Abstract
In this paper, some important dynamic characteristics such as multistability and synchronization phenomena are investigated for a game of an economic Cournot duopoly whose time evolution is received by the iteration of a noninvertible map in the plane. In the asymmetric case, the equilibrium points of game’s map are calculated, and their stability conditions are obtained. The obtained results show that the Nash equilibrium point loses its stability through flip bifurcation. Under some restrictions, the map’s coordinate axes form an invariant manifold, and hence their dynamics are studied based on a one-dimensional discrete dynamic map. In the symmetric case where both firms are identical, the map has the property of symmetry, and this implies that the diagonal forms an invariant manifold and therefore synchronization phenomena occur. Global analysis of the behavior of the noninvertible map is carried out through studying critical manifolds of the map that categorize it as type. Furthermore, global bifurcation of the basins of attraction is confirmed through contact between the critical curves and the boundaries of escaping domain.
1. Introduction
Dynamic economic games whose evolutions has been depended on discrete time maps have attracted many researchers due to the interesting dynamic behaviors and bifurcations types occurring in such maps. Analyzing the dynamic characteristics of such maps has been dated at least to [1] by Puu who passed away in 2020 after enriching this direction with many useful books and papers. In the current paper, we focus on a special type of these maps that are known as duopoly maps. In duopoly game, the market possesses only two competing firms (or players) whose target is seeking the optimal quantities in the case of Cournot by maximizing their profits, or seeking the optimal prices in the case of Bertrand. Cournot, the French economist who started the core of such kind of games in 1838, introduced a duopoly model adopting quantities as decision variables. Later in 1880s and based on Cournot model, Bertrand suggested his model known as Bertrand model and considered prices as competitive means. In 1950, the Nash equilibrium point detected by the famous American mathematician John Nash has been raised in the equilibrium theory of noncooperative games. In general, this theory provided research with significant tools in studying oligopoly models. As a result, Nash point became important aspect in studying such models and was found in several Cournot and duopoly games. Indeed, one has to highlight game theory as an important tool of applied mathematics that was used and adopted to study the interaction among players in those economic games. It has been used in studying the strategic interactions among rational competitors and decision-makers. Game theory was extensively developed in the 1950s by many scholars and was explicitly applied to all fields of social science, as well as logic, biology, economics systems, and computer science. Recently, the theory of game has extensively been applied in economics for studying competitions among competing firms. Even though static equilibrium analysis of such games is limited and has not provided valuable information on games’ evolutions, it has been investigated by many researchers (see, for example, [2]). Studying the game evolution through dynamics gives more information on the stability/instability of Nash equilibrium, the types of bifurcations by which it loses its stability, and the topological properties of basin of attraction in the phase plane. Rand [3] was the first to introduce and analyze some of these dynamic characteristics.
Firm in duopoly competition simultaneously chooses its decision or strategy taking into account two important factors: its own actions and actions by its competitor. In literature, several works have been reported for studying the dynamic characteristics of duopoly games. These dynamic characteristics included bifurcation and its types, either flip or Neimark–Sacker; peculiar shapes of attractive basins for attracting sets and chaotic attractors; and multistability phenomena. Furthermore, studying such games requires adopting specific kinds of utility consumer preferences. The most popular one that is well-known as Cobb–Douglas utility was adopted in many studies in literature. The reason is that Cobb–Douglas utility is a particular functional form of the production function. It has been widely used to represent the technological relationship between the inputs and outputs that can be produced by those inputs. Moreover, literature reported many other utility functions such as constant elasticity of substitution (CES) and Singh and Vives utility. One can get more information and properties of those utilities from literature ([4–10]).
The dynamic characteristics of duopoly models are complex and reveal interesting information on their evolutions. Therefore, many interesting results on these complex dynamics have reported many interesting results on the stability of Nash equilibrium point of these games and opened the routes to attract many researchers. Some related works and results for these models are reported here. For example, a game of two competing firms whose decision variables are different (Cournot duopoly game) has been analyzed in [11]. Other studies on duopoly and triopoly in which firms seek the optimality of production have been introduced and investigated in ([12–14]). In [15], a Cournot duopoly game whose players are rational, adopting bounded rationality rule and seeking maximization of their relative profits, has been introduced and analyzed. Using a theoretical framework, a Cournot–Bertrand game has been formalized in [8]. In [16], Cournot–Bertrand model has been studied by Naimzada et al. using a two-dimensional discrete linear map. They focused on investigating the dynamic characteristics of Nash equilibrium points and its instability through two types of bifurcations. Additionally, analyzing the game’s evolution requires representing it by a discrete time map so that firms can update of their productions. This requires some kinds of adjustment rules. Literature has presented and reported many of these rules. The first rule among those rules which has been extensively adopted is known as bounded rationality. It has been deeply adopted in many studies for process of modeling discrete time maps representing the evolutions of these games. It is described as a gradient rule because it depends on firms’ marginal profits and requires firms to carry out an estimation of their marginal profits to determine whether they increase or decrease, so they can update their outputs next period of time. There are also other mechanisms which have been used in the modeling process such as tit-for-tat rule and local monopolistic approximation mechanism ([17–27]).
Based on a linear inverse demand functions and adopting the bounded rationality mechanism, a Cournot duopoly game with players seeking the maximization of their average profits is introduced and analyzed in this manuscript. Most firms do not rank profits as the major goal. The work of modem firms is so complex that they do not think merely about profit maximization. The basis of the difference between the objectives of the neoclassical firm and the modem corporation arises from the fact that the profit maximization objective relates to the entrepreneurial behavior while modem corporations are motivated by different objectives because of the separate roles of shareholders and managers. Due to the above discussion, we have introduced a general case with different objectives. The game is described by a nonlinear discrete dynamic map that possesses four equilibrium points, three of which are boundary points, and the fourth is an interior point and coincides with Nash equilibrium point. We use in this manuscript local and global analysis to investigate the dynamics of these equilibrium points. The analysis shows that Nash point loses its local stability through flip bifurcation. The structure of the game’s map shows that it is being attracted to the origin, and hence its coordinate axes form an invariant manifold. Consequently, the map’s dynamics are studied by a one-dimensional map conjugating the famous logistic map. It is also shown that under some restrictions, the synchronization phenomena are studied and show that the diagonal in map’s phase plane forms an invariant manifold. Furthermore, our findings show that the game’s map is noninvertible and its phase plane is divided into three zones that make it belong to type.
The sections in the current paper are summarized as follows. In Section 2, the discrete form of the game’s map is introduced. The equilibrium point and its stability are investigated both analytically and numerically through local analysis in Section 3. In Section 4, the invariant manifold is studied. The basin of attraction and critical curves are analyzed in Section 5. In Section 6, the global analysis and synchronization phenomena are investigated based on some numerical experiments.
2. The Model
Suppose an economic market consisting of two competing firms (or players) whose decision variables are quantities, and . Both firms adopt linear inverse demand functions given by
The horizontal differentiated products mainly refer to difference in extrinsic features between the products, like color and shape. Such differentiation mostly adapts to differences in consumer preferences. In (1), the parameter refers to the degree of horizontal product differentiation that is taken in the interval . For , each firm behaves as a monopoly player due to the independent varieties between firms’ products. means that both products are perfect substitutes or are homogenous, and hence . In this paper, we focus on the case of . The parameter denotes maximum price in market. Suppose that the competing firms adopt different nonnegative marginal costs and hence their total costs are taken as (). Now, the total profit for each firm takes the following form.
Unlike many studies in literature, the current game assumes that both firms focus on maximization of their average profits rather than profits only. The average profits then become
The parameter , where , is directly used to measure altruism. That is to say, it explains the firms’ care of average performance. When , it means firms give care to their own profits only. Substituting (4) into (5), one getsand then the marginal average profits become
Maximizing the average profits requires firms to know some characteristics of market and the nature of competition. Knowing the characteristics of market needs complete information on the market and the strategies of firms’ rivals, and these become impossible. Therefore, firms believe that updating output productions requires watching the changes taking place in the marginal average profits. That is to say, if , this means firms will increase their production next period of time; otherwise, they decrease productions or become naive. Accordingly, firms estimate production based on bounded rationality rule that has the following form.
Adopting bounded rationality rule makes us assume that , where is known as the speed of adjustment parameter. This gives which makes relative production directly proportional to the firms marginal average profits. Equation (6) possesses some important characteristics that can be found in literature ([25–28]). Substituting (5) into (6), one simply has a two-dimensional discrete map that will be used throughout the manuscript to discuss the game’s repetition (or evolution).
The parameter denotes time steps. Setting the fixed points’ condition, in (7), one has four fixed points given by
3. Local Stability
In order to study the stability of the fixed points (8), we recall the Jacobian matrix at these points, and hence the following propositions are raised (their proofs are given in Appendix).
Proposition 1. The point is an unstable repelling node.
Proposition 2. The point is a stable point provided that and . It is an unstable saddle if and .
Proposition 3. The point is a stable point provided that and . It is an unstable saddle if and .
Proposition 4. The point that is called Nash point is locally stable provided that
Proposition 5. The point loses its stability due to flip bifurcation only.
The obtained results show that the stability condition of Nash point depends on the parameters and the speed of adjustment parameters and . Some numerical experiments are carried out here to validate the theoretical results given above. As in many studies in literature, we keep our focus on studying the influences of speed parameters on the stability of Nash while fixing the other parameters. Let us suppose the set of parameters , and (or ) on varying (or on varying ). Figures 1(a) and 1(b) show the one-dimensional period-doubling bifurcation diagram on varying with respect to the two decisional variables and . As one can see, both figures present flip bifurcation diagrams. In Figures 1(c) and 1(d), one can see the flip bifurcation on varying the speed parameter, . In Figure 1(e), the phase plane of period-8 cycle with Nash equilibrium point is depicted. Figure 1(f) shows a one-piece chaotic attractor. In Figures 1(g) and 1(h), a two-piece disconnecting chaotic attractor is given. In these figures the marginal costs are fixed.
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4. The Invariant Manifold
Looking at the map (7), one can see that the point is trapping the map. That is to say, if or , one gets or , and hence both coordinate axes become invariant axes for the map. Therefore, the invariant set made of the axes and forms an invariant manifold for . Accordingly, the dynamics of the map (7) on the invariant axes can be described by the one-dimensional map given by
It is easy to see that the map (10) is topologically equivalent tothrough the transformation given below.and , .
4.1. Dynamic Analysis
For equation (11), suppose the following function.
Function (13) shows the logistic equation, and is the parameter responsible for its dynamics. It has a first derivative given by . Its maximum value occurs at at the point . In addition, and , which means that on the open interval and for all . Moreover, it is easy to see that (11) has only two fixed points, that is, and , where . Consequently, , and then is stable if and unstable otherwise. The other fixed point is stable if and unstable if . In addition, one can see that the map (11) has two fixed points, and , where . Hence, , and then is stable if and unstable otherwise. The other fixed point is stable if and unstable if .
The simulation in Figure 2(a) presents a 1D bifurcation diagram on varying that confirms that the point is stable in . In Figure 2(b) a cobweb diagram is given at in the stability interval and shows a stable fixed point, while in Figure 2(c), an unstable behavior is plotted at , and hence the instability of the fixed point is confirmed.
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Proposition 6. Once the critical value , is reached, the trajectories of starting on the invariant axes diverge when .
Proof. The proof is straightforward as , and our discussion given above confirms the topological equivalence between the map (7) and (11) on which the trajectories of (11) diverge when .
4.2. The Symmetric Case
In this subsection, the symmetric case , and is studied. As one can see, the map defined in (7) is symmetric. That is to say, its structure does not change if the decision variables and are swapped; i.e., , where . This means that the diagonal defined by forms an invariant manifold. It implies that the dynamics of any trajectories starting on the diagonal () will get back to the diagonal for every . In this case, the dynamic behavior of map (7) under the restriction imposed on will be described by the one-dimensional map given below.
That is to say, any synchronized trajectories (i.e., for every time step ) will be governed by , and the following proposition is raised.
Proposition 7. The map (14) is unimodal and concave, and . This means that there is a threshold at . When the model becomes unimodal, one gets the following critical point.
Proof. The proof is straightforward by looking at the function that has a global maximum without any local maximum, and hence it is unimodal. The sign of the second derivative of proves that the map is concave and finally .
Now, the map has only one fixed point, that is, , and this point is locally stable if while flip bifurcation takes place at . Figure 2(d) shows a 1D bifurcation diagram where flip bifurcation takes place at .
5. Basin of Attraction
Here, some properties of the map (7) regarding its attractive basin are given. According to [29], it is possible to see that the coordinate axes and their preimages of any rank construct boundaries for initial conditions of any nondiverging trajectories. It can be seen that the dynamics of map (7) can be restricted to the invariant axes and can be rewritten in the one-dimensional map as follows.which possesses a positive fixed point, . It is easy to see that is concave and increasing function provided that . In addition, for any points in the form , or , one gets s or . Conversely, if , then is concave and unimodal. In the last case, one can get the nonnegative points and at and , respectively. If , i.e., if , this means that trajectories along the invariant axes are bounded if the initial states lie on the line segment , where is the rank-1 preimage of the point of origin. In contrast, any other trajectories along the invariant axes starting from an initial state outside become unfeasible. Furthermore, calculating the eigenvalues at a generic point on the invariant axis gives . This means that trajectories commencing close to these invariant axes with positive initial states are repelled by those axes. In Figure 3(b), the line segments , and their inverses , are depicted at the set of parameter values , and where the point is the fourth preimage point of the origin and is given by
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It is also noted that this point is the intersection of both inverses and .
5.1. Critical Curves
Besides the fact that the map (7) is nonlinear, it is also noninvertible map, which means that the rank-1 preimages of a point may not exist or may be multivalued. As one can see, the origin point possesses four rank-1 preimages, and hence the phase plane of map (7) includes zone, the zone whose points have four rank-1 preimages. It may also include , the zone whose points possess two rank-1 preimages, and , the zone with no rank-1 preimages. To investigate these zones, let us calculate the rank-1 preimages for the points along the invariant axes and take the forms and . Substituting these points into the map (7), one gets the following algebraic systems.
Solving the second equation of (18a) gives the following: or , and then one gets two case.
Case 1. Substituting in the first equation of (18a) gives the following quadratic equation.This means that the point in the form may have no preimages or two distinct real-1 preimages.
Case 2. At , one gets the algebraic systemSolving (19b), we get the following quadratic equation.This means that in this case, the point may have no preimages or two distinct real rank-1 preimages. Combining the two cases, one can conclude that such points in the invariant axes may have four, two, or no preimages. For the interior points in the form , it is complicated to categorize such points as belonging to one of the zones , or due to the difficulty of solving the algebraic system (18a) at this point. For this reason, numerical simulation is used to calculate critical curves. The critical curves are responsible for dividing the map’s phase plane into these zones. The critical curve of rank-1 is denoted by and forms a locus of coincident points of rank-1 preimages located on a set denoted by . Since the map (7) is continuously differentiable, can be defined as the locus of points where the associated Jacobian determinant is vanished; i.e., , where is defined by (A.1) in the appendix. Therefore, represents the rank-1 image of under the map given in (7); i.e., . The expression for is cumbersome and is omitted here, and we instead do numerical simulation for presenting both and . Figure 3(c) shows that consists of two branches and while Figure 3(d) presents the critical curve that is also formed by two branches, and . It is also clear that the map (7) is not invertible as its phase plane is divided into three zones, , and .
6. Global Analysis
In economic models, global analysis is required to investigate the topological structure of basin of attraction that is not recognized by local analysis and whose qualitative changes occur due to change in parameter values. The long-term behavior of variables of these economic models given initial conditions is indeed necessary in order to show the influences that may result if these initials are taken far away from a fixed point of an attracting set. The numerical analysis shows two important aspects for the map (7), the asymmetric and symmetric cases. Let us first begin analyzing the asymmetric case by adopting the following set of parameter values: , and . This set is given in Figure 4(a) which represents a period-8 cycle with Nash equilibrium point and its attractive basins. These basins are colored orange and green while the escaping (or infeasible points) domain is plotted in gray color. It is also clear that the escaping domain forms a connecting set. By increasing the speed parameter of the first firm to while keeping the other values fixed, the dynamic of map (7) turns to a chaotic attractor that consists of two disconnecting areas as shown in Figure 4(b). It can be observed that the branch of the critical curve is moving upward. With further increase in this parameter to , a one-piece chaotic attractor emerges in Figure 4(c). Its attracting domain is displayed in white color and is separated from the infeasible domain by the segments and . At , with the other parameter values being fixed, the chaotic attractor continues to appear, the critical curve becomes tangent to the boundary , and hence contact bifurcation is raised as given in Figure 4(d). This contact takes place only when one branch of the critical curve which separates zone from zone becomes tangent to . This contact with the boundary of infeasible domain occurs from one side due to asymmetry between the two branches of map (7). This contact bifurcation becomes more visible in Figure 4(e) at the parameters set and while the other parameters are fixed. As one can see, the contact bifurcation causes disconnection in the infeasible domain because an area from zone moves into zone; i.e., , which is out of the quadrilateral . This kind of contact bifurcation is well-known (see [29]). Just after it, the area from the infeasible domain enters inside the region . As a result, the points belonging to this region have now two distinct preimages. Since this region belongs to , their preimages will belong to the same region and form holes lying in the attraction domain that is bounded by the quadrilateral shape . Indeed, only at and its preimage (that is formed by two areas connected by the branch ) have a contact with coordinate axes. It is also clear that both and are entirely located in the region . Therefore, each of them has further preimages that form smaller holes of the infeasible domain located in the attracting domain and are bounded by rank-2 preimages of the segments and .
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In contrast, the symmetric case is now investigated. This case happens at , and . This means the map (7) can be rewritten in the following form.
The map (20) is symmetric on the diagonal , and studying its dynamics through local analysis around is related to synchronization (intermittency). In this regard, one can say that a dynamic behavior of the map is synonymized if there exists a step time such that for any . Therefore, in this case, the dynamics can be studied by recalling the Jacobian of map (20) given bywhere, and . Then, the eigenvalues can be given as follows.and their corresponding eigenvectors are and , respectively. In this case, , and then the following proposition is raised.
Proposition 8. The point is locally stable provided that . At , the point undergoes flip bifurcation.
Proof. Substituting into (22), we getSimple calculations make if , and if . Combining the two conditions on gives , and this completes the proof.
From the above proposition, one can deduce that the parameter plays a key role in the stability of . This means that high reactivity of each firm with respect to a marginal change in its average profit should exist. Furthermore, the importance of the above proposition comes from its generalization when an attracting cycle for the map on exists. To see that, suppose that a k-cycle given by of the map is embedded in the invariant diagonal which corresponds to the cycle . Then, the multipliers for this k-cycle are given bywhose eigenvectors are and . As shown, the stability of is governed by and is confirmed by . In contrast, a complex chaotic attractor for the map says that is asymptotically stable if and only if all trajectories belonging to are transversely attracting. This asymptotic stability condition of is given in terms of transverse Lyapunov exponent as follows.where and is the trajectories generated by the mapThe following definitions are given [29].
Definition 1. A chaotic attractor is asymptotically stable if it is Lyapunov stable; i.e., for every neighborhood of , there exists a neighborhood of such that and the attractive basin possesses a neighborhood of .
A spectrum of Lyapunov exponents can be defined according to the initial conditions as follows [30].where denotes a Lyapunov exponent evaluated at a generic trajectory in the chaotic attractor . If is nonpositive, then a set is called Lyapunov attractor. While and , the set becomes no longer Lyapunov stable and becomes Milnor attractor. It is defined as follows.
Definition 2. A closed invariant set is called a weak attractor in Milnor sense if its stable set (the basin of attraction) has positive Lebesgue measure.
Now, we give some experiments through numerical analysis to confirm the above findings. Let us assume the following set: , and . At this set, the dynamic of map (20) is represented by period-4 cycle given in Figure 4(f). Its attractive basin and the basin of Nash equilibrium point are colored green and orange, respectively, and are bounded by and and their inverses. The infeasible domain of this cycle is given in gray color. By keeping the same set of values and increasing to 4.2199558, four closed invariant rings emerge around the diagonal as depicted in Figure 4(g). All these rings are symmetric around the invariant diagonal and lie within the region only when two of them are tangent to the branch of critical curve . By further increasing to 4.2818322, the four rings convert into a period-20 cycle as shown in Figure 4(h) whose basin of attraction is colored cyan while the basin of Nash point is denoted by orange color. At , while the other parameter values are fixed, a Milnor attractor is raised on the invariant diagonal as presented in Figure 5(a). This means the dynamics of map (20) are captured by a two-piece chaotic attractor on the invariant diagonal that is in Milnor sense. This indicates that any cycle embedded in the diagonal is transversely unstable, and hence all trajectories approaching that diagonal are exploded away by transversely unstable cycles. By further increasing to 4.763885, a one-piece chaotic attractor emerges as shown in Figure 5(b). This chaotic attractor is symmetric on the diagonal because synchronization still occurs. The symmetry of this chaotic attractor is given in Figure 5(c) where its boundary at the same set of parameter values is given by the critical curves , and . In Figure 5(d), the transverse Lyapunov exponent is given. It gives better understanding on the map’s dynamics in the long term. As one can see, the bifurcation diagram corresponding to the transverse Lyapunov exponent confirms the existence of chaotic attractor on the invariant diagonal in the interval except for windows of some periodic cycles. It is also noted that in this interval of the speed parameter , the Lyapunov exponent changes its sign many times until it eventually becomes positive for large values of the speed parameter . From economic perspective, this implies that the long-term dynamics of the map (20) can be very sharp due to small changes in this parameter. In other words, firms commencing with the same initial states may be different in the long term.
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7. Conclusion
In this paper, some dynamic characteristics such as multistability, synchronization phenomena, and complex structure of some attractive basins are investigated for a nonlinear two-dimensional noninvertible map describing a Cournot duopoly game with an invariant one-dimensional manifold where synchronization takes place. It has been proved that, with identical firms, the game’s map becomes symmetric and both coordinate axes and diagonal form invariant manifolds. The main results in this paper concern the global analysis in the asymmetric and symmetric case where synchronization has taken place. Such global analysis has focused on the properties of critical curves of the noninvertible map. The phase plane of game’s map has been divided into three zones of preimages, that is, , and . We have shown that for a set of parameter values, the one-dimensional Milnor attractor to which the synchronized trajectories arise has been obtained. After that, for increasing the value of the speed parameter, the Milnor attractor is transformed into a chaotic attractor. Indeed, weak chaotic attractor in the Milnor sense has been confirmed by transverse Lyapunov exponent.
Appendix
The Jacobian of map (7) is given by
Proof of Proposition 1. At , the above Jacobian matrix becomesand hence the eigenvalues are . It is clear that , and then is unstable node.
Proof of Proposition 2. At , the Jacobian matrix (A.1) takes the formand then the eigenvalues becomeOne can see that gives and is attainable if ; then, is stable. If , one gets ; therefore, becomes unstable saddle point.
Proof of Proposition 3. The proof is similar to that of Proposition 1.
Proof of Proposition 4. Jacobian (A.1) at Nash point has a trace and determinant given byThen, Jury conditions can be obtained as follows.Since and , then the multiplication is always nonnegative, and then condition (A.6a) is always positive. Suppose that conditions (A.6b) and (A.6c) are nonnegative; then, combining them gives , and hence Nash point is stable. The proof is completed.
Proof of Proposition 5. Jacobian (A.1) at Nash point has two eigenvalues taking the following form.These eigenvalues are real since , and . Then, the Nash point loses its stability through flip bifurcation only.
Data Availability
The data are included in the manuscript.
Conflicts of Interest
The author declares no conflicts of interest regarding the content of this article.
Acknowledgments
This research was supported by Project no. RSP-2022/167, King Saud University, Riyadh, Saudi Arabia.