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This paper, considering risk aversion and fair concern, establishes a dynamic price game model of a dual-channel supply chain in which dual-channel retailer sells products through traditional channel and online channel and the online retailer only sells products through online channel. The stability of the system and the influences of different parameter values on utilities are analyzed emphatically using game theory and nonlinear dynamic theory, such as 2D and 3D bifurcation diagram, parameter plot basin, chaos attractor, and sensitivity to initial value. The results find that the system is more likely to lose stability and fall into chaos with the customer demand fluctuating greatly. The system enters into chaos through flip bifurcation with the increase of the price adjustment speed; adjusting the risk-aversion levels or the fairness concern levels of the two retailers can make the system be in a stable state or delay the occurrence of system instability. When the system is in chaos, the average utility of the online retailer will decrease and one of the dual-channel retailers will increase. Using the state feedback control method, the system can return to a stable state from chaos by selecting appropriate control parameters. The research of this paper is of great significance to the decision-makers’ price decision and supply chain operation management.

The rapid development of information technology and the Internet has provided good conditions and development space for the establishment of online channel. The dual-channel sales model is conducive to expanding market share and increasing profits, as well as exacerbating channel competition [

Behavioral factors have great influence on the operation decisions of the supply chain. Related literatures have studied the optimal decision-making, contract coordination, and profit distribution of the risk-aversion supply chain. When decision-makers have risk-aversion behavior, standard repurchase contracts or profit distribution contracts cannot coordinate the supply chain [

Behavioral studies have found that people pay great attention to fairness in real life, and many empirical or experimental studies have confirmed the existence of fair concern behavior tendencies. Under the influence of fair concern behavior, when competitors feel unfair, they will take actions to achieve the purpose of punishing the other party at the expense of their own interests [

The interest distribution between suppliers and retailers will trigger the occurrence of fair concern behavior tendency. Cui et al. [

Zhang and Ma [

The dynamic game model and its evolutionary mechanism have always been a hot issue for scholars. Puu [

In view of this, based on the dual-channel supply chain environment, considering that the two retailers have risk-aversion and fair concern behaviors, this paper will construct a dynamic price game model of dual-channel supply chain using game theory and nonlinear dynamic theory and focus on the analysis of the stability characteristics of equilibrium point and the effect of parameters on the system stability and profits. On this basis, the state feedback control method is used to control the chaotic behavior of the system. The research in this paper has great guidance significance for decision-maker’s price decision and supply chain operation management.

In this paper, we study a dual-channel supply chain in which the two retailers have risk-aversion behavior and fair concern behavior, the dual-channel retailer sells products through traditional channel and online channel, and the online retailer only sells products through online channel (as shown in Figure

Dual-channel supply chain.

where

The two retailers have limited rationality, risk-aversion behavior, and fair concern behavior

Market demand

The two retailers have the same product cost (

The demand functions of the two retailers are expressed as follows:

Suppose that the price decisions of the two retailers occur within the discrete time period

Because of the uncertainty of customer’s demand, the two retailers have financial risks and should consider the influence of the risk attitude on price decision. The exponential utility equation, which has important application in financial risk assessment and decision-making theory, is used to measure the impact of risk-aversion behavior on retailers’ utility acquisition. The exponential utility equation is expressed as follows:

Du et al. [

Cui et al. [

According to the actual market situation, the fair feeling of the dual-channel retailer is related to the comparison of the absolute profits of the two retailers on the online channel; the fair feeling of the online retailer depends on the comparison with the relative profit and the absolute profit of the dual-channel retailer. Thus, the utility function equations of the two risk-averse retailers can be obtained in a discrete time period

The marginal utilities of the two retailers can be obtained by taking the first-order partial derivatives of utility functions for

The two retailers can only get part of the market information when they make decisions. Therefore, the two retailers will adjust the price decisions of the next period according to the marginal utility of this period. When the marginal utility of this period is greater than zero, the retailer will increase the price adjustment speed in the next period; in contrast, when the marginal utility of this period is less than zero, the retailer will reduce the price adjustment speed in the next stage. The price adjustment mechanism is as follows:

A three-dimensional discrete dynamic price game model of the dual-channel supply chain is set up as follows:

By making

The following analyzes the stability of each equilibrium point; the Jacobi matrix of system (

The necessary and sufficient conditions for the stability of the equilibrium point of the system are that all the corresponding eigenvalues are less than one. The equilibrium point is unstable if the nonzero eigenvalue is greater than one.

Take

It is known from the parameter values that

The following analyzes the stability of the Nash equilibrium point (

According to the Jury conditions, the sufficient and necessary conditions for the stability of the Nash equilibrium point of the system (

By solving condition (

According to the current situation and characteristics of the dual-channel supply chain enterprises, the parameter values are as follows:

The 3D stable regions of system (

Figure

Through the analysis of the stability region of the system (

Figure

The evolution process of system (

The bifurcation diagram

LLE diagram

Figure

When the customer demand fluctuates greatly, the system is more likely to lose stability and fall into chaos, the two retailers will cut down the product prices to reduce the risk caused by the uncertain market demand, and the increase of risk-aversion level will make the stable range of price adjustment speed of the corresponding retailer larger and make that of the other retailer little.

The evolution process of system (

In the discrete dynamic system, the chaotic attractor is an inseparable set of bounded points composed by many infinitely unstable point sets, which is an important feature of the dissipative dynamical system. When the system is in a stable state, the system’s attractor is stable at fixed point; when the system goes into a chaotic state, the system’s attractor will occupy a larger space and the structure of the chaotic attractor will be more complicated (shown in Figure

The chaotic attractor when

The initial value sensitivity is another important feature of the chaotic system. When the initial value of

The sensitivity to initial value when

The 2D bifurcation diagram is a more powerful tool for numerical simulation than the 1D bifurcation diagrams. In this paper, we will use 2D bifurcation diagram to analyze the evolution process of the system’s stability region.

Figure

The 2D bifurcation diagram of system (

The plane of

The plane of

When

Adjusting the risk-aversion level or the fairness concern level of the online retailer can make the system (

The 2D bifurcation diagram of system (

The plane of

The plane of

The plane of

The plane of

From the analysis above, we can find that, with the change of parameter values, the instability of system (

Figure

The evolution process of the two retailers’ utilities with

Bifurcation diagram of utility

Average utility

Figures

The influence of risk-aversion level on the utilities of the two retailers.

The influence of fairness concern level on the utilities of the two retailers.

Figure

When the system (

The influences of

When the market is in a chaotic state, the competition becomes disordered and unpredictable and the utilities of competitors fluctuate fiercely, which is not conducive to market operation management and long-term decision-making.

As shown above, once the chaos occurs in a complex and dynamic supply chain, the two retailers will find it is not easy to maintain the market in equilibrium state. There are so many factors which affect the stability of the system, such as the price adjustment speed, risk-aversion behavior, fair concern behavior, and the ability to collect information. When the system loses the stable state, the system will be out of order and unpredictable. In order to achieve business objectives, the efficient measures should be taken to make the system return to a stable state.

The state feedback control method is widely applied to the chaos control of supply chain system [

Suppose that the initial system is

From Figure

The influence of

The two retailers hope the market is stable because it is easier for them to make decisions and pursue maximum utilities in a stable state. However, the dual-channel supply chain system is a very complex system; the changes of decision variables and parameters will make the system enter the chaotic state from the stable state. Therefore, the two retailers need to jointly formulate measures to delay or eliminate chaos, so as to achieve market stability and development.

Considering the risk-averse behavior and fairness concern behavior of the two retailers, this paper constructs a dynamic price game model of a dual-channel supply chain. Based on game theory and nonlinear dynamics theory, the influences of parameter changes on the stability of the model and the retailers’ utilities are studied. The conclusions obtained are as follows:

When the customer demand fluctuates greatly, the system is more likely to lose stability and fall into chaos and the two retailers will cut down the product’s price to reduce the risk caused by the uncertain market demand. Adjusting the risk-aversion level or the fairness concern level of the online retailer can make the system (

As the price adjustment speed increases, the system will enter chaos through flip bifurcation; when the system is in a chaotic state, the average utility of online retailer decreases and the average utility of dual-channel retailer increases.

When the system is in a stable state, increasing the risk-aversion level and fairness concern level of the two retailers can obtain higher utilities; when the system is in chaos, the utilities of the two retailers will fluctuate sharply or even lose money. For the sake of system stability, the two retailers should control the parameters in a certain range to avoid the market in chaos

Using the state feedback control method, the system can return to a stable state from chaos by selecting appropriate control parameters.

There are no data used to support this study.

The authors declare that they have no conflicts of interest.

The research was supported by Henan Province Soft Science Research Plan Project (no. 182400410054), Henan Provincial Government Decision Research Tendering Project (no. 2018B019), and Henan Provincial Social Science Planning Decision Consultation Project (no. 2018JC05).