability of O2. Figures 1(a) and 1(b) show that O2 is locally asymptotically stable for all values of k1 and k2; until we get the cycle of period 2, the point O2 becomes unstable. Indeed, the simulation shows some interesting behavior of the dynamic of map (6) around the point O2. To present that dynamics, we have to study the effects of k1 or k2 separately. Assuming the following parameters set, c1=0.7,c2=0.5, and k1=2. As k2 increases to 6.54, a period-2 cycle (represented by squares in Figure 1(c)) is born around the fixed point O2. It is plotted with its basins of attraction represented by the light green colors in Figure 1(c) while the grey color refers to divergent (or nonconvergent) points in the phase plane. This cycle turns into a cycle of period four as k2 increases to 7.14 with quite complicated basins of attraction as displayed in Figure 1(d). Other basins of attraction for the cycle of period four are formed in Figure 1(e) at k2=7.24 where nonconvergent points appeared in white color within the basins. The basins of attraction become more complicated as the period-8 starting to appear at k2=7.24 as depicted in Figure 1(f). Increasing k2 further gives rise to four disconnected pieces and two chaotic attractors and then one chaotic attractor. We give in Figures 1(g) and 1(h) examples of two and one chaotic attractors and their basins at k2=7.53 and k2=7.69, respectively. The white color in the basins denotes the nonconvergent points in the phase plane. The same discussion can be performed for the other parameter k1 under the same values of costs. This makes us to display the 2D-bifurcation diagram in the k1,k2−plane on which interesting readers can detect more about the dynamics of map (6) around O2. Figure 2(a) presents the 2D-bifurcation in the k1,k2−plane with examples of attractive basins of period-7 (Figure 2(b)), period-9 (Figure 2(c)), and period-10 (Figure 2(d)). Furthermore, the 2D-bifurcation diagram shows that the dynamics of map (6) do not permit the appearance of period-3 and period-5.Figure 1
(a) Bifurcation diagram and LLE on varyingk1. (b) Bifurcation diagram and LLE on varying k2. (c) The basins of attraction of period-2 cycle at k2=6.54. (d) The basins of attraction of period-4 cycle at k2=7.14 . (e) The basins of attraction of period-4 cycle at k2=7.24. (f) The basins of attraction of period-8 cycle at k2=7.43. (g) The basins of attraction of two chaotic attractors at k2=7.53. (h) The basins of attraction of a chaotic attractor at k2=7.69. Other parameters’ values are, c1=0.7 and c2=0.5.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 2
(a) The 2D-bifurcation in thek1,k2−plane. (b) The basins of attraction of period-7 cycle at k1=2.288 and k2=7.5362. (c) The basins of attraction of period-9 cycle at k1=2.713 and k2=7.3140. (d) The basins of attraction of period-10 cycle at k1=2.933 and k2=7.263 . Other parameters’ values are, c1=0.7 and c2=0.5.
(a)(b)(c)(d)Now, we study the case whenc1<c2 by assuming the set of parameters’ values, c1=0.5,c2=0.9. In this case, we do not want to repeat the above discussions and only we give some simulations represented by Figures 3(a) to 3(f) at this set in order to see the difference in dynamics in this case and the previous one. All these results so far show a peculiar shape of the basins of attraction and require from us to investigate other characteristics of map (6). From the above discussion and the numerical experiments, we highlight that the assumptions c1>c2 and c1<c2 affect the stability region obtained by the second condition of (14) for the interior equilibrium point in the k1,k2−plane. We have seen that at c1>c2c1=0.7,c2=0.5, we get k1∈0,4.9 and k2∈0,5.8 while in the case c1<c2c1=0.5,c2=0.9, we have k1∈0,6.166 and k2∈0,4.625 for the same condition.Figure 3
(a) The basins of attraction of the fixed pointO2 at k1=3 and k2=3. (b) The basins of attraction of period-2 cycle at k1=6.06 and k2=3. (c) The basins of attraction of period-6 cycle at k1=7.06 and k2=3. (d) The basins of attraction of six chaotic attractors at k1=7.3 and k2=3. (e) The basins of attraction of two chaotic attractors at k1=7.33 and k2=3. (f) The basins of attraction of a chaotic attractor at k1=7.68 and k2=3. Other parameters’ values are c1=0.5 and c2=0.9.
(a)(b)(c)(d)(e)(f)


### 3.3. Critical Curves and Phase Plane Zones

The structure of the basins of attraction of map (6) at any ϱ that may represent O2 or periodic cycle or any chaotic attractor is constructed by some boundaries. These boundaries are calculated as follows.Proposition 5.
The origin pointO=0,0 has four real preimages.Proof.
Settingq1t+1=0 and q2t+1=0 in (6) and solving the system algebraically, we get(18)O−10=0,0,O−11=12c1k1,0,O−12=0,12c2k2,O−13=q∘1q∘2,where O−13⊗ is obtained by solving algebraically the following system:(19)1+k114q∘2q∘1−2c1q∘1=0,1+k214q∘1q∘2−2c2q∘2=0.Proposition 6.
Letω1=O−10O−11 and ω2=O−10O−12 be the two line segments of the coordinates q1 and q2. Also, let ω1−1 and ω2−1 be their preimages. Then, the boundary of the basins of attraction of any ϱ is denoted by Ϝ and is given by(20)Ϝ=∪n=0∞T−nω1∪∪n=0∞T−nω2,where T−n refers to the set of all preimages of rank-n.These boundaries are displayed in Figure4. Furthermore, the figure shows the critical curve LC−1 that is calculated by vanishing the determinant of the Jacobian of (6). It is represented by the following curve:(21)324c2k2q2−1c1k1q13/2q21/2−4c1q12+c2q22k1k2++81−4c2k2q2q11/2q21/2+k1q2+k2q1=0.Figure 4
The critical curve and preimages of map (6) at the parameters’ values, c1=0.7,c2=0.5,k1=2, and k2=7.69.It is represented by red color in Figure4 as it contains two parts, LC−1a∪LC−1b. It is hard to plot or to get an analytical form for LC in order to identify the zones in the phase plane; however, it is simple to see that O2 has two real preimages of rank-1 and hence it belongs to Z2. Furthermore, both ω1−1 and ω2−1 have no preimages and they belong to Z0. With the preimages of the origin, we detect that the phase plane of map (6) is divided into three zones, Z4,Z2, and Z0. Therefore, map (6) is a noninvertible map.


## 3.1. Analytical Results

The map (6) can make negative or unbounded trajectories if the initial condition q0,1,q0,2 is chosen far from O=0,0. It is obvious that if qi0>1/2ciki,i=1,2, negative values for qit+1 can be obtained.Proposition 1.
The map (6) has two fixed points that are O1=0,0 and O2=1/8c1c1/c24,1/8c1c1/c234.Proof.
Settingq1t+1=q1t and q2t+1=q2t in (6) and solving algebraically, we get the two fixed points.Proposition 2.
The fixed pointO1 is an unstable point.Proof: The Jacobian matrix of map (6) at O1 becomes(10)00000000,which contains indefinite form and then the point is unstable and the proof is completed.Proposition 3.
The fixed pointO2 is locally asymptotically stable provided that(11)k1>163c2c14−c2c1k2.Proof.
AtO2, the Jacobian matrix becomes(12)1−3k18c1c24k18c2c14k28c1c241−3k28c2c14,whose trace and determinant are given by(13)τ=2−3k18c1c24−3k28c2c14,δ=1−3k18c1c24−3k28c2c14+18k1k2.
Substituting (13) in Jury conditions [40] gives(14)Δ1≔1−τ+δ=k1k28,Δ2≔1+τ+δ=4−3k14c1c24−3k24c2c14+k1k28,Δ3≔1−δ=3k18c1c24+3k28c2c14−k1k28.
Sincek1 and k2 are positive speeds of adjustment parameters then Δ1>0 is always satisfied. Δ2>0 and Δ3>0 give(15)4−34c1c24k1−34c2c14k2+18k1k2>0,38c1c24k1+38c2c14k2−18k1k2>0.
Combining the above inequalities completes the proof.Proposition 4.
The fixed pointO2 can be destabilized due to flip bifurcation only.Proof.
The Jacobian (12) has the following eigenvalues:(16)λ±=1−3k116c1c24−3k216c2c14±9c1k1−c2k22+4c1c2k1k216c1c24.
It is clear that9c1k1−c2k22+4c1c2k1k2 is always positive, and therefore, we have only two real eigenvalues. So the point becomes unstable because of flip bifurcation.


## 3.2. Numerical Results

The following numerical simulation experiments validate the above theoretical results. We start our numerical experiments by assuming the following parameters’ setc1>c2, c1=0.7,c2=0.5. At this set, O2=0.1942423762,0.2298306788 and the Jacobian becomes(17)−0.223730.344750.54388−0.37898,where we assume k1=3 and k2=4. The eigenvalues then become λ±=0.138563,−0.741273 with λ±<1 and then O2 is locally asymptotically stable. Keeping the costing shift parameters c1 and c2 as previously, we now study the influence of k1 or k2 on the stability of O2. Figures 1(a) and 1(b) show that O2 is locally asymptotically stable for all values of k1 and k2; until we get the cycle of period 2, the point O2 becomes unstable. Indeed, the simulation shows some interesting behavior of the dynamic of map (6) around the point O2. To present that dynamics, we have to study the effects of k1 or k2 separately. Assuming the following parameters set, c1=0.7,c2=0.5, and k1=2. As k2 increases to 6.54, a period-2 cycle (represented by squares in Figure 1(c)) is born around the fixed point O2. It is plotted with its basins of attraction represented by the light green colors in Figure 1(c) while the grey color refers to divergent (or nonconvergent) points in the phase plane. This cycle turns into a cycle of period four as k2 increases to 7.14 with quite complicated basins of attraction as displayed in Figure 1(d). Other basins of attraction for the cycle of period four are formed in Figure 1(e) at k2=7.24 where nonconvergent points appeared in white color within the basins. The basins of attraction become more complicated as the period-8 starting to appear at k2=7.24 as depicted in Figure 1(f). Increasing k2 further gives rise to four disconnected pieces and two chaotic attractors and then one chaotic attractor. We give in Figures 1(g) and 1(h) examples of two and one chaotic attractors and their basins at k2=7.53 and k2=7.69, respectively. The white color in the basins denotes the nonconvergent points in the phase plane. The same discussion can be performed for the other parameter k1 under the same values of costs. This makes us to display the 2D-bifurcation diagram in the k1,k2−plane on which interesting readers can detect more about the dynamics of map (6) around O2. Figure 2(a) presents the 2D-bifurcation in the k1,k2−plane with examples of attractive basins of period-7 (Figure 2(b)), period-9 (Figure 2(c)), and period-10 (Figure 2(d)). Furthermore, the 2D-bifurcation diagram shows that the dynamics of map (6) do not permit the appearance of period-3 and period-5.Figure 1
(a) Bifurcation diagram and LLE on varyingk1. (b) Bifurcation diagram and LLE on varying k2. (c) The basins of attraction of period-2 cycle at k2=6.54. (d) The basins of attraction of period-4 cycle at k2=7.14 . (e) The basins of attraction of period-4 cycle at k2=7.24. (f) The basins of attraction of period-8 cycle at k2=7.43. (g) The basins of attraction of two chaotic attractors at k2=7.53. (h) The basins of attraction of a chaotic attractor at k2=7.69. Other parameters’ values are, c1=0.7 and c2=0.5.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 2
(a) The 2D-bifurcation in thek1,k2−plane. (b) The basins of attraction of period-7 cycle at k1=2.288 and k2=7.5362. (c) The basins of attraction of period-9 cycle at k1=2.713 and k2=7.3140. (d) The basins of attraction of period-10 cycle at k1=2.933 and k2=7.263 . Other parameters’ values are, c1=0.7 and c2=0.5.
(a)(b)(c)(d)Now, we study the case whenc1<c2 by assuming the set of parameters’ values, c1=0.5,c2=0.9. In this case, we do not want to repeat the above discussions and only we give some simulations represented by Figures 3(a) to 3(f) at this set in order to see the difference in dynamics in this case and the previous one. All these results so far show a peculiar shape of the basins of attraction and require from us to investigate other characteristics of map (6). From the above discussion and the numerical experiments, we highlight that the assumptions c1>c2 and c1<c2 affect the stability region obtained by the second condition of (14) for the interior equilibrium point in the k1,k2−plane. We have seen that at c1>c2c1=0.7,c2=0.5, we get k1∈0,4.9 and k2∈0,5.8 while in the case c1<c2c1=0.5,c2=0.9, we have k1∈0,6.166 and k2∈0,4.625 for the same condition.Figure 3
(a) The basins of attraction of the fixed pointO2 at k1=3 and k2=3. (b) The basins of attraction of period-2 cycle at k1=6.06 and k2=3. (c) The basins of attraction of period-6 cycle at k1=7.06 and k2=3. (d) The basins of attraction of six chaotic attractors at k1=7.3 and k2=3. (e) The basins of attraction of two chaotic attractors at k1=7.33 and k2=3. (f) The basins of attraction of a chaotic attractor at k1=7.68 and k2=3. Other parameters’ values are c1=0.5 and c2=0.9.
(a)(b)(c)(d)(e)(f)


## 3.3. Critical Curves and Phase Plane Zones

The structure of the basins of attraction of map (6) at any ϱ that may represent O2 or periodic cycle or any chaotic attractor is constructed by some boundaries. These boundaries are calculated as follows.Proposition 5.
The origin pointO=0,0 has four real preimages.Proof.
Settingq1t+1=0 and q2t+1=0 in (6) and solving the system algebraically, we get(18)O−10=0,0,O−11=12c1k1,0,O−12=0,12c2k2,O−13=q∘1q∘2,where O−13⊗ is obtained by solving algebraically the following system:(19)1+k114q∘2q∘1−2c1q∘1=0,1+k214q∘1q∘2−2c2q∘2=0.Proposition 6.
Letω1=O−10O−11 and ω2=O−10O−12 be the two line segments of the coordinates q1 and q2. Also, let ω1−1 and ω2−1 be their preimages. Then, the boundary of the basins of attraction of any ϱ is denoted by Ϝ and is given by(20)Ϝ=∪n=0∞T−nω1∪∪n=0∞T−nω2,where T−n refers to the set of all preimages of rank-n.These boundaries are displayed in Figure4. Furthermore, the figure shows the critical curve LC−1 that is calculated by vanishing the determinant of the Jacobian of (6). It is represented by the following curve:(21)324c2k2q2−1c1k1q13/2q21/2−4c1q12+c2q22k1k2++81−4c2k2q2q11/2q21/2+k1q2+k2q1=0.Figure 4
The critical curve and preimages of map (6) at the parameters’ values, c1=0.7,c2=0.5,k1=2, and k2=7.69.It is represented by red color in Figure4 as it contains two parts, LC−1a∪LC−1b. It is hard to plot or to get an analytical form for LC in order to identify the zones in the phase plane; however, it is simple to see that O2 has two real preimages of rank-1 and hence it belongs to Z2. Furthermore, both ω1−1 and ω2−1 have no preimages and they belong to Z0. With the preimages of the origin, we detect that the phase plane of map (6) is divided into three zones, Z4,Z2, and Z0. Therefore, map (6) is a noninvertible map.


## 4. Properties of Map (9)

### 4.1. Analytical Results

Proposition 7.
The map (9) has only two fixed points that are o1=0,0 and o2=q¯1,q¯2 where(22)q¯1=143/8c13/4c21/4ω43+1−ω43/8,q¯2=141/8c11/4c23/4ω43+1−ω49/8.Proof.
Settingq1t+1=q1t and q2t+1=q2t in (9) and solving the system algebraically completes the proof.Proposition 8.
The fixed pointo1=0,0 is unstable.Proof.
The proof is the same as Proposition2.Proposition 9.
The fixed pointo2=q¯1,q¯2 is locally asymptotically stable provided that k<4−2s/45−22sℏ where(23)ℏ=155/824c1c2421−ω+2326ω223−11+223+443+15ω5/8.Proof.
The Jacobian ato2 becomes(24)J11J12J21J22,where(25)J11=1−45ℏ,J12=c12c24k1−ω+223ω,J21=s6c1c21−ω+223ω,J22=1−s,and the trace and determinant take the following form:(26)τ=1−s+kℏ244s−45,δ=2−s+452kℏ,and then,(27)Δ1=11sk12c1c2421−ω+2326ω1−ω+223ω5/8,Δ2=4−2s+kℏ22s−45,Δ3=s+kℏ245−44s.
It is clear thatΔ1>0 and Δ3>0 since s∈0,1 and ℏ is always positive. While Δ2>0 is attained, if k<4−2s/45−22sℏ. The proof is completed.
We should highlight here that ifk>4−2s/45−22sℏ, then o2 can be destabilized due to flip bifurcation only.


### 4.2. Numerical Results

We assume the followingc1>c2, c1=0.7,c2=0.5 with ω=0.2. At this set, o2=0.6068957946,0.4381254240 and the Jacobian becomes, at taking k=1.2 and s=0.2,(28)−0.911720.176540.0481270.8,and the eigenvalues are λ±=−0.916669,0.804949 with λ±<1 and then o2 is locally asymptotically stable. Increasing k above the value 4−2s/45−22sℏ gives rise to a period-doubling bifurcation, as shown in Figure 5(a). It is clear that the fixed point o2 is stable for all the values of k until k reaches the value 4−2s/45−22sℏ where the first period-2 cycle is born. As discussed in map (6), we give here some complex behaviors of map (9) at different values of the bifurcation parameter k. For instance, at k=8.026, a cycle of period-2 is emerged and is plotted with its basins of attraction in Figure 5(b). At k=8.44, a period-4 cycle is coexisted with its attractive basins in Figure 5(c). It is clear that they are quite complicated basins. The basins become more complicated when the period-8 cycle appears as given in Figure 5(d) at k=8.58. After that higher periodic cycles exist with quite complicated basins and then the map enters the chaos region at k=8.83 and k=8.9 where two chaotic attractors and then one chaotic attractor are emerged and displayed in Figures 5(e) and 5(f), respectively.Figure 5
(a) Bifurcation diagram and LLE on varyingk. (b) The basins of attraction of period-2 cycle at k=8.026. (c) The basins of attraction of period-4 cycle at k=8.44. (d) The basins of attraction of period-8 cycle at k=8.58. (e) The basins of attraction of two chaotic attractors at k=8.83. (f) The basins of attraction of a chaotic attractor at k=8.9. Other parameters’ values are c1=0.7,c2=0.5,ω=0.2, and s=0.2 .
(a)(b)(c)(d)(e)(f)The numerical simulation shows that increasingω which means the second firm gives more interesting to the market share maximization than the profit maximization reduces the stability interval of o2. The same observation when the firm gives more interesting to the traded-offs between the market share maximization and profit maximization is obtained. It is noticed that low values of the traded-off parameter s keep the stability interval of o2 larger. Furthermore, when we take values of c1 and c2 in the form c1<c2, the dynamics of map (9) do not give any new complex dynamics and hence the dynamic is still governed by the change that may be happened on both parameters ω and s.


### 4.3. Critical Curves and Phase Plane Zones

(29)2c1q14/3−q11/3−k4sc223s−1ω43+1−ω4=0,which has no real solutions as the expression under the square root is negative. Therefore, o belongs to Z1zone.Proposition 10.
The origin pointo=0,0 has only one real preimage that is o−10=0,0.Proof.
Settingq1t+1=0 and q2t+1=0 in (9), we get(30)q1+kq114q2q1−2c1q1=0,1−sq2+sω43+1−ω4q1c223=0.The first equation in (29) gives q1=0 or 1+k1/4q2/q1−2c1q1=0. Substituting q1=0 in the second equation of (29) gives q2=0, then we get o−10=0,0. Now, we have the following case:(31)1+k14q2q1−2c1q1=0,1−sq2+sω43+1−ω4q1c223=0.
Solving (30) givesProposition 11.
The fixed pointo2 has no real preimages.Proposition 12.
Any point in the form0,q,q≠0 has two real preimages.Proposition 13.
Any point in the formq,0,q≠0 has no real preimages.Now,LC−1 for map (9) is given by the following curve:(32)96sc1kq113/6q21/2−96c1kq113/6q21/2−24sq17/6q21/2+24q17/6q21/2−3skq2q12/3++3kq2q12/3−ksc223ω43+1−ω4q1=0.It is plotted in Figure6 with red color at the parameters’ values, c1=0.7,c2=0.5,k=2,ω=0.2, and s=0.2. Simulation shows that any point belonging to LC−1 has two real preimages. However, it is hard to plot LC but we can conclude that the phase plane of (9) may be divided into only two zones that are Z2 and Z0. Therefore, map (9) is a noninvertible map.Figure 6
The critical curve and zones of map (9) at the parameters’ values, c1=0.7,c2=0.5,k=2,ω=0.2, and s=0.2.


## 4.1. Analytical Results

Proposition 7.
The map (9) has only two fixed points that are o1=0,0 and o2=q¯1,q¯2 where(22)q¯1=143/8c13/4c21/4ω43+1−ω43/8,q¯2=141/8c11/4c23/4ω43+1−ω49/8.Proof.
Settingq1t+1=q1t and q2t+1=q2t in (9) and solving the system algebraically completes the proof.Proposition 8.
The fixed pointo1=0,0 is unstable.Proof.
The proof is the same as Proposition2.Proposition 9.
The fixed pointo2=q¯1,q¯2 is locally asymptotically stable provided that k<4−2s/45−22sℏ where(23)ℏ=155/824c1c2421−ω+2326ω223−11+223+443+15ω5/8.Proof.
The Jacobian ato2 becomes(24)J11J12J21J22,where(25)J11=1−45ℏ,J12=c12c24k1−ω+223ω,J21=s6c1c21−ω+223ω,J22=1−s,and the trace and determinant take the following form:(26)τ=1−s+kℏ244s−45,δ=2−s+452kℏ,and then,(27)Δ1=11sk12c1c2421−ω+2326ω1−ω+223ω5/8,Δ2=4−2s+kℏ22s−45,Δ3=s+kℏ245−44s.
It is clear thatΔ1>0 and Δ3>0 since s∈0,1 and ℏ is always positive. While Δ2>0 is attained, if k<4−2s/45−22sℏ. The proof is completed.
We should highlight here that ifk>4−2s/45−22sℏ, then o2 can be destabilized due to flip bifurcation only.


## 4.2. Numerical Results

We assume the followingc1>c2, c1=0.7,c2=0.5 with ω=0.2. At this set, o2=0.6068957946,0.4381254240 and the Jacobian becomes, at taking k=1.2 and s=0.2,(28)−0.911720.176540.0481270.8,and the eigenvalues are λ±=−0.916669,0.804949 with λ±<1 and then o2 is locally asymptotically stable. Increasing k above the value 4−2s/45−22sℏ gives rise to a period-doubling bifurcation, as shown in Figure 5(a). It is clear that the fixed point o2 is stable for all the values of k until k reaches the value 4−2s/45−22sℏ where the first period-2 cycle is born. As discussed in map (6), we give here some complex behaviors of map (9) at different values of the bifurcation parameter k. For instance, at k=8.026, a cycle of period-2 is emerged and is plotted with its basins of attraction in Figure 5(b). At k=8.44, a period-4 cycle is coexisted with its attractive basins in Figure 5(c). It is clear that they are quite complicated basins. The basins become more complicated when the period-8 cycle appears as given in Figure 5(d) at k=8.58. After that higher periodic cycles exist with quite complicated basins and then the map enters the chaos region at k=8.83 and k=8.9 where two chaotic attractors and then one chaotic attractor are emerged and displayed in Figures 5(e) and 5(f), respectively.Figure 5
(a) Bifurcation diagram and LLE on varyingk. (b) The basins of attraction of period-2 cycle at k=8.026. (c) The basins of attraction of period-4 cycle at k=8.44. (d) The basins of attraction of period-8 cycle at k=8.58. (e) The basins of attraction of two chaotic attractors at k=8.83. (f) The basins of attraction of a chaotic attractor at k=8.9. Other parameters’ values are c1=0.7,c2=0.5,ω=0.2, and s=0.2 .
(a)(b)(c)(d)(e)(f)The numerical simulation shows that increasingω which means the second firm gives more interesting to the market share maximization than the profit maximization reduces the stability interval of o2. The same observation when the firm gives more interesting to the traded-offs between the market share maximization and profit maximization is obtained. It is noticed that low values of the traded-off parameter s keep the stability interval of o2 larger. Furthermore, when we take values of c1 and c2 in the form c1<c2, the dynamics of map (9) do not give any new complex dynamics and hence the dynamic is still governed by the change that may be happened on both parameters ω and s.


## 4.3. Critical Curves and Phase Plane Zones

(29)2c1q14/3−q11/3−k4sc223s−1ω43+1−ω4=0,which has no real solutions as the expression under the square root is negative. Therefore, o belongs to Z1zone.Proposition 10.
The origin pointo=0,0 has only one real preimage that is o−10=0,0.Proof.
Settingq1t+1=0 and q2t+1=0 in (9), we get(30)q1+kq114q2q1−2c1q1=0,1−sq2+sω43+1−ω4q1c223=0.The first equation in (29) gives q1=0 or 1+k1/4q2/q1−2c1q1=0. Substituting q1=0 in the second equation of (29) gives q2=0, then we get o−10=0,0. Now, we have the following case:(31)1+k14q2q1−2c1q1=0,1−sq2+sω43+1−ω4q1c223=0.
Solving (30) givesProposition 11.
The fixed pointo2 has no real preimages.Proposition 12.
Any point in the form0,q,q≠0 has two real preimages.Proposition 13.
Any point in the formq,0,q≠0 has no real preimages.Now,LC−1 for map (9) is given by the following curve:(32)96sc1kq113/6q21/2−96c1kq113/6q21/2−24sq17/6q21/2+24q17/6q21/2−3skq2q12/3++3kq2q12/3−ksc223ω43+1−ω4q1=0.It is plotted in Figure6 with red color at the parameters’ values, c1=0.7,c2=0.5,k=2,ω=0.2, and s=0.2. Simulation shows that any point belonging to LC−1 has two real preimages. However, it is hard to plot LC but we can conclude that the phase plane of (9) may be divided into only two zones that are Z2 and Z0. Therefore, map (9) is a noninvertible map.Figure 6
The critical curve and zones of map (9) at the parameters’ values, c1=0.7,c2=0.5,k=2,ω=0.2, and s=0.2.


## 5. Comparison between the MapsT and T1

The above analysis and discussion about the two maps show that the model described by map (9) is quite complicated than those of map (6); however, map (9) gives a large stability region for the fixed point under the same set of parameters. This is clear when comparing the 1D bifurcation diagram for the two maps. This means that the firm adopting the trade-offs between the market share maximization and the profit maximization will be more stable in the market competition than those using a gradient-based mechanism such as the bounded rationality. It is also observed from the numerical experiments that when the second firm updates its outputs based on the average of market share and profit maximization, the stability region of the fixed point reduces and becomes almost like those provided by map (6) under the same set of parameters’ values. Figure 7(a) shows the bifurcation diagram at c1=0.7,c2=0.5,ω=0.5, and s=0.5. Increasing ω further to 0.7 and keeping the other parameters’ values fixed, the region of stability region of the fixed point reduces, as shown in Figure 7(b). Furthermore, Figures 7(c) and 7(d) show the influences of the parameters s and ω on the dynamics of map (9). From an economic perspective, one can see that low values of the parameter ω means the second firm pays more attention for the profit maximization (just as the first firm decision-making mechanism) than market share and hence the interior equilibrium point becomes stable for any value of s in the interval 0,1. As ω increases further, the stability interval of the equilibrium point with respect to s is reduced, as shown in Figure 7(c). Similar discussion is for the influences of the parameter ω given in Figure 7(d).Figure 7
(a) Bifurcation diagram on varyingk at c1=0.7,c2=0.5,ω=0.5, and s=0.5. (b) Bifurcation diagram on varying k at c1=0.7,c2=0.5,ω=0.7, and s=0.5. (c) Bifurcation diagram on varying s at c1=0.7,c2=0.5,k=5, and different values for ω. (d) Bifurcation diagram on varying ω at c1=0.7,c2=0.5,k=5, and different values for s.
(a)(b)(c)(d)


## 6. Conclusion

In this manuscript, we have studied two nonlinear duopoly models whose players adopted gradient-based mechanism and trade-off mechanism. We have shown that the fixed points in both models become unstable due to flip bifurcation only. The duopoly in the first model has been compared with the duopoly of the second model whose second firm uses trade-off between market share maximization and profit maximization. Our contributions showed that the second model was more stabilizing than the first model. It has been investigated that the trade-off mechanism can give better stability region for the fixed point. We found that when the second firm in the second model adopts the average between market share and profit maximization, the region of stability was reduced. Furthermore, using higher weights of the trade-off makes the stability region to shrink and hence the second firm should adopt low weights to become more stable in the market. We have also investigated that both maps are noninvertible, and for this reason, the basins of attraction of any dynamic behavior of the maps were quite complicated. In future study, we intend to apply the trade-off mechanism to oligopoly models with more than two firms.



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*Source: 6641852-2021-02-16.xml*