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.

Green technology acquisition strategies for chemical enterprises considering competition effects and spillover effects

PurposeGreen technology, characterized by its environmentally friendly attributes and sustainable practices, has emerged as a crucial tool in harmonizing the economic and ecological benefits. However, the challenge lies in selecting the most effective strategies for acquiring green technology. This paper aims to explore how chemical enterprises choose green technology acquisition strategies across diverse scenarios.Design/methodology/approachConsidering the influence of competition effects, spillover effects and their interactions on selecting green technology acquisition strategies, this paper develops three decision models (independent R&D, cooperative R&D and technology introduction). Drawing on the duopoly game theory as its theoretical framework, this paper delves into the examination of the economic and environmental benefits within distinct scenarios.FindingsCooperative R&D excels in promoting green technology R&D when spillover effects are strong, while independent R&D demonstrates superiority when spillover effects are weak. The threshold for the strength of spillover effects is related to competition effects. Additionally, cooperative R&D typically yields greater financial advantages than independent R&D and technology introduction. Moreover, the economic and environmental benefits may not be optimized simultaneously. Only enterprises that satisfy low competition and spillover effects as well as high competition and spillover effects, can achieve win-win economic and environmental benefits.Originality/valueAlthough green technology R&D and introduction are alternative strategies, they have typically been considered separately in prior literature. This study attempts to incorporate green technology R&D and introduction into a strategic system to investigate the selection of green technology acquisition strategies, taking into account competition effects, spillover effects and their interactions.

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

Local stability and convergence factors of nonlinear duopoly games with different types of players

Local stability and convergence factors of nonlinear duopoly games with different types of players

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.

R&D investment and network effects: A dynamic model of the search engine market

ABSTRACT This paper considers R&D investment for quality improvement in a dynamic duopoly game in the search engine market. The aim is to determine the circumstances under which asymmetry between search engines (in terms of quality) tends to increase or decrease over time. I show that when the future sufficiently matters, the low-quality search engine can catch up with the high-quality engine by investing in quality. In contrast, when only the present matters, the high-quality search engine wins and the structure of the market transitions to a monopoly. I also show that a shift in asymmetry between firms can result in either an increase or decrease in consumer welfare, depending on the degree of substitutability between the two search engines.

Dynamics of a Cournot game with bounded rational firms and various scale effects.

In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model's resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms' marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria if firms face economies of scale. The heterogeneous case where firms' marginal costs change at distinct rates is much more complex, thus we investigate the duopoly game with only two firms involved. Methods of symbolic computations such as triangular decomposition and partial cylindrical algebraic decomposition are employed in the analytical investigations of the equilibrium, which is nearly impossible by using the pencil-and-paper approach since the closed-form equilibrium is quite complicated. According to the computational results, we derive that two stable positive equilibria may coexist if both firms face economies of scale. Additionally, we conduct preliminary numerical simulations and find two different types of complex dynamics of the model considered in this paper: complex trajectories such as periodic and chaotic orbits may appear; the topological structure of the basins of attraction may be complex.

A Game Model Analysis of the International Digital Service Tax in an Asymmetric Market Duopoly

Against the backdrop of the rapidly expanding digital economy, multinational corporations are increasingly exploiting information asymmetry in the market to employ covert and diverse methods of tax avoidance. This poses a significant challenge to tax collection and administration. The proposal of a digital service tax aims to adapt the international tax system to the digitalization of the economy, ensuring fair and reasonable tax payments. To address these challenges, this article assumes an asymmetric international digital economy market and a domestic market characterized by duopoly. It establishes an asymmetric market duopoly game, calculates the digital gains of each country's market using the logistic function, determines the Nash equilibrium of the data service tax game, and analyzes the relationship between the digital service tax rates of strong and weak countries in the digital economy.

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.

Fractional-order quantum game chaotic map and its synchronization with application

In this paper, according to the knowledge of discrete fractional calculus and quantum game chaotic map, we first get the fractional quantum game chaotic map, which describes the dynamics of a quantum duopoly game. Then, the phase space, maximum Lyapunov exponent, and bifurcation diagram of the proposed map are obtained for the purpose of observing dynamic behaviors compared with the original quantum game chaotic map. After that, the synchronous controller is designed and the chaos synchronization of the fractional quantum game chaotic map is observed. Finally, we develop the fractional quantum game chaotic map into the chaotic sequence generator and image encryption scheme.

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.

Cournot–Bertrand Duopoly Model: Dynamic Analysis Based on a Computed Cost

In this paper, some mathematical properties and dynamic investigations of a Cournot–Bertrand duopoly game using a computed nonlinear cost are studied. The game is repeated and its evolution is presented by noninvertible map. The fixed points for this map are calculated and their stability conditions are discussed. One of those fixed points is Nash equilibrium, and the discussion shows that it can be unstable through flip and Neimark–Sacker bifurcation. The invariant manifold for the game’s map is analyzed. Furthermore, the case when both competing firms are independent is investigated. Due to unsymmetrical structure of the game’s map, global analysis gives rise to complicated basin of attraction for some attracting sets. The topological structure for these basins of attraction shows that escaping (infeasible) domain for some attracting sets becomes unconnected and the rise of holes is obtained. This confirms the existence of contact bifurcation.

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.

Nonlinear dynamic in a remanufacturing duopoly game: spectral entropy analysis and chaos control

<abstract><p>In this study, our focus is on stabilizing a competitive game involving an original equipment manufacturer (OEM) and a third-party remanufacturer (TPR). To assess the presence of chaos within the dynamics of this game, we employ various analytical tools, including spectral entropy (SE), bifurcation diagrams, and Lyapunov exponents. The unpredictable nature of chaotic dynamics significantly influences the market and has negative implications for the strategic decisions of both firms. Our approach to counteracting this chaotic behaviour and stabilizing the system revolves around the implementation of the Ott, Grebogi, and Yorke (OGY) method. Crucially, our analysis highlights that the marginal costs ($ c_n $ and $ c_r $) incurred by the OEM and TPR emerge as pivotal factors in achieving stabilization within the game. To provide a tangible demonstration of the effectiveness of our proposed stabilization strategy in the context of this competitive environment, we conducted numerical simulations.</p></abstract>

Comparison between Simultaneous and Sequential Movements in a Duopoly Game

In general, a Stackelberg competition benefits the leader and the consumer due to making the first move in a sequential game. However, in this paper, sequential movement does not offer an advantage over simultaneous movement in a static duopoly under quantity competition with an isoelastic demand function. By introducing dynamics into the duopoly model through numerical simulations of simultaneous and sequential movements towards the Cournot equilibrium, there is evidence of first-mover advantage. However, simultaneous movement is advantageous over sequential movement in terms of better profits for both firms and lower prices for the consumer.

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.

Strategic investment under uncertainty: why multi-option firms lose the preemption run

We consider a dynamic duopoly game where firms choose both the timing and size of their investments. The existing real options literature predominantly consists of contributions where firms have a single option to invest. This paper relaxes this assumption by giving Firm A multiple options to undertake further investments with the purpose to expand whereas Firm B only holds the option to enter the market. In this asymmetric setting we get the surprising result that, in equilibrium, Firm B invests first. If Firm A invests first, Firm A and Firm B keep on being involved in preemption games for subsequent investments until Firm B enters the market, which leads to inefficiently early investments of Firm A. When Firm B invests first, then only one preemption game is played, which leads to Firm A being free to choose its unrestricted optimal investment moments.

Mixed-strategy equilibrium of the symmetric production in advance game: The missing case

The mixed-strategy equilibrium of the symmetric production-in-advance type capacity-constrained Bertrand-Edgeworth duopoly game has not been derived analytically over the entire range of intermediate capacities in the literature. In this paper we derive for the missing region a symmetric mixed-strategy equilibrium analytically.

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.