Abstract
The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.
1. Introduction
In this paper, investigations on the dynamics of nonlinear economic games whose competing players adopt linear inverse demand and horizontal differentiation with computed quadratic cost function derived from the Cobb–Douglas utility function are studied. The time evolution of this game is expressed by a two-dimensional nonlinear discrete-time map using the bounded rationality. Literature has reported several analyses of such economic games which have attracted many researchers because of the complicated behaviors of these maps. In the current paper, we focus on a special type of those maps that are known as duopoly maps. The focus of this paper is on studying duopoly games with quantity setting, and for future studies, we urge interested readers and researchers to investigate the cases of Cournot–Bertrand mixed game and triopoly under the conditions imposed on the current game. The duopoly game has been characterized by a market possessing only two competing firms (or players) seeking the optimal quantities that maximize players’ profits. After Cournot, the famous French economist, who introduced the first duopoly model with quantity setting as decision variables, many works have been raised to study the dynamics of such games. Analyzing the dynamics of such games has been dated to [1] who died last year after enriching literature with many useful books and papers. In 1950, the Nash equilibrium point explored by John Nash, a famous American mathematician, has been raised in the theory of equilibrium points in noncooperative games. The Nash equilibrium point represented the optimum output of a game in which no player has an incentive to deviate from their chosen strategy after considering an opponent’s choice. Studying the dynamic characteristics of duopoly games requires first to calculate the Nash point and then studying its stability. In fact, one should highlight the theory of games as an important applied mathematics tool that has been adopted to analyze the interaction amongst players in such economic games. Game theory has been used for investigating strategic interactions amongst decision-makers and rational competitors. It was extensively developed in the 1950s by many scholars and was explicitly applied to several fields such as social science, logic relations, computer science, biology, and economics’ models. Recently, it has been applied to economic models for studying the dynamic competition amongst competing firms. However, static equilibrium analysis of such games is limited and has not given valuable information on the time evolution of such games, although it has been studied by many researchers (see, for example, [2]). Dynamic studies give more valuable information on the stability/instability of Nash equilibrium, future predictions of the game’s behavior, types of bifurcations by which the Nash point may lose its stability, and the topological structure of the phase plane describing the dynamics around the Nash equilibrium point. In literature, it has been reported that Rand [3] was the first person who analyzed some of these dynamic characteristics.
Literature has reported many works that have analyzed the dynamic characteristics of duopoly games. These dynamic characteristics have included bifurcation types such as flip and Neimark–Sacker, some attracting sets and chaotic attractors whose basin of attractions are represented by peculiar shapes, and synchronization and multistability phenomena. Moreover, adopting specific inverse demand functions requires using specific types of utility consumer preferences. One of the most important utilities in literature is known as the Cobb–Douglas utility which was used in several studies in the literature and is adopted in the current work. The reason is that the Cobb–Douglas utility represents a production function and has been widely used to represent the technological relationship between the inputs and production outputs. There are other utility functions that have been reported in the literature and used in such games, for example, constant elasticity of substitution (CES) and Singh and Vives utility. Interested readers on the properties of these utilities are advised to check references [4–10].
The complexity behind duopoly models and their dynamic characteristics reveal interesting information on game’s evolution and its future predictions. Such interesting results on the complexity have been raised in the literature and opened new routes for investigations. Here, we report some related works and results from the literature. For instance, a Cournot duopoly game that is known as a mixed game (in such a game, the two decision variables are represented by quantity and price) has been analyzed in [11]. Other related works on duopoly and triopoly in which competing players or firms have sought optimality of production have been introduced and analyzed in [12–14]. In [15], a Cournot duopoly game with rational competing players adopting the bounded rationality mechanism and seeking the maximization of relative profits has been studied. Based on a theoretical framework, a mixed-type game has been formalized and analyzed in [8]. In [16], another mixed-type game has been introduced and studied by Naimzada et al. based on a two-dimensional discrete nonlinear map. They focused on analyzing the dynamics of Nash equilibrium and its instability through two different types of bifurcations. On the other hand, studying the complex dynamic characteristics of such a game requires building a map which is used to describe the time evolution of the game. From an economic perspective, this map is a tool by which firms can update their productions. Forming such maps depends on some adjustment rules. Amongst those rules, literature has extensively reported the famous one which is known as the bounded rationality approach. Many duopoly games have been deeply used this approach for the process of modelling the discrete maps used to represent the time evolutions of these games. This approach is called a gradient rule as it depends on the marginal profits of competing firms. It requires firms to perform an estimation for their marginal profits to see whether they are increased or decreased, and consequently, they may update their outputs next time step. Besides this approach, there are other mechanisms that have been adopted in the modelling process such as the tit-for-tat rule and local monopolistic approximation mechanism [17–27]. Other recent studies on the dynamics of such games have been recently reported in the literature [28–32]. Recent studies on the complicated dynamics of such games have been reported in literature [33–40]. In addition, the case of Cournot–Bertrand on where decision variables are quantity and price has been recently studied in [41].
In literature, many cost functions have been assumed to be linear or nonlinear, and in this manuscript, we introduce a computed quadratic cost function that is used for constructing our game’s model. Our aim in this paper is to deepen the complex dynamic characteristics of the nonlinear Cournot duopoly game considering that competing players (or firms) operate with quadratic cost function (a case of decreasing returns-to-scale technology) with incomplete market information. This paper’s game is presented by a two-dimensional (2D) nonlinear discrete dynamic map with four fixed points, three of which are boundary points while the fourth is an interior point representing a Nash equilibrium point. The complex dynamic behavior of the map is entirely analyzed based on the local and global investigations. These investigations show that the Nash equilibrium point can be destabilized through flip bifurcation, and under certain parameters, the values of Neimark–Sacker bifurcations are raised after the period-4 cycle. Furthermore, under some restrictions, synchronization and multistability are studied showing that the diagonal set in the phase plane forms an invariant manifold. In addition, the obtained results show that the game’s map is noninvertible and its phase plane is formed by three zones that are , and . Overall, the proposed game and its characteristics generalize some existing works in the literature [9, 10, 15].
In brief, the current paper consists of many sections. Section 2 introduces the market competition including the computation of cost and profit. In Section 3, the 2D game’s map is formed and its characteristics such as critical curves and noninvertible properties are discussed. The map’s fixed points and their stability are given in Section 4. In Section 5, the invariant manifold for the map is investigated through a one-dimensional map. In Section 6, the basin of attraction, global analysis, and the case of independent firms are analyzed. Finally, the conclusion is presented in Section 7.
2. Market Competition
The market competition proposed in this paper consists of two firms labeled by and . Both firms produce the quantities, and , respectively. The market price is restricted to the following inverse demand functions (prices):where denotes the maximum price in the market. The parameter has an important influence on the dynamics of the competition. It denotes the degree of product differentiation or product substitution. If , one gets two identical inverse demand functions and hence homogeneous goods are provided to the market. If , the two prices are independent and the case of two monopolistic markets is raised. If [), the complementarity between the two competing firms is obtained. We will discuss the analysis of competition in the case of substitutability on which .
2.1. Computation of Cost and Profit Functions
In order to compute the quantities produced by the two firms, we recall the Cobb–Douglas [42] production function given bywhere is a constant and refers to the total factor productivity, denotes the total labor, and is the total capital. is a constant, and for simplicity, it is assumed to be 0.5. The total cost imposed by a firm can be computed aswhere refers to the wage per unit of labor, while denotes the rental price per unit of capital. Therefore, by substituting (2) in (3), the total cost for each firm takes the following form:where the marginal cost is and . Now, the profit is given bywhere is the total revenue. By using (1) and (4) in (5), the profits become
3. The Competition Model
There are many mechanisms that have been used in the literature to build the discrete dynamic model describing such games. In this paper, the bounded rationality mechanism is adopted. It depends on the marginal profits of firms. The marginal profits then become as follows:
In such games, firms watch their marginal profits whether they are increased or decreased. If , it means that firms are ready to increase their production in the next period of time, and otherwise, they decrease production or become naive. So, the updating of productions is described by the following mechanism:
Let us assume that , where is the parameter of speed of adjustment. It means that the relative production is directly proportional to . By substituting (7) in (8), one gets the discrete map that is used to describe the evolution of the proposed game (or the game’s repetition) as follows:where the time steps are denoted by the parameter . Setting gives , which describes each firm’s optimal (profit-maximizing) quantity of output. This can be thought of as describing a firm’s best response to the other firm’s level of output. Therefore, by using this symmetrical relationship between firms, we find the equilibrium quantity by fixing . The equilibrium levels make firms have no incentive to change their level of output.
3.1. Critical Curves and Noninvertible Property
The critical curves are used to describe the decision space and are responsible for dividing it into zones. The rank-1 of the critical curve is expressed by which forms a locus containing all rank-1 preimages located on a set denoted by . As one can see that map (9) is of class (continuously differentiable). This means that the set can be defined as the locus of points such that the Jacobian determinant associated with it is vanished. So, is given bywhere is the Jacobian matrix. Therefore, represents the rank-1 image of under the map , i.e., . For map (9), is expressed by the following relation:where
Both and are plotted at two different sets of parameter values in Figure 1. Figure 1(a) shows that consists of two parts and . Figure 1(b) shows also that consists of two parts and . Both figures are plotted at the values of the parameters, , and . Figures 1(c) and 1(d) present those critical curves at the parameters values, , and . One can also see that map (9) is a noninvertible map since the two branches of the critical curve divide the decision space into three zones , and . These zones possess the set of points with 0, 2, and 4 positive real number preimages. In order to declare that let us calculate the real rank-1 preimages for the origin point. By substituting in map and by solving the obtained algebraic system, one gets
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This means that the origin point belongs to zone . For convenience, let and be two segments on the invariant axes and . Let also and be their rank-1 preimages, respectively. So at any points and , their rank-1 preimages will satisfy the following algebraic systems:
So both and can be presented by
As one can see in Figure 2, those lines intersect at point .
4. Fixed Points and Stability
By setting in map (9), the following fixed points are obtained:
Since and , it is clear that all the fixed points are positive. The following propositions are raised, and their proofs are given in Appendix A.
Proposition 1. The boundary equilibrium point is an unstable repelling node.
Proposition 2. The boundary equilibrium point is a saddle point if . Otherwise, it is an unstable node.
Proposition 3. The boundary equilibrium point is a saddle point if . Otherwise, it is an unstable node.
Proposition 4. The point is called the Nash equilibrium point and is asymptotically stable if
Proposition 5. The Nash point loses its stability due to flip bifurcation if
Figure 2 presents the basin of attraction of Nash point which looks like a regular convex polygon whose vertices are the four preimages of the origin. It can be seen that both and divide the phase plane into two parts and . The first part contains all points which generate bounded trajectories converging to the Nash point, i.e., , while the second part consists of all points which generate unbounded trajectories colored by gray. It can also be seen that the boundaries of and are equal, i.e., .
4.1. Local Bifurcation
Map (9) presents a dynamical system that contains two important parameters and . These parameters are called the speed of adjustment parameters and any slight change in them may raise a great change in the dynamics of the system. These parameters are selected to be the bifurcation parameters. Let us assume the following values, , and (or ) on varying (or on varying ). Figures 3(a) and 3(b) show the influence of the speed parameters on the stability of the Nash point. Both parameters undergo a flip bifurcation diagram. Fixing those parameters and changing the value of present its impact on the stability region of Nash point. Numerical experiments show that increasing close to 1 makes the region of stability with respect to the speed parameters increase. On the other hand, at the values, , and (or ) on varying (or on varying ), Figures 3(c) and 3(d) show that the period-4 cycle emanating from the flip bifurcation may undergo a Neimark–Sacker one as both the speed parameters increase. Keeping the value of speed parameters relatively small keeps system (9) in a stable state. Furthermore, as increases to be close to 1, the region of stability with respect to the speed parameters is also extended. Conversely, as the parameter increases while the other parameter values are fixed, the stability region is decreased with respect to the speed parameters. To end this local analysis, we plot in Figure 4 the influence of the parameter on the map’s dynamics at the values, and .
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5. The Invariant Manifold
The point (0, 0) for map (9) has an important aspect. If the map initiates from this point it will be directly trapped to this point. This means that if or , then or and hence the coordinates axes and become invariant. Those axes construct an invariant manifold for , and then, its dynamics can be described by a one-dimensional map as follows:
We see that map (19) is topologically equivalent tothrough the linear transformation given asand .
5.1. Dynamic Analysis
For map (20), let us consider the following function:
This function presents the well-known logistic equation with a parameter causing its dynamics. The first derivative, attains a function maximum value occurring at and the point . It is clear that and , which means that [0, 1) and for all . It is also simple to see that map (20) has two fixed points, and , which are nonnegative provided that . One can see that and then the fixed point is stable if , otherwise it is unstable. The point, , is stable if , otherwise it is unstable if . Figure 5(a) demonstrates the bifurcation diagram of map (20) on varying parameter . As mentioned above, its dynamics is similar to the well-known standard logistic map. Figures 5(b) and 5(c) illustrate the cobweb diagrams of map (20) at different values of the parameter . When , the fixed point becomes stable but when , a transcritical bifurcation emerges. At , the fixed point gets stable resulting in a stable period-2 cycle at as shown in Figure 5(b). At , the period-2 cycle becomes unstable around the fixed point as shown in Figure 5(c) on which a chaotic trajectory is born.
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Proposition 6. At the critical value, , the trajectories of beginning on the invariant axes and diverge when .
Proof. The proof is straightforward. Using and , we complete the proof.
5.2. The Symmetric Case
Map (9) possesses an important aspect. At the assumption , the map becomes symmetric. It means that if and are swapped, the structure of the map does not change, i.e., , where . Consequently, the diagonal set defined by constructs an invariant manifold. This implies that the dynamics of any trajectories which start on the diagonal set will come back to the diagonal set at each time step . So the dynamics of map (9) can be studied by a one-dimensional map under the restriction given on as follows:
Now, for any synchronized trajectories (i.e., at every time period t), they will be governed by and one gets the following proposition.
Proposition 7. Map (23) is characterized as unimodal and concave and and . It means that it possesses a threshold point that occurs at . The unimodal aspect means that there is a critical point given by
Proof. It is easy to see that the second derivative of is nonpositive for all and hence has only one global maximum value at without any local maximum. This means that the function is unimodal and concave with .
Now, the map possesses only one nonzero fixed point that is which is locally stable if , while flip bifurcation occurs at . Figure 6 shows the one-dimensional bifurcation diagram where flip bifurcation occurs at .
6. Basin of Attraction
Let us now investigate additional properties for map (19) such as its basin. The map can be rewritten in the following form:
Following [43], the invariant axes and their preimages of any rank can be used to form the boundaries of nondiverging trajectories. Map (25) possesses a positive fixed point . One can easily see that is a concave function and an increasing function if . In addition, for any point or , it gives or . In contrast, if , then is concave and unimodal. In the latter case, one can get the nonnegative points and at and , respectively. If , i.e., if , then it means that the bounded trajectories along the invariant axes and are bounded provided that the initial states lie on the segment , where presents rank-1 preimages for the origin point. Conversely, any other trajectories starting on those axes for initials outside become infeasible (or nondiverging) trajectories. Moreover, the eigenvalues at a point on those invariant axes become . For example, at a point , one gets and , which means that trajectories with positive initial states beginning close to those axes are repelled by them. Figure 7 presents the basin of attraction of the point and period-4 cycle marked by the colors brown and blues, respectively. This cycle is born at the values , and . The other colors that are gray and white are for the divergence and nonconvergence points. It is easy to see that the lines and their inverses separate the divergence and nonconvergence points.
6.1. Global Analysis
In economic models, it is necessary to globally investigate the topological structure of the basin of attraction. This structure does not appear when carrying out local analysis. Analyzing the behavior of map (9) in the long-run whose variables take intials values not close to the equilibrium point is necessary in order to see the qualitative changes that may be occurred in the topological structure. Numerically, two cases for map (9) that are the asymmetric and symmetric cases are investigated in this subsection. First, the asymmetric case is studied at two different sets of parameter values as follows. We start with the first set, . Figure 8(a) shows the two-dimensional bifurcation diagram in the . The gray color illustrates the stability region of the equilibrium point while the other colors present different types of period cycles marked by . Assuming and , Figure 8(b) shows the basin of period-7 cycle marked by squares. Its basin is colored in light blue while the yellow color is for the basin of the equilibrium point . It can be seen from the figure that the infeasible region colored in gray is disconnected. It is clear that some holes of this region lie within the basin of cycle 7. This is because of the contact bifurcation that took place. This contact bifurcation is discussed in Figure 8(c) in detail. Figure 8(c) presents the basin of attraction of a chaotic attractor (marked by green) that occurred at the values, . This chaotic attractor is distributed in the zones, and . As one can see, the infeasible region (or the escaping domain) is disconnected. It is clear that the whole region does not lie in and part of it denoted by enters the region . The part is constructed due to the intersection between the critical curve branch and the boundary line . It consists of two parts connected by the branch . This part belongs to the zone whose points possess two distinct real rank-1 preimages and are responsible for the disconnection of the escaping domain and the coexistence of main hole that entirely lies in . The main hole is formed by two parts and connected by the branch . Each point belonging to the main hole will have four distinct real rank-2 preimages. Numerical simulation shows that points in the main hole are responsible for constructing the small holes denoted by , , and . So it is obvious from the Figures 8(b) and 8(c) that contact bifurcation has occurred and consequently holes from the infeasible domain are born. It is also seen that the structure of the basins for the period-7 cycle and chaotic attractor is bounded by the quadrilateral shape . Such complicated structures make the future evolution of the game unpredictable in case players take initial states inside those holes.
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Then, we investigate the dynamics of map (9) using another set of values. Let us assume the values, . At this set, the dynamics of map (9) are represented by four closed rings around the equilibrium point due to Neimark–Sacker taking place after the period-4 cycle. Keeping the set of values fixed but by increasing to 1.29, the four rings turn into a four-band chaotic attractor. The basin of attraction of this chaotic attractor with the equilibrium point is plotted in Figure 9(b). As one can see that the origin point belongs to and the chaotic attractor is distributed in zones and . By further increasing the speed parameter to 1.3, the chaotic attractor is converted into period-16 cycle. This cycle has a basin of attraction bounded by the quadrilateral shape . This period cycle is turned into a four-band chaotic attractor as increases to 1.32 as shown in Figure 9(d). Keeping the other values fixed and by increasing the speed parameter to the value a two-band chaotic attractor emerges whose basin is given in Figure 9(e). As one can see that as the speed parameter increases the critical branch is approaching to the boundary . When the speed parameter reaches the value 1.449, one gets a one-band chaotic attractor whose attractive basin is given in Figure 9(f) and then becomes tangent to the boundary . This tangentiality causes the contact bifurcation and makes the infeasible domain disconnected due to the holes coexisting in the attracting domain.
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In contrast, let us now investigate the symmetric case of map (9). This case occurred by assuming that and hence map (9) can be rewritten as follows:
It is easy to see that map (26) is symmetric on the diagonal . In this case, there is no mathematical meaning for the equilibrium points except the point which belongs to the diagonal . The local stability of this point around is related to synchronization (intermittency). Simply, one can say that a synchronous dynamic behavior is obtained if there exists a period of time such that for any . So, the Jacobian of map (26) is given bywhere the eigenvalues are represented asand the corresponding eigenvectors are (1, 1) and , respectively. Thus, the following proposition is obtained.
Proposition 8. The local stability of the equilibrium point is achieved if . A flip bifurcation is raised if .
Proof. Substituting in (28), one gets the following eigenvalues:Here, and give and , respectively. Thus, combining the two conditions completes the proof.
The following figures simulate the abovementioned proposition. Let us assume and . Figure 10(a) shows that at , the period-2 cycle and hence the one-dimensional flip bifurcation is obtained on varying the speed parameter . Figure 10(b) presents the time series for both decision variables. One can observe from Figure 10(c) (which represents the displacement with time ) that the transient behaviors of map (25) may be described by bursts that appear outside the diagonal . Therefore, intermittency will occur in map (25) as shown in Figure 10(d). Figure 10(e) shows a complex chaotic attractor raised entirely on the diagonal at the parameters values, with bursts out of the diagonal . With a slight increase in the speed parameter with at the same parameter values, the weak attractor given in Figure 10(d) turns into a complex chaotic attractor depicted in Figure 10(e). As one can see, the equilibrium point lies on the diagonal and the chaotic attractor is not symmetric around the diagonal due to bursts raised on both sides as shown in Figure 10(f).
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From the abovementioned proposition and analysis, it is clear that the speed parameter affects the stabilization of the equilibrium point . It is noted that the high reactivity of the competing firms to the marginal profit should coexist. Moreover, suppose a cycle given by of the map embedded into the invariant diagonal corresponding to the cycle . Therefore, this cycle has multipliers given bywith corresponding eigenvectors (1, 1) and . So, the stability of is guaranteed by and is confirmed by . For a complex chaotic attractor of the map (25), is asymptotically stable if and only if all trajectories of are transversely attracting. This asymptotic stability condition of is obtained based on the transverse Lyapunov exponent as follows:where and are the trajectories constructed by the map, and
The definitions are given ([42, 43]) as follows.
Definition 9. An asymptotically stable chaotic attractor is Lyapunov stable if for every neighborhood of there exists a neighborhood of such that and its basin has a neighborhood of .
Definition 10. A spectrum of Lyapunov exponent is defined based on the initial conditions as follows:where denotes a Lyapunov exponent obtained at a generic trajectory in . If, then a set is called a Lyapunov attractor. When and , then a set is no longer Lyapunov stable and gets a Milnor attractor. Milnor attractor is defined as follows.
Definition 11. A closed invariant set is called a weak attractor in Milnor sense if its stable set (the basin of attraction) has a positive Lebesgue measure.
6.2. Independent Firms
In this subsection, we study a particular case in which competing firms behave as if they were monopolistic firms with independent products, that is . In this case, map (25) can be rewritten as follows:which represents a diagonal map. It is easy to see that the abovementioned map conjugates the logistic map, with . The eigenvalues of the abovementioned map and are symmetric. This means that any attracting set along the diagonal will have identical eigenvalues. Due to this property, any period-doubling bifurcation that has taken place along the diagonal (which is also associated with the cascade bifurcation with the well-known logistic map) will be followed by a simultaneous period-doubling bifurcation that has occurred along the symmetric direction. Hence the following proposition is raised.
Proposition 12. The equilibrium point is locally stable if . It loses its stability through period-doubling bifurcation at .
Proof. Substituting in (29) gives and and this completes the proof.
At , one gets and hence period-2 cycle is born. A further increase in gives rise to high periodic cycles and hence flip bifurcation is obtained. Figure 11 shows the basin of attraction of two different period-4 cycles. The first one occurs at the values, , and and lies entirely on the diagonal. As increases to 1.27, the second one is born but on both sides from the diagonal. This is due to the phenomena of multistability.
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7. Conclusion
Through this paper, the synchronization and multistability phenomena for a simple Cournot duopoly game whose cost has been computed based on the Cobb–Douglas utility function have been investigated. As in many relevant papers in the literature, the fixed points of the proposed game are obtained for the asymmetric case and their conditions of stability are investigated. Our obtained results have confirmed that the Nash equilibrium point may be unstable through flip bifurcation and under certain parameter values, Neimark–Sacker bifurcation has been born after the period-4 cycle. Some complex structures of basins for the nonlinear two-dimensional map describing the game have been discussed. Our discussion has shown that the game’s map is noninvertible and its plane has been divided into three zones of preimages that are , and . Furthermore, under certain parameter values the dynamics of the game’s map behave as a weak chaotic attractor in Milnor’s sense. This weak attractor has been distributed on the diagonal at which the synchronized trajectories arise. Any slight increase in the speed parameter has affected this weak attractor and converted it into a complicated one. Finally, the case when competing firms behave as if they were monopolistic firms with independent products has been studied showing the coexistence of two different period-4 cycles due to multistability.
The obtained results in this manuscript and their potential applications may be extended in future works for biological models and economic games with three competitors under different types of adjustment mechanisms. Furthermore, it may be also applied to economic games whose main interests are to optimize their objectives which include profits and social welfare.
Appendix
Map (9) admits the following Jacobian matrix:
Proof of Proposition 1. At , the Jacobian matrix (A.1) becomesIt is quite obvious that (A.2) is a diagonal matrix and hence the corresponding eigenvalues are with eigenvectors (1, 0) and (0, 1). From the nonnegativity of the adjustment speed parameter, and from the auxiliary parameter , it is clear that , hence is an unstable repelling node.
Proof of Proposition 2. At , the Jacobian matrix (A.1) becomesIt is clear that the matrix (A.3) is an upper triangular matrix whose eigenvalues are given byThe corresponding eigenvectors are (1, 0) along the and , respectively. It is clear that gives and if , then is a saddle point. If , then is an unstable node.
Proof of Proposition 3. The proof is similar to Proposition 2.
Proof of Proposition 4. The Jacobian (A.1) at Nash point becomeswhereand the trace and determinant are given byUsing (A.6) and (A.7) Jury conditions, [30] can be obtained as follows:Due to the nonnegativity of the auxiliary parameters , and , the condition (A.8a) is always positive. Simple calculations show that if the condition (A.8c) is greater than zero, this means that the condition (A.8b) is greater than zero and hence is asymptotically stable.
Proof of Proposition 5. Now, suppose that the condition (A.8b) is less than zero and the condition (A.8c) is kept nonnegative then using simple calculations, the point loses its stability due to flip bifurcation if the following condition is achieved:
Data Availability
The data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Researchers Supporting Project number (RSP2024R167), King Saud University, Riyadh, Saudi Arabia.