Dynamic Analysis and Chaos Control of Bertrand Triopoly Based on Differentiated Products and Heterogeneous Expectations

This article investigates the dynamics of a mixed triopoly game in which a state-owned public firm competes against two private firms. In this game, the public firm and private firms are considered to be boundedly rational and naive, respectively. Based on both quantity and price competition, the game’s equilibrium points are calculated, and then the local stability of boundary points and the Nash equilibrium points is analyzed. Numerical simulations are presented to display the dynamic behaviors including bifurcation diagrams, maximal Lyapunov exponent, and sensitive dependence on initial conditions. The chaotic behavior of the two models has been stabilized on the Nash equilibrium point by using the delay feedback control method. The thresholds under price and quantity competition are also compared.

“…From Schur-Cohn stability criterion, the coefficients of the characteristic polynomial need to satisfy the following conditions [29,30]:…”

Section: Local Stability Of Equilibrium Pointsmentioning

confidence: 99%

Section: Chaos Controlmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Complex Dynamics of Mixed Triopoly Game with Quantity and Price Competition

Wang

Zhang

Huang

et al. 2021

“…Main theory Shaked and Sutton [7] Product differentiation + three-stage game model Chang [8] Product differentiation + collusion Goldberg [9] Product differentiation + oligopoly competition Zhang et al [10] Product differentiation + two-stage stochastic programming model Zhao and Du [11] Product differentiation + heterogeneous expectations Meng et al [12] Product differentiation + horizontally competitive + carbon tax Li and Chen [13], Raza and Govindaluri [14] Product differentiation + supply chain Ivanova and Ushchev [15] Product differentiation + CES model…”

Section: Researchermentioning

confidence: 99%

Research on the Influence of Bounded Rationality and Product Differentiation on the Stability of Steel Industry Market

Duan

Zeng-lin

et al. 2020

At present, the problems of homogenization and low quality in China’s iron and steel industry are particularly prominent and the ability of the enterprises to cope with change is insufficient. Adopting product differentiation strategy and dynamic adjustment strategy can allow steel enterprises and the industry to better adapt to future changes. By introducing the product differentiation degree (substitution coefficient) and the bounded rationality strategy to simulate these two strategic means, this paper constructs an extended two-stage dynamic game model to analyse the dynamic game scenarios and steel market stability in China. As new findings, we report the following: (1) The system is more likely to fall into an unbalanced state when multiple enterprises adopt the policy of dynamic output adjustment simultaneously. (2) Enterprises with large output and small output have different output adjustment policies. When enterprises with big-scale output adopt a bit larger adjustment policies, enterprises with small output will be strongly impacted, and the available adjustment space will be sharply compressed. (3) The gradual increase in the difference between products reduces the stability of the market. (4) When product differentiation and bounded rationality strategies coexist, the steel market may fall into an unbalanced state when the degree of product difference increases excessively and the enterprise adopts more drastic output adjustment policies. Therefore, there are pros and cons to product differentiation strategy and bounded rationality adjustment strategy. When each steel oligopoly enterprise formulates a production plan, it needs to comprehensively consider the output changes of the other enterprises and carefully weigh the strategic issues.

“…Ma and Si [4] introduced a continuous Bertrand duopoly game model with a two-stage delay and investigated the influence of delay parameters on the dynamic characteristics of the system. Zhao [5] studied the dynamic properties of a Bertrand game model with three oligarchs in which enterprises have heterogeneous expectations. Askar and Alkhedhairi [6] introduced two different Bertrand duopoly models where the first one is the competition of price in which each player wants to maximize its relative profit, and the second model is the classic Bertrand competition in which the players want to maximize their profits.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and Stability Analysis of a Stackelberg Mixed Duopoly Game with Price Competition in Insurance Market

Wei

Wang

et al. 2021

This paper investigates the dynamical behaviors of a Stackelberg mixed duopoly game with price competition in the insurance market, involving one state-owned public insurance company and one private insurance company. We study and compare the stability conditions for the Nash equilibrium points of two sequential-move games, public leadership, and private leadership games. Numerical simulations present complicated dynamic behaviors. It is shown that the Nash equilibrium becomes unstable as the price adjustment speed increases, and the system eventually becomes chaotic via flip bifurcation. Moreover, the time-delayed feedback control is used to force the system back to stability.