Cournot Duopoly Game Research Articles

Exploring complex dynamics in a Stackelberg Cournot duopoly game model

Abstract In this study, we investigate the dynamics of a Cournot duopoly game model with bounded rational players, incorporating a leader-follower mechanism where the first player acts as a leader and the second as a follower, aware of the leader’s production. We examine the existence and stability of all fixed points in the model and use center manifold and bifurcation theory to analyze the occurrence and direction of period-doubling and Neimark-Sacker bifurcations at the positive fixed point. To control bifurcation and chaos, feedback control and hybrid control methods are applied. Numerical examples are provided to confirm our theoretical results and reveal the model’s complex dynamics. Our results highlight the critical role of the leader’s strategic decisions, particularly the adjustment speed parameter v 1, in driving the system from stability to chaos, affecting both firms and leading to significant shifts in market dynamics.

A discrete fractional order cournot duopoly game model with relative profit delegation: Stability, bifurcation, chaos, 0-1 testing and control

Due to the memory effect, fractional order dynamical systems provide more realistic results compared with ordinary counterparts. In this study, we consider a Cournot-duopoly game model with relative profit delegation in the sense of Caputo fractional derivative. To describe richer dynamical behavior such as chaos in the model, a discrete dynamical system is needed. As a result of the discretization method based on the use of piecewise constant arguments, we obtain a two dimensional system of difference equations. The stability conditions of all equilibrium points of the discrete dynamical system are given comprehensively. The existence of the flip bifurcation in the system has been demonstrated theoretically. Lyapunov exponents and 0–1 test chaos imply that chaotic structures are formed as a result of this bifurcation. In addition, we present the chaos control technique such as Pyragas method to eliminate chaos in the model. All theoretical results dealing with the stability, bifurcation and chaos in the model are stimulated by numerical simulations.

Complex dynamics of a quantum Cournot duopoly game with two different objectives

Complex dynamics of a quantum Cournot duopoly game with two different objectives

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

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.

A Stochastic Discrete Fractional Cournot Duopoly Game: Modeling, Stability, and Optimal Control

A stochastic discrete fractional Cournot duopoly game model with a unique interior Nash equilibrium is developed in this study. Some sufficient criteria of the Lyapunov stability in probability for the proposed model at the interior Nash equilibrium are derived using the Lyapunov theory. The proposed model’s finite time stability in probability is then investigated using a nonlinear feedback control approach at the interior Nash equilibrium. The stochastic Bellman theory is also used to explore the locally optimum control problem. Furthermore, bifurcation diagrams, time series, and the 0-1 test are used to investigate the chaotic dynamics of this model. Finally, numerical examples are given to illustrate the obtained results.

Two-qubit entangling operators as chaos control in a discrete dynamic Cournot duopoly game.

The current trend in economics research is to incorporate quantum mechanical concepts to increase the security of business models. This interdisciplinary field of study represents real-world market dynamics more closely than do its classical counterparts. In this paper, we shed light on the significance of the two-qubit entangling operators in controlling chaos. We introduce a modified Eisert-Wilkens-Lewenstein scheme in a nonlinear Cournot duopoly game with complete and incomplete information. By doing so, the following interesting results are obtained: To begin, monopoly in a duopoly game can be avoided with the use of special perfect entanglers. Also, the stability analysis shows that there exists a class of entangling operators which can stabilize an unstable system and vice versa. Second, numerical analysis highlights the two-qubit entangling operators which can stabilize a chaotic system or at least delay chaos. Finally, we show that with an appropriate choice of initial state and speed of adjustments, entangling operators can decrease the sensitivity of the system. In short, while we know the importance of entangling operators in quantum game theory, in this paper we indicate the significance of operators in the context of a chaotic system.

A robust route to randomness in a simple Cournot duopoly game where ambiguity aversion meets constant expectations

In this paper we investigate the dynamics of a duopoly game with ambiguity aversion regarding uncertainty in demand and constant expectations concerning competitor production. The focus is on an asymmetric Cournot game where players engage in robust optimization and have different beliefs about the possible realizations of the random parameters of the price function. The players’ ambiguity aversion introduces multiple equilibria and instability that otherwise would not be present. The investigation of the global dynamics of the game reveals the emergence, through border-collision bifurcations, of periodic and chaotic dynamics.

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash equilibrium and the expression of commodity price, firm production, and variable profit under some key competition intensities. Furthermore, we analyze the local stability of the Nash equilibrium and derive that the equilibrium may lose its stability only through a 1:4 resonance bifurcation. Numerical simulations are conducted, through which we find that the Nash equilibrium transitions to complex dynamics through a cascade of period-doubling bifurcations. Phase portraits are also provided to illustrate more details of the dynamics, which confirm the previous theoretical finding that the Nash equilibrium loses its stability through a 1:4 resonance bifurcation.

Effect of players’ expectations and memory in a quantum Cournot game

The study of Game Theory is widely developed currently and has many applications in several fields such as economics, psychology, biology, etc. The use of quantum theory is a demonstrated way to improve the results comparing to the classic games due to the entanglement between the players. As well, it is known the positive effect of implementing long term memory in the stability of a dynamical system. In this context, memory is applied to the model of a quantum dynamical Cournot duopoly game considering two different cases, depending on the players’ expectations and the mechanisms they used to maximize their profits. In this work we analyse the game with homogeneous players, considering two boundedly rational players, and the game with heterogeneous expectations, where one of the players is boundedly rational and the other one is a naive player, comparing the results obtained in both cases. Firstly, we come to the conclusion that neither memory nor the type of players (heterogeneous or homogeneous) produces variations in the stable fixed points comparing to the quantum, or even, classic Cournot duopoly game and, therefore, Nash equilibrium is preserved. Secondly, it is observed that the game with homogeneous players, which is not deeply studied previously in quantum games, can improve the local stability of the system versus the game with heterogeneous players, under certain conditions. Finally, it is shown the role of memory as an effective mechanism of chaos control. This achievement is a remarkable economic advantage, since enables the system to reach the stability faster. Throughout this article, these statements are proved analytically and supported widely with several numerical simulations, using different values of the memory factor.

Lyapunov Stability in the Cournot Duopoly Game

This paper studies Lyapunov stability at a point and its application in the Cournot duopoly game. We first explore the relationship between Lyapunov stability at a point, nonsensitivity and non-Devaney chaos and find that a dynamical system is nonsensitive and non-Devaney chaotic if there is a point in this system such that it is Lyapunov stable at that point. We next prove a group of equivalent characterizations of Lyapunov stability at a point to deduce the composite theorems and product theorems of Lyapunov stability at a point, and then we prove three equivalent characterizations of the Cournot duopoly system to demonstrate that this system is Lyapunov stable at its unique nonzero fixed point (Cournot equilibrium point) when the unit costs of the Cournot double oligarchies satisfy certain conditions. Therefore, we conclude that the Cournot duopoly system is safe relative to both sensitivity and Devaney chaos. The robustness of our results are also verified by conducting numerical simulations in the Cournot duopoly game.

Chaotic dynamics of a quantum Cournot duopoly game with heterogeneous expectations

On the basis of the bounded rationality and adaptive expectations, we investigate the dynamics of quantum Cournot duopoly game with heterogeneous players by using the Li et al. scheme. Further, we discuss the effect of quantum entanglement on local stability of the system. The results show that: (i) The special form of quantum Cournot duopoly game is the classical Cournot duopoly game, and the quantum Cournot duopoly game with naïve expectation is also a special form of quantum Cournot duopoly game with adaptive expectation. (ii) The increase or decrease of the stable region is dependent on the degree of quantum entanglement. It is related to the difference of relative marginal cost. (iii) The numerical simulation reveals the complexity characteristics of quantum dynamic Cournot game system through the analysis of stability region, bifurcation, strange attractor and sensitivity to initial conditions, and shows the difference from the classical dynamic Cournot model.

Complexity And Chaos Control of A Cournot Duopoly Game With Relative Profit Maximization And Heterogeneous Expectations

The paper considers a Cournot-type duopoly game, in which linear demand and cost functions are used. The two players produce differentiated goods and offer them at discrete times on a common market. In the cost functions of the players, in addition to the production cost, the cost of transporting the products is also included. Each firm does not care only about its profits but also about the percentage of its opponents’ profits, using a generalized relative profit function. In this model, the players follow different strategies. More specifically, the first player is characterized as a bounded rational player while the second player follows an adaptive mechanism. The existence of the Nash Equilibrium is proved, and its stability conditions are found. The complexity that appears for some values of the game’s parameters is shown. A mechanism by which the chaotic behavior of the discrete dynamical system is presented, importing a new parameter m. The algebraic results are verified, and the apparent complexity is shown by plotting bifurcation diagrams and strange attractors, computing the Lyapunov numbers, and checking the system’s sensitivity on its initial conditions.
 Keywords: Cournot duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavior, chaos control

Fostering collusion through action revision in duopolies

We design an experiment to study the implications of allowing players to revise their actions in a Cournot duopoly game. Payoffs are determined only by the quantities selected at the end in a real-time revision game. On the other hand, in a stochastic revision game, opportunities to adjust quantities arrive randomly, and the quantities selected at the last revision opportunity are implemented. Contrasting results emerge: while real-time revision results in choices that are more competitive than the static Cournot-Nash, significantly lower quantities are implemented when revisions are stochastic. The results hold for different arrival rates of revision opportunities. Our findings demonstrate that the ability to revise actions can sustain partial cooperation. Although quantity adjustment choices display substantial heterogeneity, the main implication of the theory of revision games put forth by Kamada and Kandori (2020) is supported.

Nonlinear Dynamics of a Quantum Cournot Duopoly with Bounded Rationality and Relative Profit Maximization

Based on boundary rationality and relative profit maximization, quantum game theory is applied to develop a dynamic model of the quantum Cournot duopoly game. We investigate the effect of quantum properties on game stability and nonlinear dynamics. The results show that quantum entanglement can improve the model’s stability and control the generation of bifurcation and chaos when comparing the classical game and the game without considering relative profit maximization. It facilitates the player’s choice of quantum entanglement to control the chaos that emerges in the production output. Numerical simulations verify the chaotic properties of the game by means of bifurcations, maximum Lyapunov exponents, and phase diagrams. The results show that quantum entanglement has different effects on different games.

Complex Dynamics of a Cournot Quantum Duopoly Game with Memory and Heterogeneous Players.

Previous authors tend to consider a certain range of values of the parameters involved in a game, not taking into account other possible values. In this article, a quantum dynamical Cournot duopoly game with memory and heterogeneous players (one of them is boundedly rational and the other one, a naive player) is studied, where the quantum entanglement can be greater than one and the speed of adjustment can be negative. In this context, we analyzed the behavior of the local stability and the profit in those values. Considering the local stability, it is observed that the stability region increases in the model with memory regardless of whether the quantum entanglement is greater than one or whether the speed of adjustment is negative. However, it is also shown that the stability is greater in the negative than in the positive zone of the speed of adjustment and, therefore, it improves the results obtained in previous experiments. This increase of stability enables higher values of speed of adjustment and, as a result of that, the system reaches the stability faster, resulting in a remarkable economic advantage. Regarding the behavior of the profit with these parameters, the principal effect shown is that the application of memory causes a certain delay in the dynamics. Through this article, all these statements are analytically proved and widely supported with several numerical simulations, using different values of the memory factor, the quantum entanglement, and the speed of adjustment of the boundedly rational player.

Complex dynamics of R&D competition with one-way spillover based on intellectual property protection

In order to study the R&D competition with one-way spillover based on intellectual property protection, a R&D-Cournot duopoly game between a foreign-invested enterprise and a domestic-invested enterprise is set up in this paper. It is discussed the local stabilities near all the equilibrium points, and it is diplayed the stability conditions of each equilibrium point. Regarding the dynamical behaviors of the model, we mainly carry out our research from two aspects. The first issue we concerned about is the dynamical behaviors when there is only one enterprise engaged in R&D. We found that the coordinate axes are invariant sets, on which the dynamical behaviors are governed by two different one-dimensional maps. The dynamical behaviors of both enterprises engaged in R&D is another topic of our research. We find that not only the local bifurcations can cause the sudden change of dynamics, but also the global bifurcations can lead to the “death” of attractors or the disappearance of coexistence of multiple attractors. Moreover, the stability exchange between different attractors caused by the transcritical bifurcation is also common in our model. The stability and evolutionary pattern of the kinetic evolutionary behavior revealed with the change of the adjustment parameters is closely related to the long-term effects of strengthening intellectual property protection. It is of high practical value to study the impact of intellectual property protection intensity on R&D investment of high-tech enterprises.

Stability and global dynamics of a quantum Cournot duopoly game with isoelastic demand

Stability and global dynamics of a quantum Cournot duopoly game with isoelastic demand

Global dynamics of an oligopoly competition model with isoelastic demand and strategic delegation

In this paper, we propose a dynamic Cournot duopoly game model based on isoelastic demand and strategic delegation. The local stability conditions of each equilibrium point of the system are discussed. We also investigate the influence of parameters on system stability, and we find that with smaller speed of adjustment, relatively large price elasticity, relatively small profit weight coefficient and marginal cost, the duopoly is more stable. Then the problem of coexistence of attractors is studied through the basin of attraction, and it is found that the jump phenomenon in the one-dimensional bifurcation diagram is due to coexistence of attractors. Finally, the structure of the basins of attraction and the global dynamics of the system are analyzed by using critical curves. It is found that when a critical curve or an attractor contact with the boundary of the basin of attraction, global bifurcations occur in the system.

On the dynamics of Cournot duopoly game with private firms: Investigations and analysis

In this paper, we introduce a Cournot duopoly game whose players’ inverse prices are isoelastic functions and are derived from a simple quadratic utility function. The proposed game consists of two competing firms seeking the optimum of their productions by maximizing weighted objectives that are profit and social welfare under the same marginal costs. Based on those objectives, the bounded rationality is adopted to construct the model describing the game’s evolution. A unique fixed point which is Nash equilibrium point is obtained for the game. The stability conditions for this point are discussed and shown that the equilibrium point can be unstable through flip bifurcation. In our discussion, we show that the model’s map is noninvertible and belongs to Z3−Z1 type. Numerical experiments for the dynamics of the map are performed and shown that the map possesses several stable attractors. Furthermore, the shapes of some attractive basins for the map are analyzed providing the existence of the so-called lobes.