On Comparing between Two Nonlinear Cournot Duopoly Models

)e comparison between two nonlinear duopoly models constructed based on symmetric utility function that is derived from Cobb–Douglas is investigated in this paper. )e first model consists of two firms which update their outputs using gradient-based mechanism called bounded rationality. )e 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. )e 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.


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.
ese 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]. ere 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]). e 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][6][7][8][9][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. e 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][13][14][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. ese 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. e 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][24][25][26][27][28][29][30]. e current paper belongs to both homogeneous and heterogeneous duopoly games in which games' players adopt similar and different decisional mechanisms. e 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. e 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. e 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. Section 2 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.

The Game of Competition
We consider a game of two firms (or players) whose decision variables in the market are quantities. We take q 1 and q 2 as the quantities produced by each firm. We recall the Cobb-Douglas function of two variables appearing in many economic models as follows: is 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 � ���� q 1 q 2 √ returns (p 1 /p 2 ) � (q 2 /q 1 ) or p 1 q 1 � p 2 q 2 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 transformation u � lnU. is means that (zu/zq 1 ) � (a/q 1 ) and (zu/zq 2 ) � (b/q 2 ) and hence the marginal utility of each quantity is positive. (ii) It is a concave utility function.
is means that (z 2 u/zq 1 zq 2 ) � 0 and hence the marginal of the quantity q 1 is independent of the quantity q 2 . So, the quantities cannot represent substitutes or complements. (iv) It is a homothetic function.
is means that if We assume the symmetric case, a � b � (1/2), with the budget constraint p 1 q 1 + p 2 q 2 � 1 where p i is the price of q i , i � 1, 2. Now, we have the following maximization problem: Solving (2) gives the following prices: At the economic market, each competitor wants to maximize its profit given by where C(q i ) � c i q 2 i 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, (zC(q i )/zq i ) � 2c i q i , and c i , i � 1, 2 is a constant parameter. e marginal profits take the following form: 2 Complexity Here, we discuss two situations. e first situation assumes that both firms use a gradient-based mechanism such as the bounded rationality defined in the literature [4,[35][36][37]. Using this mechanism, we get the following discrete dynamical system: where k i , i � 1, 2 is a speed of adjustment parameter. e 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. e market share means it seeks the optimum output by putting π 2 � 0 and then we get the optimum, , while the profit maximization gives Since it is traded off between those, we get the following: 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: where s ∈ (0, 1) is a constant weight. So, we have another model with heterogeneous players given by It is obvious that if s � 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.
is 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].

Analytical Results.
e map (6) can make negative or unbounded trajectories if the initial condition

Proposition 1.
e map (6) has two fixed points that are ).

Proposition 2. e fixed point O 1 is an unstable point.
Proof: e Jacobian matrix of map (6) at O 1 becomes which contains indefinite form and then the point is unstable and the proof is completed.

Proposition 3. e fixed point O 2 is locally asymptotically stable provided that
Proof. At O 2 , the Jacobian matrix becomes whose trace and determinant are given by Substituting (13) in Jury conditions [40] gives Since k 1 and k 2 are positive speeds of adjustment parameters then Δ 1 > 0 is always satisfied. Δ 2 > 0 and Δ 3 > 0 give Combining the above inequalities completes the proof. Proof.
e Jacobian (12) has the following eigenvalues: It is clear that 9 where we assume k 1 � 3 and k 2 � 4. e eigenvalues then become λ ± � (0.138563, −0.741273) with |λ ± | < 1 and then O 2 is locally asymptotically stable. Keeping the costing shift parameters c 1 and c 2 as previously, we now study the influence of k 1 or k 2 on the stability of O 2 . Figures 1(a) and 1(b) show that O 2 is locally asymptotically stable for all values of k 1 and k 2 ; until we get the cycle of period 2, the point O 2 becomes unstable. Indeed, the simulation shows some interesting behavior of the dynamic of map (6) around the point O 2 . To present that dynamics, we have to study the effects of k 1 or k 2 separately. Assuming the following parameters set, c 1 � 0.7, c 2 � 0.5, and k 1 � 2. As k 2 increases to 6.54, a period-2 cycle (represented by squares in Figure 1(c)) is born around the fixed point O 2 .
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. is cycle turns into a cycle of period four as k 2 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 k 2 � 7.24 where nonconvergent points appeared in white color within the basins. e basins of attraction become more complicated as the period-8 starting to appear at k 2 � 7.24 as depicted in Figure 1(f ). Increasing k 2 further gives rise to four disconnected pieces and two chaotic attractors and then one chaotic attractor. We give in Figures 1(g) 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. Now, we study the case when c 1 < c 2 by assuming the set of parameters' values, c 1 � 0.5, c 2 � 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 c 1 > c 2 and c 1 < c 2 affect the stability region obtained by the second condition of (14) for the interior equilibrium point in the (k 1 , k 2 ) − plane. We have seen that at c 1 > c 2 (c 1 � 0.7, c 2 � 0.5), we get k 1 ∈ (0, 4.9) and k 2 ∈ (0, 5.8) while in the case c 1 < c 2 (c 1 � 0.5, c 2 � 0.9), we have k 1 ∈ (0, 6.166) and k 2 ∈ (0, 4.625) for the same condition.

Critical Curves and Phase Plane Zones.
e structure of the basins of attraction of map (6) at any ϱ that may represent O 2 or periodic cycle or any chaotic attractor is constructed by some boundaries. ese boundaries are calculated as follows.
Proof. Setting q 1 (t + 1) � 0 and q 2 (t + 1) � 0 in (6) where O (3) −1 ⊗ is obtained by solving algebraically the following system: −1 be the two line segments of the coordinates q 1 and q 2 . Also, let ω −1 1 and ω −1 2 be their preimages. en, the boundary of the basins of attraction of any ϱ is denoted by 5 and is given by where T − n refers to the set of all preimages of rank-n.
ese boundaries are displayed in Figure 4. 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: It is represented by red color in Figure 4 as it contains two parts, 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 O 2 has two real preimages of rank-1 and hence it belongs to Z 2 . Furthermore, both ω −1 1 and ω −1 2 have no preimages and they belong to Z 0 . With the preimages of the origin, we detect that the phase plane of map (6) is divided into three zones, Z 4 , Z 2 , and Z 0 . erefore, map (6) is a noninvertible map.

Numerical Results.
We assume the following (c 1 > c 2 ), and the eigenvalues are λ ± � (−0.916669, 0.804949) with |λ ± | < 1 and then o 2 is locally asymptotically stable. Increasing k above the value (4 − 2s/(45 − 22s)Z) gives rise to a period-doubling bifurcation, as shown in Figure 5(a). It is clear that the fixed point o 2 is stable for all the values of k until k reaches the value (4 − 2s/(45 − 22s)Z) 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. e 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. e numerical simulation shows that increasing ω which means the second firm gives more interesting to the market share maximization than the profit maximization reduces the stability interval of o 2 . e 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 o 2 larger. Furthermore, when we take values of c 1 and c 2 in the form c 1 < c 2 , 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.

Critical Curves and Phase Plane Zones.
which has no real solutions as the expression under the square root is negative. erefore, o belongs to Z 1 zone.

Comparison between the Maps
T and T 1 e 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. is is clear when comparing the 1D bifurcation diagram for the two maps. is 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) (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).

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.
e 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.

Data Availability
e datasets generated during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e author declares no conflicts of interest.