Duopoly Game Research Articles

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.

Collective Sensitivity, Collective Accessibility, and Collective Kato’s Chaos in Duopoly Games

By using the uniform continuity of two onto maps, this paper further explores stronger forms of Kato’s chaos, sensitivity, and accessibility of Cournot maps. In particular, the sensitivity, the collective sensitivity, the accessibility, and the collective accessibility of the compositions of two reaction functions are studied. It is observed that a Cournot onto map H on a product space is sensitive (collectively sensitive, collectively accessible, accessible, or collectively Kato chaotic) if and only if the restriction of the map H2 to the MPE-set is sensitive as well. Several examples are given to show the necessity of the reaction functions being continuous onto maps.

Chaotic Dynamics in Joint Price QoS Game with Heterogeneous Internet Service Providers

This paper tries to investigate the complex characteristics of the communication market where Internet service providers (ISP) lease network access services and compete to serve a large pool of subscribers. For this purpose, we analyze the dynamics of a mixed duopoly game with two decision parameters: price and quality of service (QoS). We calculate and discuss the stability of each equilibrium solution by using the nonlinear system. A numerical simulation is used to show the flip bifurcation to chaos by the decisions of ISPs with different statuses. We discovered that the Nash equilibrium loses stability when the speed of adjustment and transmission fee increase. We show that the system parameter changes the stability of the communication market. In addition, we use a control method to keep the communication market in a stable state.

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.

A Heterogeneous Duopoly Game under an Isoelastic Demand and Diseconomies of Scale

In this paper, we investigate a duopolistic market with heterogeneous firms under the assumptions of an isoelastic demand and quadratic costs. We obtain the sufficient and necessary condition of the local stability of the Cournot–Nash equilibrium and analytically compare it with that of the analogue model under linear rather than quadratic costs. By approaches of symbolic computation, we prove that diseconomies of scale have an effect of stabilizing the game provided that the cost parameters are large enough. Moreover, by means of numerical simulations, we find that our model loses its stability only through a period-doubling bifurcation, which is different from its analogue having two possible routes to chaotic dynamics.

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

Horizontal outsourcing and price competition: The role of sole sourcing commitment

Previous studies on horizontal outsourcing between competing duopolists emphasize cost factors, such as economies of scale and/or variable cost advantages in Cournot markets, as potential explanations. Our paper studies horizontal outsourcing when two competing firms engage in Bertrand competition, and highlights the important role of sole sourcing commitment. We adopt the framework of a multistage duopoly game that comprises of an incumbent and an entrant. The incumbent has the capability to make a key component in‐house, and the entrant, who is a rival of the incumbent in the downstream market, can source the component from either the incumbent or a supplier not participating in the end‐product market. We find that if the entrant commits to sole sourcing, horizontal outsourcing can occur when the incumbent has a component cost advantage or even a small cost disadvantage over the alternative supplier. Specifically, if the component cost gap is small, horizontal outsourcing may soften downstream market competition and benefit both firms at the expense of inducing higher prices for the consumers. If the incumbent has a significant cost advantage, horizontal outsourcing may lead to an increased downstream price war by expanding the total supply of end products. Without sole sourcing commitment, horizontal outsourcing occurs only when the incumbent has a cost advantage, and it always strengthens downstream price competition. By contrast, if the firms engage in downstream Cournot competition, sole sourcing commitment has no impact on the adoption of horizontal outsourcing, and the entrant sources from her rival only when the incumbent enjoys a significant cost advantage. Finally, we also study various extensions to check and confirm the robustness of our main results to key model assumptions.

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.

Endogenous spillovers in a non‐tournament R&D duopoly under uncertainty

Including R&D risk, this paper considers the choices of R&D spillovers in a simple non‐tournament cost‐reducing R&D duopoly game with Bertrand competition. It turns out that the two firms never disclose any of their R&D information when considering their R&D non‐cooperatively. However, if the firms decide their R&D cooperatively, we show, though they would always fully share their information when the risk of R&D is low, they would not disclose any of their R&D information when the market competition is fierce, the R&D risk is high and the R&D is efficient.

Quantum Cournot duopoly game with general isoelastic demand and asymmetric production costs

This letter explores the quantum version of a Cournot duopoly game with general isoelastic demand and asymmetric production costs by applying Li-Du-Massar's minimal quantization rules, and it especially analyzes the existence region of quantum equilibrium, and the influences of quantum entanglement , difference in marginal costs (k) and elasticity of demand on the optimal profits of both firms. The results show that the existence region decreases with γ and k increasing. A larger elasticity of demand can destroy the profits of both firms. If positive γ and k are more favourable to the profits of two firms. If the first firm's profit increases with γ increasing for fixed k, but decreases with k increasing for fixed γ. The second firm's profit increases with k increasing for any fixed γ. As to the influences of γ on the second firm's profit, when k is less than a critical value, it increases with γ increasing, otherwise it decreases with γ increasing for fixed k.

Dynamics, synchronization control of a class of discrete quantum game chaotic map

In this work, A four-dimensional (4D) discrete chaotic map model based on quantum-Cournot duopoly game is proposed, and the dynamics of this map using bounded rationality are studied. It is observed that the stable region of mapping equilibrium increases with the increase of quantum entanglement γ. and the occurrence time of bifurcation and chaos delays with the increase of mapped Nash equilibrium. In addition, A novel synchronization control approach which adopts both linear and nonlinear feedback strategies is proposed by constructing a dynamic controller. The simulation results verify the effectiveness of the proposed method.

Dynamic investigations in a Stackelberg model with differentiated products and bounded rationality

This work focuses on a Stackelberg duopoly game characterized by firms with differentiated products and bounded rationality. Both firms adopt myopic mechanism to update their production decisions in each discrete period. For the purpose of stressing hierarchy between both firms, the leader firm is supposed to estimate marginal profit by only thinking over his own interfering factors. Since global stability analysis of Nash equilibrium point becomes desirable in complex behaviors research, we cover it as well besides local stability. Numerical simulations are carried out to verify some essential appearances of previous theoretical analyses. Such aspects include maximum Lyapunov exponent, strange attractor, basin of attraction and sensitivity on initial conditions. In addition, we calculate fractal dimension of the non-integral dimension shapes in chaotic state. As to the chaos emerged under certain conditions, effective control method has been utilized to force them back towards stability.

Understanding Dynamics and Bifurcation Control Mechanism for a Fractional-Order Delayed Duopoly Game Model in Insurance Market

Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies.

(In)significance of entangling operators in the noisy duopoly games

(In)significance of entangling operators in the noisy duopoly games

A green path towards sustainable development: Optimal behavior of the duopoly game model with carbon neutrality instruments

AbstractCarbon emission is a global issue, and China is facing many problems to resolve it, especially in emission‐generating companies (EGCs). To maintain sustainable development, it is necessary to reduce carbon emissions for the well‐being of society, and carbon neutrality can play an important role. With China targeting the goal of carbon neutrality to reduce carbon emissions, this research aims to bridge the gap in the literature by considering carbon neutrality instruments in a duopoly market. The study sets up a mathematical model of the duopoly game to achieve desired objectives and then applies a simulation‐based optimization technique to examine the mathematical model under different experiments. After manipulating various scenarios to find the optimal values of carbon neutrality instruments, the results indicate that green bond provision, emission tax, and emission quota play a vital role in carbon neutrality. Different combinations of carbon neutrality instruments reduce various levels of carbon emissions. We further show that a combination of emission tax upon EGCs plays a key role in reducing carbon emissions during the fifth scenario. In the second scenario, a combination of green bond provision and emission tax reduces carbon emissions, but is lower than the fifth scenario. This research offers a reference for policymakers to move towards carbon neutrality, and it is beneficial for the government to decide the optimal values of carbon neutrality instruments.

Differential quantum duopoly games

We investigate the quantization of differential duopoly games, which is a dynamic quantum game with a constraint of ordinary differential equation. By solving such differential quantum game, we analyze the properties of the equilibrium solution. The numerical simulations show that the quantum objective functions are negatively correlated with the entanglement level.

A competition of duopoly game whose players are public: Dynamic investigations

Based on a generalized nonlinear price function and consumer surplus, a Cournot duopoly game whose players want to maximize their profits and social welfare is introduced in this paper. Just like studies in literature, the game is modeled by a nonlinear two-dimensional discrete map. Due to the general price function, the dynamics for the model such as local and global dynamic behaviors, synchronization and multi-stability are investigated at some special cases. The studies and the formation of game in this paper generalize other studies in literature. The general forms of the equilibrium points including Nash one are calculated. Analytically, the conditions of stability for those points are investigated. Focusing on Nash point and its dynamics, it is confirmed that the map’s manifold can be studied using a one-dimensional map that is similar to the famous logistic map. It is confirmed that the game’s map is noninvertible and belongs to the Z4−Z2−Z0 type. The global bifurcation is analyzed using critical curves for some attracting sets and chaotic attractors. The symmetry of the game makes the diagonal line form an invariant set and hence synchronization is discussed.

Complex dynamics investigations of a mixed Bertrand duopoly game: synchronization and global analysis

Complex dynamics investigations of a mixed Bertrand duopoly game: synchronization and global analysis