scholarly journals Analysis of Price Stackelberg Duopoly Game with Bounded Rationality

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Lian Shi ◽  
Yun Le ◽  
Zhaohan Sheng
Keyword(s):  
Bounded Rationality ◽  
Stackelberg Game ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Price Adjustment ◽  
Stable Region ◽  
Largest Lyapunov Exponent ◽  
Existence And Stability ◽  
Theoretical And Numerical Analysis ◽  
Stackelberg Duopoly

The classical Stackelberg game is extended to boundedly rational price Stackelberg game, and the dynamic duopoly game model is described in detail. By using the theory of bifurcation of dynamical systems, the existence and stability of the equilibrium points of this model are studied. And some comparisons with Bertrand game with bounded rationality are also performed. Stable region, bifurcation diagram, The Largest Lyapunov exponent, strange attractor, and sensitive dependence on initial conditions are used to show complex dynamic behavior. The results of theoretical and numerical analysis show that the stability of the price Stackelberg duopoly game with boundedly rational players is only relevant to the speed of price adjustment of the leader and not relevant to the follower’s. This is different from the classical Cournot and Bertrand duopoly game with bounded rationality. And the speed of price adjustment of the boundedly rational leader has a destabilizing effect on this model.

scholarly journals Complexity Analysis of a 3-Player Game with Bounded Rationality Participating in Nitrogen Emission Reduction

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Jixiang Zhang ◽  
Xuan Xi
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Bounded Rationality ◽  
Control Method ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Stable Region ◽  
Nitrogen Emissions ◽  
Nash Equilibrium Point ◽  
The Stability

In this paper, a decision-making competition game model concerning governments, agricultural enterprises, and the public, all of which participate in the reduction of nitrogen emissions in the watersheds, is established based on bounded rationality. First, the stability conditions of the equilibrium points in the system are discussed, and the stable region of the Nash equilibrium is determined. Then, the bifurcation diagram, maximal Lyapunov exponent, strange attractor, and sensitive dependence on the initial conditions are shown through numerical simulations. The research shows that the adjustment speed of three players’ decisions may alter the stability of the Nash equilibrium point and lead to chaos in the system. Among these decisions, a government’s decision has the largest effect on the system. In addition, we find that some parameters will affect the stability of the system; when the parameters become beneficial for enterprises to reduce nitrogen emissions, the increase in the parameters can help control the chaotic market. Finally, the delay feedback control method is used to successfully control the chaos in the system and stabilize it at the Nash equilibrium point. The research of this paper is of great significance to the environmental governance decisions and nitrogen reduction management.

scholarly journals Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Junhai Ma ◽  
Hongliang Tu
Keyword(s):  
Nash Equilibrium ◽  
Bounded Rationality ◽  
Electricity Market ◽  
Complex Dynamics ◽  
Fractal Dimensions ◽  
Practical Significance ◽  
Stable Region ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Game Model

According to a triopoly game model in the electricity market with bounded rational players, a new Cournot duopoly game model with delayed bounded rationality is established. The model is closer to the reality of the electricity market and worth spreading in oligopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Its complex dynamics is demonstrated by means of the largest Lyapunov exponent, bifurcation diagrams, phase portraits, and fractal dimensions. Since the output adjustment speed parameters are varied, the stability of Nash equilibrium gives rise to complex dynamics such as cycles of higher order and chaos. Furthermore, by using the straight-line stabilization method, the chaos can be eliminated. This paper has an important theoretical and practical significance to the electricity market under the background of developing new energy.

The chaotic dynamics of a quantum Cournot duopoly game with bounded rationality

2020 ◽  
Vol 18 (06) ◽  
pp. 2050029
Author(s):  
Xinli Zhang ◽  
Deshan Sun ◽  
Wei Jiang
Keyword(s):  
Quantum Entanglement ◽  
Stability Region ◽  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Largest Lyapunov Exponent ◽  
Cournot Duopoly ◽  
The Stability ◽  
The Impact ◽  
Rational Players

This paper analyzes the chaotic dynamics of a quantum Cournot duopoly game with bounded rational players by applying quantum game theory. We investigate the impact of quantum entanglement on the stability of the quantum Nash equilibrium points and chaotic dynamics behaviors of the system. The result shows that the stability region decreases with the quantum entanglement increasing. The adjustment speeds of bounded rational players can lead to chaotic behaviors, and quantum entanglement accelerates the bifurcation and chaos of the system. Numerical simulations demonstrate the chaotic features via stability region, bifurcation, largest Lyapunov exponent, strange attractors, sensitivity to initial conditions and fractal dimensions.

scholarly journals Complexity and Entropy Analysis of a Multi-Channel Supply Chain Considering Channel Cooperation and Service

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 970 ◽  
Author(s):  
Qiuxiang Li ◽  
Xingli Chen ◽  
Yimin Huang
Keyword(s):  
Supply Chain ◽  
Stackelberg Game ◽  
Equilibrium Points ◽  
Service Level ◽  
Stability Domain ◽  
Largest Lyapunov Exponent ◽  
Game Model ◽  
Nash Game ◽  
Profit Distribution ◽  
The Stability

In this paper, based on the background of channel cooperation and service of the supply chain, this paper constructs a Nash game model and a Stackeberg game model in the multi-channel supply chain considering an online-to-store channel (OSC). Based on maximizing the profits and the bounded rationality expectation rule (BRE), this paper builds a dynamic game model, respectively, and analyzes the stability of the equilibrium points by mathematical analysis and explores the influences of parameters on stability domain and entropy of the system by using bifurcation diagram, the entropy diagram, the largest Lyapunov exponent and the chaotic attractor etc. Besides, the influences of service level and profit distribution rate on system’s profit are discussed. The theoretical results show that the greater the service level and profit distribution rate are, the smaller the stability domain of the system is; the system will go into chaotic state and the system’s entropy will increase when operators adjust her/his price decision quickly; when the manufacturer or the retailer keeps service level in the appropriate value which is conducive to maximizing her/his profit; the manufacturer should carefully set the service level of OSC to ensure the system’s profit; in Nash game model, the stability of the system weakens than that in Stackelberg game model. Furthermore, this paper puts forward some suggestions to help the manufacturer and retailer in multi-channel supply chain to do the better decision.

scholarly journals The Complex Dynamics of Bertrand-Stackelberg Pricing Models in a Risk-Averse Supply Chain

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Junhai Ma ◽  
Qiuxiang Li
Keyword(s):  
Supply Chain ◽  
Complex Dynamics ◽  
Stackelberg Game ◽  
Price Adjustment ◽  
Stable Region ◽  
Uncertain Demand ◽  
Risk Averse ◽  
Pricing Models ◽  
Increase With Increase ◽  
Adjustment Speed

We construct dynamic Bertrand-Stackelberg pricing models including two manufacturers and a common retailer in a risk-averse supply chain with the uncertain demand. The risk-averse supply chain follows these strategies: Bertrand game between the two manufacturers and Stackelberg game between the manufacturer and the retailer. We study the effect of the price adjustment speed, the risk preference, and the uncertain demand on the stability of the risk-averse supply chain using bifurcation, power spectrum, attractor, and so forth. It is observed that there exists slip bifurcation when the price adjustment speed across some critical value, the stable region, and total profit of the risk-averse supply chain will increase with increase ofRM1and decrease with increase ofσ. The profit of the supply chain and the two manufacturers will decrease and the weaker (retailer) is a beneficiary when the supply chain is in chaos. The fluctuation in the supply chain can be gradually controlled by the control of the price adjustment speed.

scholarly journals Complexity Dynamic Character Analysis of Retailers Based on the Share of Stochastic Demand and Service

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Junhai Ma ◽  
Weiya Di ◽  
Hao Ren
Keyword(s):  
Stochastic Demand ◽  
Stable State ◽  
Service Level ◽  
Price Adjustment ◽  
Stable Region ◽  
Largest Lyapunov Exponent ◽  
Market Demand ◽  
Price Fluctuation ◽  
Adjustment Speed ◽  
The Stability

Apart from the price fluctuation, the retailers’ service level becomes another key factor that affects the market demand. This paper depicts a modified price and demand game model based on the stochastic demand and the retailer’s service level which influences the market demand decided by customers’ preference, while the market demand is stochastic in this model. We explore how the price adjustment speed affects the stability of the supply chain system with respect to service level and stochastic demand. The dynamic behavior of the system is researched by simulation and the stability domain and the bifurcation phenomenon are shown clearly. The largest Lyapunov exponent and the chaotic attractor are also given to confirm the chaotic characteristic of the system. The simulation results indicate that relatively small price adjustment speed may maintain the system at stable state. With the price adjustment speed gradually increasing, the price system gets unstable and finally becomes chaotic. This chaotic phenomenon will perturb the product market and this phenomenon should be controlled to keep the system stay in the stable region. So the chaos control is done and the chaos can be controlled completely. The conclusion makes significant contribution to the system referring to the price fluctuation based on the service level and stochastic demand.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

A novel methodology to explore the periodic gait of a biped walker under uncertainty using a machine learning algorithm

Robotica ◽  
2021 ◽  
pp. 1-16
Author(s):  
Namjung Kim ◽  
Bongwon Jeong ◽  
Kiwon Park
Keyword(s):  
Limit Cycle ◽  
Systematic Approach ◽  
Learning Algorithm ◽  
Initial Conditions ◽  
Angular Displacement ◽  
External Disturbance ◽  
Rehabilitation Robotics ◽  
Stable Region ◽  
Gait Rehabilitation ◽  
External Disturbances

Abstract In this paper, we present a systematic approach to improve the understanding of stability and robustness of stability against the external disturbances of a passive biped walker. First, a multi-objective, multi-modal particle swarm optimization (MOMM-PSO) algorithm was employed to suggest the appropriate initial conditions for a given biped walker model to be stable. The MOMM-PSO with ring topology and special crowding distance (SCD) used in this study can find multiple local minima under multiple objective functions by limiting each agent’s search area properly without determining a large number of parameters. Second, the robustness of stability under external disturbances was studied, considering an impact in the angular displacement sampled from the probabilistic distribution. The proposed systematic approach based on MOMM-PSO can find multiple initial conditions that lead the biped walker in the periodic gait, which could not be found by heuristic approaches in previous literature. In addition, the results from the proposed study showed that the robustness of stability might change depending on the location on a limit cycle where immediate angular displacement perturbation occurs. The observations of this study imply that the symmetry of the stable region about the limit cycle will break depending on the accelerating direction of inertia. We believe that the systematic approach developed in this study significantly increased the efficiency of finding the appropriate initial conditions of a given biped walker and the understanding of robustness of stability under the unexpected external disturbance. Furthermore, a novel methodology proposed for biped walkers in the present study may expand our understanding of human locomotion, which in turn may suggest clinical strategies for gait rehabilitation and help develop gait rehabilitation robotics.

scholarly journals Complex Characteristics of Multichannel Household Appliance Supply Chain with the Price Competition

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Junhai Ma ◽  
Wandong Lou
Keyword(s):  
Supply Chain ◽  
Supply Chains ◽  
Price Competition ◽  
Domain Of Attraction ◽  
Bullwhip Effect ◽  
Price Adjustment ◽  
Stable Region ◽  
Supply Chain System ◽  
Household Appliance ◽  
Internet Channel

This paper studies the complex characteristics caused by the price competition in multichannel household appliance supply chains. We consider a two-level household appliance supply chain system consisting of a manufacturer with an Internet channel and a retailer with a traditional channel and an Internet channel. Each channel’s price-setting follows the bounded rational decision process in order to obtain the optimal profit or more market share. Considering that the price competition often leads to the demand and order fluctuation, we also investigate the bullwhip effect of the multichannel supply chains on the basis of the order-up-to-inventory policy. From the numerical simulation, we find a system in a chaotic state will suffer larger bullwhip effect than a stable system, and the manufacturer’s Internet channel is helpful to mitigate the bullwhip effect. Our results provide some useful managerial inspirations for the household manufacturer and retailers. Firstly, each channel should make their retail price with a suitable price adjustment speed in the stable region, and each time pricing cannot exceed the domain of attraction. Secondly, the manufacturer can adopt a more radical pricing strategy in their Internet channel to mitigate the bullwhip effect. Thirdly, the price adjustment should be reviewed and be appropriately reduced if the price adjustment is too large.

A new hyperchaotic particle motion system and its control

2019 ◽  
Vol 30 (12) ◽  
pp. 2050004
Author(s):  
Ning Cui ◽  
Junhong Li
Keyword(s):  
Particle Motion ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Multiple Equilibrium ◽  
Hopf Bifurcations ◽  
Hyperchaotic System ◽  
Motion System ◽  
The Rich ◽  
Theoretical Analyses ◽  
Multiple Equilibrium Points

This paper formulates a new hyperchaotic system for particle motion. The continuous dependence on initial conditions of the system’s solution and the equilibrium stability, bifurcation, energy function of the system are analyzed. The hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane are also investigated. It shows that the rich dynamic behaviors of the system, including the degenerate Hopf bifurcations and nondegenerate Hopf bifurcations at multiple equilibrium points, the irregular variation of Hamiltonian energy, and the hyperchaotic attractors. These results generalize and improve some known results about the particle motion system. Furthermore, the constraint of hyperchaos control is obtained by applying Lagrange’s method and the constraint change the system from a hyperchaotic state to asymptotically state. The numerical simulations are carried out to verify theoretical analyses and to exhibit the rich hyperchaotic behaviors.

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