# Dynamic Game and Coordination Strategy of Multichannel Supply Chain Based on Brand Competition

Authors: Junhai Ma; Fang Zhang; Binshuo Bao
Journal: Complexity
 (2019)

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## Abstract

In this paper, two noncooperative dynamic pricing strategies are used in a supply chain. Two dynamic Stackelberg game models have been built involving both a manufacturer and a retailer assumed to be the leader in order. In the two models, the manufacturer sells national-brand (NB) product to an independent retailer or directly to consumers through a direct channel. The retailers sell a store-brand (SB) product when they sell the NB product coming from the manufacturer. Thus, there is competition both in different channels and in products with different brands. To analyze the complexity of the model, parameter bifurcation diagrams and strange attractor diagrams have been therefore plotted. The results show that the game leader has advantages when the market is stable, but it turns disadvantageous if the state falls into unstable as the game follower can quickly adjust the strategy to seize the market. The wholesale price and the direct selling price are high that they incur larger profits if the manufacturer is dominant, but it gets worse when the adjustment speed increases. While in the model where the retailer plays a dominant role, the increase in the adjustment speed is unfavorable to retailer. By controlling the total cost of the direct channel and increasing channel competition strength and brand competition strength, the manufacturers can increase their profits in the game dominated by the retailer. In addition, the stable region within the system will be narrow since the market is sensitive to the channel competition, brand competition, and advertising indifference.


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## Body

## 1. Introduction

The e-commerce has already been rapidly developed and widely applied with the spread of the Internet around the world. More and more manufacturing enterprises set up their own network of direct sales channels; thus, their hybrid channels include both traditional distribution channels and e-channels. These network platforms can help the manufacturers to meet the demands of consumers without the hands of retailers.A considerable amount of study has been done on a dual-channel supply chain. For instance, with the consideration of shortage caused by random yields, Xiao and Shi [1] establish a dual-channel supply chain framework and study the pricing and channel priority strategy. They find it is the decentralization that really counts in determining the channel priority strategy. Kurata et al. [2] analyze the competition in multiple distribution channels. They explore cross-brand and cross-channel pricing policies and find that brand loyalty building is profitable for both a national-brand and a store-brand. Cai et al. [3] study a dual-channel supply chain game with a price discount. They show that a consistent pricing scheme can reduce the channel conflict by inducing more profit to the retailer. To investigate the optimal decisions for a dual-channel, Soleimani et al. [4] discuss the pricing strategies both in centralized and decentralized scenarios considering production cost disruptions. They figure out the optimal decisions and reveal that if the demand disruption (cost disruption) is enhanced, the optimal prices will also be increased for a given cost disruption (demand disruption). Also by researching a dual-channel model, Batarfi et al. [5] point out dual-channel strategy is better off than the single-channel strategy. Liu and Xiao [6] build a dyadic closed-loop supply chain based on green consumer and corporate social responsibility in which consumers and firms show environmental responsibility behaviors. The price and collection rate decision and reverse channel strategy are analyzed. Zheng et al. [7] build a dual-channel closed loop supply chain that consists of a manufacturer, a retailer, and a collector. The effect of forward channel competition and power structure is investigated, and a centralized and three decentralized models are analyzed. He et al. [8] construct a dual-channel closed loop supply chain in which manufacturers can distribute new products by independent retailers and sell remanufactured products by a third part enterprise. The channel structure and pricing decisions for the manufacturer and government’s subsidy with competing remanufactured products are discussed. Giri and Dey [9] extend a closed-loop supply chain which included one collector, one recycler, and one manufacturer with a backup supplier considering uncertainty of collection of used products. And the performance of the model increases compared to that of the conditional closed loop supply chain.Many research studies show that if the manufacturers set up a direct channel to sell products, they can improve their profits largely, but this behavior can reduce the profit of the retailer. Hence, retailer will in turn take some measures to reduce the loss. The retailers often sell a store-brand (SB) product while they sell the national-brand (NB) product coming from the manufacturer. SB products and NB products compete in the market. Existing literatures [10–13] show that when retailers provide the SB products, they can not only reduce the dependence to the NB products but also enhance the market competitiveness. A previous research [10] studies the competition between many NB products and a single SB product. Narasimhan and Wilcox [11] calculate the pricing decision and the corresponding changes of the profit of the manufacturers and retailers when they provide the SB product in the market. The article points out that when retailers do not sell their SB products, leading manufacturers can make the retailers’ profit down to zero through setting a wholesale price, so the retailer has to minimize losses by selling SB products. The literature [12] points out that selling the SB products can weaken the “double marginal effect.” Tyagi [13] indicates that introducing the SB products is good for retailers but is unfavorable for manufacturers.The aforementioned studies make some research on introducing the SB products but has seldom discussed the two problems in one thesis: multichannel competition and the NB product and the SB product competition. In addition, there has been no literature using dynamic game theory to discuss the aforementioned competition in the supply chain. The multistage dynamic repeated game model combined with system dynamics theory has been studied in some literature recently. Zhang and Ma [14] considered two different pricing policies in a dual-channel supply chain with a fair caring retailer. Many dynamic behaviors, such as the bifurcation, periodic cycles, and strange attractors of the systems, are shown in their papers. In this paper, we consider a dynamic pricing model in which the manufacturer has the option of exploiting the direct channel to compete with the retail channel, whereas the retailer has the option of introducing a SB product to compete with the NB product. In order to improve the competitiveness, the retailer will advertise for the SB product. Two noncooperative Stackelberg game models have been established: in one the manufacturer acts as the leader and in the other the retailer acts as the leader. We carried out numerical simulations on two models with different conditions and analyzed the complex dynamic behaviors, such as bifurcation and chaos [15–17].The structure of this paper is arranged in the following order. In Section2, we introduce the assumptions and notations. Section 3 investigates a Stackelberg model which is led by the manufacturer and reveals numerical simulations for this scenario. In Section 4, we discuss a retailer leading game framework. In Section 5, a comparison of the prices and profits between the two models are exhibited. Finally, some conclusions have been shown.


## 2. Model Notations and Assumptions

In this paper, we study a multichannel supply chain which includes a manufacturer and a retailer. The manufacturers produces a national-brand (NB) product at a marginal costc1 and distributes it through their wholly owned direct channel at a price p0 as well as an independent retailer channel at a wholesale price w. The retailers will resell the product through their own channel at price p1. At the same time, the retailers sell the SB product to the customers at a price p2. Different products and different channels compete in the supply chain. We develop two noncooperative Stackelberg game models. The supply chain is shown in Figure 1.Figure 1We consider the demand for products in the market as a linear function of the product price because linear demand functions are tractable and widely used in supply chain analysis (Ma and Xie [18]; Dai et al. [19]; and Yao et al. [20]). Assume that consumers are divided into two categories: manufacturer loyalty consumers and retailer loyalty consumers. Consumers loyal to the manufacturer buy products only from the manufacturer. Consumers loyal to the retailer choose products from the retailer. Two types of consumers are both sensitive to price. The demands in our models are as follows:(1)D0=a0−b0p0+ηp1,D1=a1−b1p1+ηp0+ηp2−θA,D2=a2−b2p2+ηp1+θA.For the convenience, we assumeA=B, and then the demand functions are(2)D0=a0−b0p0+ηp1,D1=a1−b1p1+ηp0+ηp2−θB,D2=a2−b2p2+ηp1+θB.Here, the parameterαi>0i=0,1,2 represents the potential market size. bi>0i=0,1,2 is the price elasticity coefficient. γ>0 is the channel competition coefficient. η>0 is the brand competition coefficient bi>γ,ηi=0,1,2, i.e., the ownership-price effect is greater than the cross-price effect. In addition, θ is the demand sensitivity to the advertising level of the retailer. The price and demand functions should be satisfied: ic1<w<p1, iic1<c0<p0, iiic2<p2, and ivDi>0i=0,1,2.


## 3. Manufacturer Stackelberg (MS)

### 3.1. Model Construction

In this model, the manufacturer is the leader and the retailer is the follower, and they form a Stackelberg game. The manufacturer decides the wholesale pricew and the direct selling price p0 of the NB product. According to the decision-making of the manufacturer, the retailer makes decision of the retail price p1 of the NB product, the price p2 of the SB product, and the advertising investment A.The profit function of the manufacturer is as follows:(3)πRM=D0p0−c0+D1w−c1.The profit function of the retailer is as follows:(4)πrM=D1p1−w+D2p2−c2−B2.Taking the first-order partial derivatives ofπrM with respect to p1,p2,andB, respectively, and taking the second-order derivatives further, we can get the Hessian matrix:(5)−2b12η−θ2η−2b2θ−θθ−2.Note thatb1b2>η2 when(6)−4b1b2+4η2+b1θ2+b2θ2−2ηθ2<0.The retailer’s profit function is concave and has a unique maximum solution. The retailer’s best response to the wholesale pricew and the direct sale price p0 setting by the manufacturer is given in the following:(7)p1=−4η+θ2−2a2−2b2c2+2wη+c2θ2−wθ2−4b2−θ2−2a1+b1w+p0γ−c2η+−c2+wθ244b1b2−4η2−b1θ2−b2θ2+2ηθ2,(8)p2=−4a1η−4p0γη+4c2η2+a1θ2+b2c2θ2+p0γθ2−3c2ηθ2+wηθ2+a2−4b1+θ2−b14b2c2−2c2θ2+wθ22−4b1b2+4η2+b1θ2+b2θ2−2ηθ2,(9)B=−b1b2c2+b1b2w−b2p0γ+a2b1−η+p0γη+c2η2−wη2+a1−b2+ηθ4b1b2−4η2−b1θ2−b2θ2+2ηθ2.Substituting (7)–(9) into (3) and then taking the first-order partial derivatives of πm with respect to w and p0, respectively, we can obtain the following(10)∂πmM∂w=4c0γη2−8p0γη2−4c2η3+2b2c0γθ2-2b2p0γθ2+a2ηθ2−b2c2ηθ2−3c0γηθ2+4p0γηθ2−c1η2θ2+c2η2θ2+2wη2θ2+b12c1−2w4b2−θ2+a1b14b2−θ2+η−4η+θ2+b18wη2−a2θ2+c0γθ2−2p0γθ2−c2ηθ2−4wηθ2+2c1η−2η+θ2+b2−4c0γ+8p0γ+4c2η+c2θ2⋅8b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−1,(11)∂πmM∂p0=2b0c0−2p04b1b2−4η2−b1θ2−b2θ2+2ηθ2+a08b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−γ4b2c0γ−8b2p0γ−4a2η−4c1η2+8wη2+a2θ2−b2c2θ2+2b2wθ2−c0γθ2+2p0γθ2+c1ηθ2+c2ηθ2−4wηθ2+b1c1−2w4b2−θ2+a1−4b2+θ2⋅8b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−1.Taking the second-order derivatives further, we can get the Hessian matrix:(12)H1γγH2.See note 4 From (6), we have θ2b1−η2−4b1b1b2−η2<θ2b1−η2−θ2b1b1+b2−2η<θ2η2−b1b2<0.So, while satisfying(13)θ2b1−η2−4b1b1b2−η24b1b2−4η2−b1θ2−b2θ2+2ηθ2−γγ2b0+γ2−4b2+θ24η2−4b1b2+b1θ2+b2θ2−2ηθ2−γγ2>0,the manufacturer’s profit function is concave and has a unique maximum solution because the Hessian matrix is negative definite. So, we get the optimal wholesale price wM and the optimal direct sale price p0M by letting (10) and (11) be equal to zero. Furthermore, substituting wM and p0Minto (7)–(9), we have the optimal retail price p1Mandp2M and the optimal advertising investment AM. See the appendix for a list of functions wM,p0M,p1M,p2M,andBM.Now we develop a dynamic Stackelberg game model. In fact, while making price decision, restricted by the objective conditions such as the decision-making ability, the decision makers cannot get the whole market information and their price decision is also not completely rational. Suppose that the decision in next period can be adjusted with bounded rationality (Ma et al. [17, 21]) and is based on partial estimation of the marginal profits of the current period; that is, if the marginal profits in period t is positive, the manufacturer will raise the price in period t+1. Otherwise, the manufacturer will reduce it. The model can be constructed as follows:(14)ωt+1=ωt+α1ωt∂πmMω,p0∂ωt,p0t+1=p0t+α2p0t∂πmMω,p0∂p0t.System (14) gives the manufacturer’s dynamic price decision and the decision variables directly relate to the positive parameter αii=1,2, which represents the price adjustment speed. ∂πmMw,p0/∂wt and ∂πmMw,p0/∂p0t are the marginal profits, and they can be obtained by equations (10) and (11). Furthermore, the retailer’s price decision can be obtained by wt and p0t:(15)p1t=−4η+θ2−2a2−2b2c2+2wtη+c2θ2−wtθ2−4b2−θ2−2a1+b1wt+p0tγ−c2η+−c2+wtθ244b1b2−4η2−b1θ2−b2θ2+2ηθ2,(16)p2t=−4a1η−4p0tγη+4c2η2+a1θ2+b2c2θ2+p0tγθ2−3c2ηθ2+wtηθ2+a2−4b1+θ2−b14b2c2−2c2θ2+wtθ22−4b1b2+4η2+b1θ2+b2θ2−2ηθ2,(17)Bt=−b1b2c2+b1b2wt−b2p0tγ+a2b1−η+p0tγη+c2η2−wtη2+a1−b2+ηθ4b1b2−4η2−b1θ2−b2θ2+2ηθ2.


### 3.2. Model Analysis

We have four equilibrium solutions of system (14), described by(18)E1=0,0,E2=4c0γη2−4c2η3+2b2c0γθ2+a2ηθ2−b2c2ηθ2−3c0γηθ2−c1η2θ2+c2η2θ2+b12c14b2−θ2+a1b14b2−θ2+η−4η+θ2+b1−a2−c0γ+c2ηθ2+2c1η−2η+θ2+b2−4c0γ+c24η+θ2⋅−2η2θ2+b128b2−2θ2+4b1η−2η+θ2−1,0,E3=0,2b0c04b1b2−4η2−b1θ2−b2θ2+2ηθ2+a08b1b2−8η2−2b1θ2−2b2θ2+4ηθ2+γ4a1b2−4b1b2c1−4b2c0γ+4a2η+4c1η2−a1θ2−a2θ2+b1c1θ2+b2c2θ2+c0γθ2−c1ηθ2−c2ηθ2⋅2γ2−4b2+θ2+b08b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−1,EM=ωM,p0M.E
i
i
=
1,2,3 are boundary equilibrium solutions, while EMis the only Stackelberg equilibrium solution. Its expression is shown as (A.1) and (A.2) in Appendix. In order to guarantee the practical significance, we assume that all equilibriums are positive.The Jacobian matrix of system (14) is given by(19)J=1+α1h1α2p0γα1wγ1+α2h2,h2=2H1ω+γp0+12a1+b1c1+ηc2−γc0+θ2b1b2−η2c1+c2+a2η−b1+a1+γc0b2−η24b1b2−4η2−b1θ2−b2θ2+2ηθ2,h2=2H2p0+γω+a0+b0c0−γ4c1b1b2−η2+c1θ2η−b1+c2θ2η−b2+a2θ2−4η2+4b2−θ2γc0−a124b1b2−4η2−b1θ2−b2θ2+2ηθ2.The stability of equilibrium points will be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Then, we substitute the value ofEii=1,2,3 into (19), and we have the following proposition.Proposition 1.
All the boundary equilibrium pointsEii=1,2,3 are unstable under our assumptions and conditions (6) and (13).
The significance of boundary equilibrium is that all players can predict the emergence of this equilibrium, and it is unprofitable for anyone to deviate from the equilibrium. For enterprises, unstable boundary equilibrium is not conducive to the stability of the overall system. We investigate the stability ofER by utilizing Jury’s stability condition [22]:(20)i1+TrJ+DetJ>0,ii1−TrJ+DetJ>0,iii1−DetJ>0,where TrJ and DetJ are the trace and determinant of J, respectively. The condition (20) gives a stable region in the place of the adjustment parameters α1andα2.


### 3.3. The Experimental Calculation and Numerical Simulation

In this section, numerical simulations are carried to show the influence of parameters. We will research how the adjust parameters, channel competition coefficient, brand competition coefficient, and advertising coefficient affect the system stability domain, the product prices, and the long-term average profit by Matlab simulation software. Based on the model assumptions and conditions (6) and (13), we choose some parameters as follows: a0=30; a1=55; a2=35; b0=2.2; b1=4.05; b2=2.35; γ=0.7; η=0.6; θ=0.5; c0=7; c1=5; and c2=7. We have the equilibrium prices of the system: ωM=11.406; p0M=13.3228; p1M=15.0705; p2M=13.4116; and BM=0.6868. The manufacturer and retailer’s optimal profit of stable domain are πmM=141.49andπrM=122.32.


#### 3.3.1. The Stability Region Diagrams of the System

The stability of the equilibriumEM and the two-dimensional bifurcation diagrams of the system are presented in Figure 2. We assign different colors to different areas, namely, stable steady area (blue), stable cycles of periods 2 (green), 3 (pink), 4(red), 5 (dark blue), 6 (brown), 7 (yellow-green), 8 (red pink), chaos (yellow), and divergence (gray). We can see from the figure that when 0<α1<0.042,0<α2<0.034 (blue area), the system is stable. The two-dimensional bifurcation diagrams of the system show that they have two paths of the system entering into the chaos. One is that when the parameter α1,α2 located in the region of blue, green, red, and red pink moves to yellow step by step, the system enters into chaos through flip bifurcation. Another case is that the system from 2 cycles enters into chaos through Neimark–Sacker bifurcation if the parameters from the green area directly enter into the yellow area.Figure 2
The stability region diagrams of the system.
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(b)


#### 3.3.2. The Influence of Parametersγ, η, and θ on the Stable Regions

Figure3 shows the stable regions of the Stackelberg equilibrium point with different channel competition coefficients (γ), different brand competition coefficients (η), and different advertising coefficients (θ). In Figure 3(a), when γ=0.7,γ=1.1,andγ=1.5, we assign blue, yellow, and green to the stable region, respectively. It shows that the stable region of the system decreases with the increase of γ. That is to say, the fiercer the channel competition, the more unstable the system. In Figure 3(b), when η=0.6,η=1,andη=1.4, the stable region is shown by blue, pink, and gray, respectively. We can see from the figure that the stable region of the system decreases with the increase of the brand competition coefficient η. In Figure 3(c), when θ=0.5,θ=0.9,andθ=1.3, we assign blue, red, and brown to the stable region, respectively. The figure shows that the more sensitive the market is, the less the stability of the system is.Figure 3
The influence of the parametersδ and b2 on the stable regions.
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#### 3.3.3. Complex Dynamics Analysis on the System

Figure4 gives the bifurcation diagrams, the corresponding largest Lyapunov exponent plot of system (14) and chaos attractor. We assign black curve to wholesale price w, blue curve to direct selling price p0, red curve to retail price p1, green curve to retail price p2, and yellow curve to advertising investment B. Figures 4(a) and 4(b) show the bifurcation diagrams of price and the corresponding largest Lyapunov exponent plot of the system with α2=0.01 and α1 varying from 0 to 0.065. We can see that with the change of the α1, the system enters the chaos from the stable region through flip bifurcation. When α1=0.044, the system appears in the first double period orbit. The system becomes chaotic eventually at α1=0.057. Figures 4(c) and 4(d) give the bifurcation diagrams of price and the corresponding largest Lyapunov exponent plot of the system with α2 at 0, 0.065 and α1 at 0.01. We can see that the system appears the first bifurcation at α2=0.035 and appears the second bifurcation at α2=0.043, and then through a series of bifurcation the system enters into the chaos eventually. Figures 4(e) and 4(f) show the chaotic attractor when α1=0.06,α2=0.01 and α1=0.01,α2=0.05. Figures 4(g)–4(l) give the case that the system enters into the chaos through Neimark–Sacker bifurcation. It shows that when α2=0.038, with the change of α1 the system enters into chaos from the period-two orbit directly. Figures 4(g) and 4(h) show the bifurcation diagram and the corresponding largest Lyapunov exponent plot of the wholesale price w when α2=0.08. Figures 4(i)–4(l) show the process from the limit cycle to the strange attractor of the system (18).Figure 4
Bifurcation diagrams and chaos attractor.
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#### 3.3.4. The Influence of Parameters on the Profits

Figure5 shows the influence of adjustment parameters αii=1,2 and advertising parameter θ on the profits of the manufacturer and the retailer. The black curve represents the manufacturer's profits and the blue curve represents the retailer's profits. Figure 5 shows the influence of αii=1,2 on the long-term average profit of the supply chain. In stable state, the profits of the manufacturer are more than the profits of the retailer. With the increase in adjustment speed, in Figure 5(a), when α1>0.0441, the manufacturer’s profits begin to decrease but the retailer’s profits begin to increase, and the retailer’s profits exceed the manufacturer’s profits when α1>0.0441. Similarly, Figure 5(b) shows that the profits of the manufacturer decrease while the profits of retailer increase when α1>0.035. Therefore, we conclude that, in the dynamic Stackelberg game, the higher adjustment speed of the wholesale or the direct price has adverse effect on the leader manufacturer, but it is favorable for the follower retailer. Figures 5(c) and 5(d) show the influence of advertising parameter θ on the long-term average profit in the system’s stable region (α1=0.01andα2=0.01) (Figure 5(c)) and bifurcation region (α1=0.01andα2=0.04) (Figure 5(d)), respectively. As shown, with the increase in θ, both in stable region and unstable region, the profit of the manufacturer decreases and the profit of the retailer increases. Therefore, it is unfavorable for the leader manufacturer but favorable for the follower retailer in this model when market is more sensitive to the advertising investment.Figure 5
The influence of parameters on the profits.
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(d)


## 3.1. Model Construction

In this model, the manufacturer is the leader and the retailer is the follower, and they form a Stackelberg game. The manufacturer decides the wholesale pricew and the direct selling price p0 of the NB product. According to the decision-making of the manufacturer, the retailer makes decision of the retail price p1 of the NB product, the price p2 of the SB product, and the advertising investment A.The profit function of the manufacturer is as follows:(3)πRM=D0p0−c0+D1w−c1.The profit function of the retailer is as follows:(4)πrM=D1p1−w+D2p2−c2−B2.Taking the first-order partial derivatives ofπrM with respect to p1,p2,andB, respectively, and taking the second-order derivatives further, we can get the Hessian matrix:(5)−2b12η−θ2η−2b2θ−θθ−2.Note thatb1b2>η2 when(6)−4b1b2+4η2+b1θ2+b2θ2−2ηθ2<0.The retailer’s profit function is concave and has a unique maximum solution. The retailer’s best response to the wholesale pricew and the direct sale price p0 setting by the manufacturer is given in the following:(7)p1=−4η+θ2−2a2−2b2c2+2wη+c2θ2−wθ2−4b2−θ2−2a1+b1w+p0γ−c2η+−c2+wθ244b1b2−4η2−b1θ2−b2θ2+2ηθ2,(8)p2=−4a1η−4p0γη+4c2η2+a1θ2+b2c2θ2+p0γθ2−3c2ηθ2+wηθ2+a2−4b1+θ2−b14b2c2−2c2θ2+wθ22−4b1b2+4η2+b1θ2+b2θ2−2ηθ2,(9)B=−b1b2c2+b1b2w−b2p0γ+a2b1−η+p0γη+c2η2−wη2+a1−b2+ηθ4b1b2−4η2−b1θ2−b2θ2+2ηθ2.Substituting (7)–(9) into (3) and then taking the first-order partial derivatives of πm with respect to w and p0, respectively, we can obtain the following(10)∂πmM∂w=4c0γη2−8p0γη2−4c2η3+2b2c0γθ2-2b2p0γθ2+a2ηθ2−b2c2ηθ2−3c0γηθ2+4p0γηθ2−c1η2θ2+c2η2θ2+2wη2θ2+b12c1−2w4b2−θ2+a1b14b2−θ2+η−4η+θ2+b18wη2−a2θ2+c0γθ2−2p0γθ2−c2ηθ2−4wηθ2+2c1η−2η+θ2+b2−4c0γ+8p0γ+4c2η+c2θ2⋅8b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−1,(11)∂πmM∂p0=2b0c0−2p04b1b2−4η2−b1θ2−b2θ2+2ηθ2+a08b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−γ4b2c0γ−8b2p0γ−4a2η−4c1η2+8wη2+a2θ2−b2c2θ2+2b2wθ2−c0γθ2+2p0γθ2+c1ηθ2+c2ηθ2−4wηθ2+b1c1−2w4b2−θ2+a1−4b2+θ2⋅8b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−1.Taking the second-order derivatives further, we can get the Hessian matrix:(12)H1γγH2.See note 4 From (6), we have θ2b1−η2−4b1b1b2−η2<θ2b1−η2−θ2b1b1+b2−2η<θ2η2−b1b2<0.So, while satisfying(13)θ2b1−η2−4b1b1b2−η24b1b2−4η2−b1θ2−b2θ2+2ηθ2−γγ2b0+γ2−4b2+θ24η2−4b1b2+b1θ2+b2θ2−2ηθ2−γγ2>0,the manufacturer’s profit function is concave and has a unique maximum solution because the Hessian matrix is negative definite. So, we get the optimal wholesale price wM and the optimal direct sale price p0M by letting (10) and (11) be equal to zero. Furthermore, substituting wM and p0Minto (7)–(9), we have the optimal retail price p1Mandp2M and the optimal advertising investment AM. See the appendix for a list of functions wM,p0M,p1M,p2M,andBM.Now we develop a dynamic Stackelberg game model. In fact, while making price decision, restricted by the objective conditions such as the decision-making ability, the decision makers cannot get the whole market information and their price decision is also not completely rational. Suppose that the decision in next period can be adjusted with bounded rationality (Ma et al. [17, 21]) and is based on partial estimation of the marginal profits of the current period; that is, if the marginal profits in period t is positive, the manufacturer will raise the price in period t+1. Otherwise, the manufacturer will reduce it. The model can be constructed as follows:(14)ωt+1=ωt+α1ωt∂πmMω,p0∂ωt,p0t+1=p0t+α2p0t∂πmMω,p0∂p0t.System (14) gives the manufacturer’s dynamic price decision and the decision variables directly relate to the positive parameter αii=1,2, which represents the price adjustment speed. ∂πmMw,p0/∂wt and ∂πmMw,p0/∂p0t are the marginal profits, and they can be obtained by equations (10) and (11). Furthermore, the retailer’s price decision can be obtained by wt and p0t:(15)p1t=−4η+θ2−2a2−2b2c2+2wtη+c2θ2−wtθ2−4b2−θ2−2a1+b1wt+p0tγ−c2η+−c2+wtθ244b1b2−4η2−b1θ2−b2θ2+2ηθ2,(16)p2t=−4a1η−4p0tγη+4c2η2+a1θ2+b2c2θ2+p0tγθ2−3c2ηθ2+wtηθ2+a2−4b1+θ2−b14b2c2−2c2θ2+wtθ22−4b1b2+4η2+b1θ2+b2θ2−2ηθ2,(17)Bt=−b1b2c2+b1b2wt−b2p0tγ+a2b1−η+p0tγη+c2η2−wtη2+a1−b2+ηθ4b1b2−4η2−b1θ2−b2θ2+2ηθ2.


## 3.2. Model Analysis

We have four equilibrium solutions of system (14), described by(18)E1=0,0,E2=4c0γη2−4c2η3+2b2c0γθ2+a2ηθ2−b2c2ηθ2−3c0γηθ2−c1η2θ2+c2η2θ2+b12c14b2−θ2+a1b14b2−θ2+η−4η+θ2+b1−a2−c0γ+c2ηθ2+2c1η−2η+θ2+b2−4c0γ+c24η+θ2⋅−2η2θ2+b128b2−2θ2+4b1η−2η+θ2−1,0,E3=0,2b0c04b1b2−4η2−b1θ2−b2θ2+2ηθ2+a08b1b2−8η2−2b1θ2−2b2θ2+4ηθ2+γ4a1b2−4b1b2c1−4b2c0γ+4a2η+4c1η2−a1θ2−a2θ2+b1c1θ2+b2c2θ2+c0γθ2−c1ηθ2−c2ηθ2⋅2γ2−4b2+θ2+b08b1b2−8η2−2b1θ2−2b2θ2+4ηθ2−1,EM=ωM,p0M.E
i
i
=
1,2,3 are boundary equilibrium solutions, while EMis the only Stackelberg equilibrium solution. Its expression is shown as (A.1) and (A.2) in Appendix. In order to guarantee the practical significance, we assume that all equilibriums are positive.The Jacobian matrix of system (14) is given by(19)J=1+α1h1α2p0γα1wγ1+α2h2,h2=2H1ω+γp0+12a1+b1c1+ηc2−γc0+θ2b1b2−η2c1+c2+a2η−b1+a1+γc0b2−η24b1b2−4η2−b1θ2−b2θ2+2ηθ2,h2=2H2p0+γω+a0+b0c0−γ4c1b1b2−η2+c1θ2η−b1+c2θ2η−b2+a2θ2−4η2+4b2−θ2γc0−a124b1b2−4η2−b1θ2−b2θ2+2ηθ2.The stability of equilibrium points will be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Then, we substitute the value ofEii=1,2,3 into (19), and we have the following proposition.Proposition 1.
All the boundary equilibrium pointsEii=1,2,3 are unstable under our assumptions and conditions (6) and (13).
The significance of boundary equilibrium is that all players can predict the emergence of this equilibrium, and it is unprofitable for anyone to deviate from the equilibrium. For enterprises, unstable boundary equilibrium is not conducive to the stability of the overall system. We investigate the stability ofER by utilizing Jury’s stability condition [22]:(20)i1+TrJ+DetJ>0,ii1−TrJ+DetJ>0,iii1−DetJ>0,where TrJ and DetJ are the trace and determinant of J, respectively. The condition (20) gives a stable region in the place of the adjustment parameters α1andα2.


## 3.3. The Experimental Calculation and Numerical Simulation

In this section, numerical simulations are carried to show the influence of parameters. We will research how the adjust parameters, channel competition coefficient, brand competition coefficient, and advertising coefficient affect the system stability domain, the product prices, and the long-term average profit by Matlab simulation software. Based on the model assumptions and conditions (6) and (13), we choose some parameters as follows: a0=30; a1=55; a2=35; b0=2.2; b1=4.05; b2=2.35; γ=0.7; η=0.6; θ=0.5; c0=7; c1=5; and c2=7. We have the equilibrium prices of the system: ωM=11.406; p0M=13.3228; p1M=15.0705; p2M=13.4116; and BM=0.6868. The manufacturer and retailer’s optimal profit of stable domain are πmM=141.49andπrM=122.32.


### 3.3.1. The Stability Region Diagrams of the System

The stability of the equilibriumEM and the two-dimensional bifurcation diagrams of the system are presented in Figure 2. We assign different colors to different areas, namely, stable steady area (blue), stable cycles of periods 2 (green), 3 (pink), 4(red), 5 (dark blue), 6 (brown), 7 (yellow-green), 8 (red pink), chaos (yellow), and divergence (gray). We can see from the figure that when 0<α1<0.042,0<α2<0.034 (blue area), the system is stable. The two-dimensional bifurcation diagrams of the system show that they have two paths of the system entering into the chaos. One is that when the parameter α1,α2 located in the region of blue, green, red, and red pink moves to yellow step by step, the system enters into chaos through flip bifurcation. Another case is that the system from 2 cycles enters into chaos through Neimark–Sacker bifurcation if the parameters from the green area directly enter into the yellow area.Figure 2
The stability region diagrams of the system.
(a)
(b)


### 3.3.2. The Influence of Parametersγ, η, and θ on the Stable Regions

Figure3 shows the stable regions of the Stackelberg equilibrium point with different channel competition coefficients (γ), different brand competition coefficients (η), and different advertising coefficients (θ). In Figure 3(a), when γ=0.7,γ=1.1,andγ=1.5, we assign blue, yellow, and green to the stable region, respectively. It shows that the stable region of the system decreases with the increase of γ. That is to say, the fiercer the channel competition, the more unstable the system. In Figure 3(b), when η=0.6,η=1,andη=1.4, the stable region is shown by blue, pink, and gray, respectively. We can see from the figure that the stable region of the system decreases with the increase of the brand competition coefficient η. In Figure 3(c), when θ=0.5,θ=0.9,andθ=1.3, we assign blue, red, and brown to the stable region, respectively. The figure shows that the more sensitive the market is, the less the stability of the system is.Figure 3
The influence of the parametersδ and b2 on the stable regions.
(a)
(b)
(c)


### 3.3.3. Complex Dynamics Analysis on the System

Figure4 gives the bifurcation diagrams, the corresponding largest Lyapunov exponent plot of system (14) and chaos attractor. We assign black curve to wholesale price w, blue curve to direct selling price p0, red curve to retail price p1, green curve to retail price p2, and yellow curve to advertising investment B. Figures 4(a) and 4(b) show the bifurcation diagrams of price and the corresponding largest Lyapunov exponent plot of the system with α2=0.01 and α1 varying from 0 to 0.065. We can see that with the change of the α1, the system enters the chaos from the stable region through flip bifurcation. When α1=0.044, the system appears in the first double period orbit. The system becomes chaotic eventually at α1=0.057. Figures 4(c) and 4(d) give the bifurcation diagrams of price and the corresponding largest Lyapunov exponent plot of the system with α2 at 0, 0.065 and α1 at 0.01. We can see that the system appears the first bifurcation at α2=0.035 and appears the second bifurcation at α2=0.043, and then through a series of bifurcation the system enters into the chaos eventually. Figures 4(e) and 4(f) show the chaotic attractor when α1=0.06,α2=0.01 and α1=0.01,α2=0.05. Figures 4(g)–4(l) give the case that the system enters into the chaos through Neimark–Sacker bifurcation. It shows that when α2=0.038, with the change of α1 the system enters into chaos from the period-two orbit directly. Figures 4(g) and 4(h) show the bifurcation diagram and the corresponding largest Lyapunov exponent plot of the wholesale price w when α2=0.08. Figures 4(i)–4(l) show the process from the limit cycle to the strange attractor of the system (18).Figure 4
Bifurcation diagrams and chaos attractor.
(a)
(b)


### 3.3.4. The Influence of Parameters on the Profits

Figure5 shows the influence of adjustment parameters αii=1,2 and advertising parameter θ on the profits of the manufacturer and the retailer. The black curve represents the manufacturer's profits and the blue curve represents the retailer's profits. Figure 5 shows the influence of αii=1,2 on the long-term average profit of the supply chain. In stable state, the profits of the manufacturer are more than the profits of the retailer. With the increase in adjustment speed, in Figure 5(a), when α1>0.0441, the manufacturer’s profits begin to decrease but the retailer’s profits begin to increase, and the retailer’s profits exceed the manufacturer’s profits when α1>0.0441. Similarly, Figure 5(b) shows that the profits of the manufacturer decrease while the profits of retailer increase when α1>0.035. Therefore, we conclude that, in the dynamic Stackelberg game, the higher adjustment speed of the wholesale or the direct price has adverse effect on the leader manufacturer, but it is favorable for the follower retailer. Figures 5(c) and 5(d) show the influence of advertising parameter θ on the long-term average profit in the system’s stable region (α1=0.01andα2=0.01) (Figure 5(c)) and bifurcation region (α1=0.01andα2=0.04) (Figure 5(d)), respectively. As shown, with the increase in θ, both in stable region and unstable region, the profit of the manufacturer decreases and the profit of the retailer increases. Therefore, it is unfavorable for the leader manufacturer but favorable for the follower retailer in this model when market is more sensitive to the advertising investment.Figure 5
The influence of parameters on the profits.
(a)
(b)
(c)
(d)


## 3.3.1. The Stability Region Diagrams of the System

The stability of the equilibriumEM and the two-dimensional bifurcation diagrams of the system are presented in Figure 2. We assign different colors to different areas, namely, stable steady area (blue), stable cycles of periods 2 (green), 3 (pink), 4(red), 5 (dark blue), 6 (brown), 7 (yellow-green), 8 (red pink), chaos (yellow), and divergence (gray). We can see from the figure that when 0<α1<0.042,0<α2<0.034 (blue area), the system is stable. The two-dimensional bifurcation diagrams of the system show that they have two paths of the system entering into the chaos. One is that when the parameter α1,α2 located in the region of blue, green, red, and red pink moves to yellow step by step, the system enters into chaos through flip bifurcation. Another case is that the system from 2 cycles enters into chaos through Neimark–Sacker bifurcation if the parameters from the green area directly enter into the yellow area.Figure 2
The stability region diagrams of the system.
(a)
(b)


## 3.3.2. The Influence of Parametersγ, η, and θ on the Stable Regions

Figure3 shows the stable regions of the Stackelberg equilibrium point with different channel competition coefficients (γ), different brand competition coefficients (η), and different advertising coefficients (θ). In Figure 3(a), when γ=0.7,γ=1.1,andγ=1.5, we assign blue, yellow, and green to the stable region, respectively. It shows that the stable region of the system decreases with the increase of γ. That is to say, the fiercer the channel competition, the more unstable the system. In Figure 3(b), when η=0.6,η=1,andη=1.4, the stable region is shown by blue, pink, and gray, respectively. We can see from the figure that the stable region of the system decreases with the increase of the brand competition coefficient η. In Figure 3(c), when θ=0.5,θ=0.9,andθ=1.3, we assign blue, red, and brown to the stable region, respectively. The figure shows that the more sensitive the market is, the less the stability of the system is.Figure 3
The influence of the parametersδ and b2 on the stable regions.
(a)
(b)
(c)


## 3.3.3. Complex Dynamics Analysis on the System

Figure4 gives the bifurcation diagrams, the corresponding largest Lyapunov exponent plot of system (14) and chaos attractor. We assign black curve to wholesale price w, blue curve to direct selling price p0, red curve to retail price p1, green curve to retail price p2, and yellow curve to advertising investment B. Figures 4(a) and 4(b) show the bifurcation diagrams of price and the corresponding largest Lyapunov exponent plot of the system with α2=0.01 and α1 varying from 0 to 0.065. We can see that with the change of the α1, the system enters the chaos from the stable region through flip bifurcation. When α1=0.044, the system appears in the first double period orbit. The system becomes chaotic eventually at α1=0.057. Figures 4(c) and 4(d) give the bifurcation diagrams of price and the corresponding largest Lyapunov exponent plot of the system with α2 at 0, 0.065 and α1 at 0.01. We can see that the system appears the first bifurcation at α2=0.035 and appears the second bifurcation at α2=0.043, and then through a series of bifurcation the system enters into the chaos eventually. Figures 4(e) and 4(f) show the chaotic attractor when α1=0.06,α2=0.01 and α1=0.01,α2=0.05. Figures 4(g)–4(l) give the case that the system enters into the chaos through Neimark–Sacker bifurcation. It shows that when α2=0.038, with the change of α1 the system enters into chaos from the period-two orbit directly. Figures 4(g) and 4(h) show the bifurcation diagram and the corresponding largest Lyapunov exponent plot of the wholesale price w when α2=0.08. Figures 4(i)–4(l) show the process from the limit cycle to the strange attractor of the system (18).Figure 4
Bifurcation diagrams and chaos attractor.
(a)
(b)


## 3.3.4. The Influence of Parameters on the Profits

Figure5 shows the influence of adjustment parameters αii=1,2 and advertising parameter θ on the profits of the manufacturer and the retailer. The black curve represents the manufacturer's profits and the blue curve represents the retailer's profits. Figure 5 shows the influence of αii=1,2 on the long-term average profit of the supply chain. In stable state, the profits of the manufacturer are more than the profits of the retailer. With the increase in adjustment speed, in Figure 5(a), when α1>0.0441, the manufacturer’s profits begin to decrease but the retailer’s profits begin to increase, and the retailer’s profits exceed the manufacturer’s profits when α1>0.0441. Similarly, Figure 5(b) shows that the profits of the manufacturer decrease while the profits of retailer increase when α1>0.035. Therefore, we conclude that, in the dynamic Stackelberg game, the higher adjustment speed of the wholesale or the direct price has adverse effect on the leader manufacturer, but it is favorable for the follower retailer. Figures 5(c) and 5(d) show the influence of advertising parameter θ on the long-term average profit in the system’s stable region (α1=0.01andα2=0.01) (Figure 5(c)) and bifurcation region (α1=0.01andα2=0.04) (Figure 5(d)), respectively. As shown, with the increase in θ, both in stable region and unstable region, the profit of the manufacturer decreases and the profit of the retailer increases. Therefore, it is unfavorable for the leader manufacturer but favorable for the follower retailer in this model when market is more sensitive to the advertising investment.Figure 5
The influence of parameters on the profits.
(a)
(b)
(c)
(d)


## 4. Retailer Stackelberg (RS)

### 4.1. Model Construction

In this case, the retailer plays the dominant role in the supply chain and becomes the leader of the Stackelberg game. The retailer first decides the retail pricep1 of the NB product, the retail price p2 of the SB product, and the advertising investment B. The manufacturer decides the wholesale price w and direct selling price p0 based on the retailer’s decision. We assume the unit retail margin is u, and then p1=w+u. Similar to the model MS, the optimization problem is solved with backward induction.The profit function of the manufacturer and the retailer are as follows:(21)πmR=D0p0−c0+D1w−c1,(22)πrR=D1u+D2p2−c2−B2.Taking the first-order partial derivatives ofπmR with respect to w and p0, respectively, and taking the second-order derivatives further, we can get the Hessian matrix:(23)−2b12γ2γ−2b0.Note that−2b1<0 and 4b0b1−4γ2>0. So, the manufacturer’s profit function is concave and has a unique maximum solution. We can work out the manufacturer's best reaction function by ∂πmR/∂w=0 and ∂πmR/∂p0=0. That is,(24)w=a1b0+b0b1c1−b0b1u+a0γ−c1γ2+uγ2+b0p2η−Bb0θ2b0b1−γ2=c1−p1+a1b0+a0γ+b0p2η−Bb0θb0b1−γ2,(25)p0=c02+a0b1+a1γ+γηp2−γθB2b0b1−γ2.Substituting (24) and (25) into (22) and then taking the first-order partial derivatives of πrR with respect to p1,p2, and B, respectively, we can obtain the following(26)∂πrR∂p1=a1+b1c1+c0γ−c1η−4b1p1+2ηp2−Bθ+2a1b0b1+2a0b1γ+2b0b1ηp2−Bθb0b1−γ2,(27)∂πrR∂p2=a2+b2c2−c1η2−2b2p2+2ηp1+θB+b02b1η−ηp2+θB−b0b1ηa0γ+a1b0b0b1−γ22+ηa0γ+b0b1c1+2a1b0+b0c0γ2−b0b1+γ2+2b0b1ηp1−b0η2p2+b0ηθBb0b1−γ2,(28)∂πrR∂B=12−b0b1+γ22γ3−a0+c1−2c2+p1−p2γ+2a1b02b0b1−γ2+b02b12b1c1−c2−3p1+p2+c0γ+4p2η−b0γ−3a0b1+γb13c1−4c2−8p1+4p2+c0γ+2p2ηθ+B−4γ4−4b02b1b1+θ2+2b0γ24b1+θ2.Taking the second-order derivatives further, we can get the Hessian matrix:(29)−4b1f1f2f1f3f4f2f4f5,where(30)f1=4b0b1η−2γ2ηb0b1−γ2,f2=−3b0b1+γ2θb0b1−γ2,f3=−2b2+b0−2b0b1+γ2η2−b0b1+γ22,f4=γ4−b0γ22b1+η+b02b1b1+2ηθ−b0b1+γ22,f5=−2γ4−2b02b1b1+θ2+b0γ24b1+θ2−b0b1+γ22.While satisfying(31)8b0b1b1b2−η2+4γ2−2b1b2+η2b0b1−γ2>0,(32)1−b0b1+γ222γ4−8b1b2+4η2+2b1θ2+b2θ2−2ηθ2+2b02b1η2θ2+b12−8b2+2θ2+b18η2+b2θ2−4ηθ2+b0γ2−η2θ2+8b124b2−θ2−4b16η2+b2θ2−3ηθ2<0,the retailer’s profit function is concave and has a unique maximum solution because the Hessian matrix is negative definite. So, we get the optimal retail price p1R,p2R, and BR. Furthermore, we also get the optimal wholesale price wR and direct selling price p0R (see Appendix A.16–A.20).Similar to the MS model, we develop a dynamic Stackelberg game model. Suppose that the decision in next period can be adjusted with bounded rationality and is based on partial estimation of the marginal profits of the current period. The model can be constructed as follows:(33)p1t+1=p1t+β1p1t∂πrRp1,p2,B∂p1t,p2t+1=p2t+β2p2t∂πrRp1,p2,B∂p2t,Bt+1=Bt+β3Bt∂πrRp1,p2,B∂Bt.System (45) gives the retailer’s dynamic price decision, and the decision variables directly relate to the positive parameterβii=1,2,3 which represents the price adjustment speed. ∂πRRp1,p2,B/∂p1t, ∂πRRp1,p2,B/∂p2t, and ∂πRRp1,p2,B/∂Bt are the marginal profits, and they can be obtained from (26)–(28). Furthermore, the manufacturer's price decision can be worked out based on p1t,p2t, and Bt.(34)wt=c1−p1t+a1b0+a0γ+b0ηp2t−b0θBtb0b1−γ2,p0t=c02+a0b1+a1γ+γηp2t−γθBt2b0b1−γ2.


### 4.2. Model Analysis

The eight equilibrium solutions of system (33) (E1,E2,…,E8) can be shown as follows (here only giving the nonzero equilibrium solutions):(35)E7=4a13b02b12−4b0b1γ2+γ4+γ3a0−8b1+θ2+γ4b1c1+4c0γ−4c2η−c1θ2+2c2θ2+b02b14b12c1+c0γ−4c2ηθ2+b14c0γ−4c2η−2c1θ2+6c2θ2−b0γa0b1−8b1+θ2+γ8b12c1+c0γ−2c2ηθ22+b18c0γ−8c2η−3c1θ2+8c2θ2⋅2−b0b1+γ228b1−θ2−1,0,a1−b0b1+γ2+γ2−b1c1+4b1c2+c0γ−c2η+b0b1b1c1−4b1c2−c0γ+3c2ηθ⋅b0b1−γ28b1−θ2−1,ER=p1R,p2R,BR.E
i
i
=
1,2
,
…
,
7 are boundary equilibrium solutions, while ER is the only Stackelberg equilibrium solution. Its expression is shown (A.16)–(A.18) in Appendix. In order to guarantee the practical significance, we assume that all equilibriums are positive.The Jacobian matrix of system (33) is given by(36)1+β1t1β2f1p2β3f2Bβ1f1p11+β2t2β3f4Bβ1f2p1β2f4p21+β3t3,where(37)t1=a1+b1c1−ηc2+γc0+2ηp2−8b1p1−Bθ+2b1a1b0+a0γ+b0ηp2−b0Bθb0b1−γ2,t2=12−b0b1+γ222a2−b0b1+γ22+γ32b2c2−4p2γ+a0η−c1γη+4p1γη+2Bγθ+b02b1−η4a1+c0γ+8p2η−4Bθ+2b1b2c2−4b2p2−c1η+4p1η+Bθ+b0γγη2a1+c0γ+4p2η−2Bθ−b14b2c2−4p2γ+3a0η−3c1γη+12p1γη+4Bγθ,t3=12−b0b1+γ22γ3−a0+c1−2c2+p1−p2γ+2a1b02b0b1−γ2+b02b12b1c1−c2−3p1+p2+c0γ+4p2η−b0γ−3a0b1+γb13c1−4c2−8p1+4p2+c0γ+2p2ηθ−4B2γ4+2b02b1b1+θ2−b0γ24b1+θ2.The stability of equilibrium points will be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Then, we substitute the value ofEii=1,2…,7 into (36) and we have the following proposition.Proposition 2.
All the boundary equilibrium pointsEii=1,2…,7 are unstable under our assumptions and conditions (31) and (32).
We investigate the stability ofEM by utilizing Jury’s stability condition [22]. Assuming the characteristic polynomial for the Jacobian matrix J is(38)fλ=λ3+Aλ2+Bλ+C.
Then, the local stability domain can be determined by the following conditions:(39)i:f1=A+B+C+1>0,ii:f−1=A−B+C−1<0,iii:C2−1<0,iv:1−C22−B−AC2>0.


### 4.3. The Experimental Calculation and Analysis

In this section, numerical simulations are carried to show the influence of parameters. We will study how the adjust parameters, advertising coefficients, and cost parameters affect the product prices and the long-term average profit. We first choose some values for some parameters based on actual competition:a0=30; a1=55; a2=35; b0=2.2; b1=4.05; b2=2.35; γ=0.7; η=0.6; θ=0.5; c0=7; c1=5; and c2=7 (the values are the same as the model of MS). These parameters should meet the model assumptions and conditions (34) and (A.16). We can calculate the equilibrium prices of the system: p1R=15.0997; p2R=13.4153; BR=0.6879; wR=8.7782; and p0R=13.3215. The manufacturer and retailer’s optimal profit of stable region are πmR=112.289andπrR=150.898.


#### 4.3.1. The Stability Region and the Bifurcation Diagrams of the System

In Figure6, the stability of the equilibrium region is presented. We can see that the system is stable when β1<0.008, β2<0.02, and β3<1.4. A high adjustment speed will make the system leave the stable area.Figure 6
The stability region of the system.Figure7(a) presents the parameter basin with respect to the parameters β1,β2β3=0.1 and assigns different colors to different periods (see the MS model). From this two-dimensional bifurcation diagram of the system, we can see that the system enters into chaos through period-doubling bifurcation and Neimark–Sacker bifurcation. Figure 7(b) gives the bifurcation diagrams of the retail price p1 with β2=0.018,β3=0.1, and β1 at 0, 0.012. Figure 7(c) gives the limit cycle diagram when β1=0.0103,β2=0.018,andβ3=0.1. Figure 7(d) is the corresponding largest Lyapunov exponent plot.Figure 7
Two-dimensional bifurcation diagrams of the system.


#### 4.3.2. Chaos Attractor and Sensitivity to Initial Values

Figure8 shows the chaotic attractor and the butterfly effect. Figure 8(a) is the chaotic attractor of system (33) when β1 = 0.011, β2 = 0.01, and β3 = 0.1. Figures 8(b)–8(d) show the sensitivity to initial values of system (33) when p1=15.09969,p2=13.4153,andB=0.68792. In Figure 8(b), blue curve is the retail price p1 with the initial valuesp1=15.09969, and red curve is the retail price p1 with the initial valuesp1=15.09968. We can see from the figure that when the retail price p1 initial value changed 0.00001, the system tends to be significantly different after about 20 cycles iteration. Similarly, Figures 8(c)–8(d) give the case when the retail price p2 initial value changed 0.0001 and the advertising investment B changed 0.00001. It can be seen that a small change of the initial value will cause quite different results in the end.Figure 8
Chaos attractor and the sensitivity to initial value.
(a)
(b)
(c)
(d)


#### 4.3.3. The Influence of Parameters on the Profits

Figures9(a)–9(d) give the influence of βii=1,2,3 on the long-term average profit of the manufacturer (black curve) and the retailer (blue curve). As shown, with the increase of the βii=1,2,3, the profit of the manufacturer and the retailer is stable at first, and when the system enters into the bifurcation region, the retailer’s profits begin to decrease. But, the result is opposite for the manufacturer. Therefore, we conclude that in the RS model, the higher adjustment speed of the price is adverse to the leader retailer, but it is favorable to the follower manufacturer.Figures9(e)–9(j) show the influence of the cost c0, channel competition coefficient γ, and brand competition coefficient η on the long-term average profit in the system stable region (β1=0.001,β2=0.01,andβ3=0.1) and the bifurcation region (β1=0.01,β2=0.01,andβ3=0.1), respectively. From Figures 9(e) and 9(f) , we can see that both profits are in the stable region and the bifurcation region separately. So, in the RS model, the manufacturers can improve their profit through controlling the cost of the direct channel. From Figures 9(g) and 9(h), we can see that channel competition coefficient γ has obvious effects on profits. In the stable region, with the increase of γ, both manufacturer’s profit and retailer’s profit are rising; the profit of the manufacturer rise significantly greater than that of the retailer; when γ>0.005, the profit of the manufacturer and the profit of the retailer are the same; when γ>0.005, the profit of the manufacturer exceed the profit of the retailer. In the bifurcation region with the increase of γ, the manufacturer’s profit is still increasing while the retailer’s profit decreases rapidly. It shows that in the RS model, the manufacturer who sets up a direct sales channel is more advantageous than the retailer when the market is sensitive to channel competition. Figures 9(i) and 9(j) give the influence of the brand competition coefficient η on the profit. We can see from the figure, in the stable region, with the increase of η, both manufacturer’s and retailer's profit will increase, but when the system enters into the bifurcation region, the manufacturer’s profit is still increasing while the retailer’s profit decreases rapidly. So, in a word, controlling the cost and increasing the channel competitive strength and brand competition strength will make the manufacturer increase its profit in the game dominated by the retailer.Figure 9
The influence of parameters on the profits.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)


## 4.1. Model Construction

In this case, the retailer plays the dominant role in the supply chain and becomes the leader of the Stackelberg game. The retailer first decides the retail pricep1 of the NB product, the retail price p2 of the SB product, and the advertising investment B. The manufacturer decides the wholesale price w and direct selling price p0 based on the retailer’s decision. We assume the unit retail margin is u, and then p1=w+u. Similar to the model MS, the optimization problem is solved with backward induction.The profit function of the manufacturer and the retailer are as follows:(21)πmR=D0p0−c0+D1w−c1,(22)πrR=D1u+D2p2−c2−B2.Taking the first-order partial derivatives ofπmR with respect to w and p0, respectively, and taking the second-order derivatives further, we can get the Hessian matrix:(23)−2b12γ2γ−2b0.Note that−2b1<0 and 4b0b1−4γ2>0. So, the manufacturer’s profit function is concave and has a unique maximum solution. We can work out the manufacturer's best reaction function by ∂πmR/∂w=0 and ∂πmR/∂p0=0. That is,(24)w=a1b0+b0b1c1−b0b1u+a0γ−c1γ2+uγ2+b0p2η−Bb0θ2b0b1−γ2=c1−p1+a1b0+a0γ+b0p2η−Bb0θb0b1−γ2,(25)p0=c02+a0b1+a1γ+γηp2−γθB2b0b1−γ2.Substituting (24) and (25) into (22) and then taking the first-order partial derivatives of πrR with respect to p1,p2, and B, respectively, we can obtain the following(26)∂πrR∂p1=a1+b1c1+c0γ−c1η−4b1p1+2ηp2−Bθ+2a1b0b1+2a0b1γ+2b0b1ηp2−Bθb0b1−γ2,(27)∂πrR∂p2=a2+b2c2−c1η2−2b2p2+2ηp1+θB+b02b1η−ηp2+θB−b0b1ηa0γ+a1b0b0b1−γ22+ηa0γ+b0b1c1+2a1b0+b0c0γ2−b0b1+γ2+2b0b1ηp1−b0η2p2+b0ηθBb0b1−γ2,(28)∂πrR∂B=12−b0b1+γ22γ3−a0+c1−2c2+p1−p2γ+2a1b02b0b1−γ2+b02b12b1c1−c2−3p1+p2+c0γ+4p2η−b0γ−3a0b1+γb13c1−4c2−8p1+4p2+c0γ+2p2ηθ+B−4γ4−4b02b1b1+θ2+2b0γ24b1+θ2.Taking the second-order derivatives further, we can get the Hessian matrix:(29)−4b1f1f2f1f3f4f2f4f5,where(30)f1=4b0b1η−2γ2ηb0b1−γ2,f2=−3b0b1+γ2θb0b1−γ2,f3=−2b2+b0−2b0b1+γ2η2−b0b1+γ22,f4=γ4−b0γ22b1+η+b02b1b1+2ηθ−b0b1+γ22,f5=−2γ4−2b02b1b1+θ2+b0γ24b1+θ2−b0b1+γ22.While satisfying(31)8b0b1b1b2−η2+4γ2−2b1b2+η2b0b1−γ2>0,(32)1−b0b1+γ222γ4−8b1b2+4η2+2b1θ2+b2θ2−2ηθ2+2b02b1η2θ2+b12−8b2+2θ2+b18η2+b2θ2−4ηθ2+b0γ2−η2θ2+8b124b2−θ2−4b16η2+b2θ2−3ηθ2<0,the retailer’s profit function is concave and has a unique maximum solution because the Hessian matrix is negative definite. So, we get the optimal retail price p1R,p2R, and BR. Furthermore, we also get the optimal wholesale price wR and direct selling price p0R (see Appendix A.16–A.20).Similar to the MS model, we develop a dynamic Stackelberg game model. Suppose that the decision in next period can be adjusted with bounded rationality and is based on partial estimation of the marginal profits of the current period. The model can be constructed as follows:(33)p1t+1=p1t+β1p1t∂πrRp1,p2,B∂p1t,p2t+1=p2t+β2p2t∂πrRp1,p2,B∂p2t,Bt+1=Bt+β3Bt∂πrRp1,p2,B∂Bt.System (45) gives the retailer’s dynamic price decision, and the decision variables directly relate to the positive parameterβii=1,2,3 which represents the price adjustment speed. ∂πRRp1,p2,B/∂p1t, ∂πRRp1,p2,B/∂p2t, and ∂πRRp1,p2,B/∂Bt are the marginal profits, and they can be obtained from (26)–(28). Furthermore, the manufacturer's price decision can be worked out based on p1t,p2t, and Bt.(34)wt=c1−p1t+a1b0+a0γ+b0ηp2t−b0θBtb0b1−γ2,p0t=c02+a0b1+a1γ+γηp2t−γθBt2b0b1−γ2.


## 4.2. Model Analysis

The eight equilibrium solutions of system (33) (E1,E2,…,E8) can be shown as follows (here only giving the nonzero equilibrium solutions):(35)E7=4a13b02b12−4b0b1γ2+γ4+γ3a0−8b1+θ2+γ4b1c1+4c0γ−4c2η−c1θ2+2c2θ2+b02b14b12c1+c0γ−4c2ηθ2+b14c0γ−4c2η−2c1θ2+6c2θ2−b0γa0b1−8b1+θ2+γ8b12c1+c0γ−2c2ηθ22+b18c0γ−8c2η−3c1θ2+8c2θ2⋅2−b0b1+γ228b1−θ2−1,0,a1−b0b1+γ2+γ2−b1c1+4b1c2+c0γ−c2η+b0b1b1c1−4b1c2−c0γ+3c2ηθ⋅b0b1−γ28b1−θ2−1,ER=p1R,p2R,BR.E
i
i
=
1,2
,
…
,
7 are boundary equilibrium solutions, while ER is the only Stackelberg equilibrium solution. Its expression is shown (A.16)–(A.18) in Appendix. In order to guarantee the practical significance, we assume that all equilibriums are positive.The Jacobian matrix of system (33) is given by(36)1+β1t1β2f1p2β3f2Bβ1f1p11+β2t2β3f4Bβ1f2p1β2f4p21+β3t3,where(37)t1=a1+b1c1−ηc2+γc0+2ηp2−8b1p1−Bθ+2b1a1b0+a0γ+b0ηp2−b0Bθb0b1−γ2,t2=12−b0b1+γ222a2−b0b1+γ22+γ32b2c2−4p2γ+a0η−c1γη+4p1γη+2Bγθ+b02b1−η4a1+c0γ+8p2η−4Bθ+2b1b2c2−4b2p2−c1η+4p1η+Bθ+b0γγη2a1+c0γ+4p2η−2Bθ−b14b2c2−4p2γ+3a0η−3c1γη+12p1γη+4Bγθ,t3=12−b0b1+γ22γ3−a0+c1−2c2+p1−p2γ+2a1b02b0b1−γ2+b02b12b1c1−c2−3p1+p2+c0γ+4p2η−b0γ−3a0b1+γb13c1−4c2−8p1+4p2+c0γ+2p2ηθ−4B2γ4+2b02b1b1+θ2−b0γ24b1+θ2.The stability of equilibrium points will be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Then, we substitute the value ofEii=1,2…,7 into (36) and we have the following proposition.Proposition 2.
All the boundary equilibrium pointsEii=1,2…,7 are unstable under our assumptions and conditions (31) and (32).
We investigate the stability ofEM by utilizing Jury’s stability condition [22]. Assuming the characteristic polynomial for the Jacobian matrix J is(38)fλ=λ3+Aλ2+Bλ+C.
Then, the local stability domain can be determined by the following conditions:(39)i:f1=A+B+C+1>0,ii:f−1=A−B+C−1<0,iii:C2−1<0,iv:1−C22−B−AC2>0.


## 4.3. The Experimental Calculation and Analysis

In this section, numerical simulations are carried to show the influence of parameters. We will study how the adjust parameters, advertising coefficients, and cost parameters affect the product prices and the long-term average profit. We first choose some values for some parameters based on actual competition:a0=30; a1=55; a2=35; b0=2.2; b1=4.05; b2=2.35; γ=0.7; η=0.6; θ=0.5; c0=7; c1=5; and c2=7 (the values are the same as the model of MS). These parameters should meet the model assumptions and conditions (34) and (A.16). We can calculate the equilibrium prices of the system: p1R=15.0997; p2R=13.4153; BR=0.6879; wR=8.7782; and p0R=13.3215. The manufacturer and retailer’s optimal profit of stable region are πmR=112.289andπrR=150.898.


### 4.3.1. The Stability Region and the Bifurcation Diagrams of the System

In Figure6, the stability of the equilibrium region is presented. We can see that the system is stable when β1<0.008, β2<0.02, and β3<1.4. A high adjustment speed will make the system leave the stable area.Figure 6
The stability region of the system.Figure7(a) presents the parameter basin with respect to the parameters β1,β2β3=0.1 and assigns different colors to different periods (see the MS model). From this two-dimensional bifurcation diagram of the system, we can see that the system enters into chaos through period-doubling bifurcation and Neimark–Sacker bifurcation. Figure 7(b) gives the bifurcation diagrams of the retail price p1 with β2=0.018,β3=0.1, and β1 at 0, 0.012. Figure 7(c) gives the limit cycle diagram when β1=0.0103,β2=0.018,andβ3=0.1. Figure 7(d) is the corresponding largest Lyapunov exponent plot.Figure 7
Two-dimensional bifurcation diagrams of the system.


### 4.3.2. Chaos Attractor and Sensitivity to Initial Values

Figure8 shows the chaotic attractor and the butterfly effect. Figure 8(a) is the chaotic attractor of system (33) when β1 = 0.011, β2 = 0.01, and β3 = 0.1. Figures 8(b)–8(d) show the sensitivity to initial values of system (33) when p1=15.09969,p2=13.4153,andB=0.68792. In Figure 8(b), blue curve is the retail price p1 with the initial valuesp1=15.09969, and red curve is the retail price p1 with the initial valuesp1=15.09968. We can see from the figure that when the retail price p1 initial value changed 0.00001, the system tends to be significantly different after about 20 cycles iteration. Similarly, Figures 8(c)–8(d) give the case when the retail price p2 initial value changed 0.0001 and the advertising investment B changed 0.00001. It can be seen that a small change of the initial value will cause quite different results in the end.Figure 8
Chaos attractor and the sensitivity to initial value.
(a)
(b)
(c)
(d)


### 4.3.3. The Influence of Parameters on the Profits

Figures9(a)–9(d) give the influence of βii=1,2,3 on the long-term average profit of the manufacturer (black curve) and the retailer (blue curve). As shown, with the increase of the βii=1,2,3, the profit of the manufacturer and the retailer is stable at first, and when the system enters into the bifurcation region, the retailer’s profits begin to decrease. But, the result is opposite for the manufacturer. Therefore, we conclude that in the RS model, the higher adjustment speed of the price is adverse to the leader retailer, but it is favorable to the follower manufacturer.Figures9(e)–9(j) show the influence of the cost c0, channel competition coefficient γ, and brand competition coefficient η on the long-term average profit in the system stable region (β1=0.001,β2=0.01,andβ3=0.1) and the bifurcation region (β1=0.01,β2=0.01,andβ3=0.1), respectively. From Figures 9(e) and 9(f) , we can see that both profits are in the stable region and the bifurcation region separately. So, in the RS model, the manufacturers can improve their profit through controlling the cost of the direct channel. From Figures 9(g) and 9(h), we can see that channel competition coefficient γ has obvious effects on profits. In the stable region, with the increase of γ, both manufacturer’s profit and retailer’s profit are rising; the profit of the manufacturer rise significantly greater than that of the retailer; when γ>0.005, the profit of the manufacturer and the profit of the retailer are the same; when γ>0.005, the profit of the manufacturer exceed the profit of the retailer. In the bifurcation region with the increase of γ, the manufacturer’s profit is still increasing while the retailer’s profit decreases rapidly. It shows that in the RS model, the manufacturer who sets up a direct sales channel is more advantageous than the retailer when the market is sensitive to channel competition. Figures 9(i) and 9(j) give the influence of the brand competition coefficient η on the profit. We can see from the figure, in the stable region, with the increase of η, both manufacturer’s and retailer's profit will increase, but when the system enters into the bifurcation region, the manufacturer’s profit is still increasing while the retailer’s profit decreases rapidly. So, in a word, controlling the cost and increasing the channel competitive strength and brand competition strength will make the manufacturer increase its profit in the game dominated by the retailer.Figure 9
The influence of parameters on the profits.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)


## 4.3.1. The Stability Region and the Bifurcation Diagrams of the System

In Figure6, the stability of the equilibrium region is presented. We can see that the system is stable when β1<0.008, β2<0.02, and β3<1.4. A high adjustment speed will make the system leave the stable area.Figure 6
The stability region of the system.Figure7(a) presents the parameter basin with respect to the parameters β1,β2β3=0.1 and assigns different colors to different periods (see the MS model). From this two-dimensional bifurcation diagram of the system, we can see that the system enters into chaos through period-doubling bifurcation and Neimark–Sacker bifurcation. Figure 7(b) gives the bifurcation diagrams of the retail price p1 with β2=0.018,β3=0.1, and β1 at 0, 0.012. Figure 7(c) gives the limit cycle diagram when β1=0.0103,β2=0.018,andβ3=0.1. Figure 7(d) is the corresponding largest Lyapunov exponent plot.Figure 7
Two-dimensional bifurcation diagrams of the system.


## 4.3.2. Chaos Attractor and Sensitivity to Initial Values

Figure8 shows the chaotic attractor and the butterfly effect. Figure 8(a) is the chaotic attractor of system (33) when β1 = 0.011, β2 = 0.01, and β3 = 0.1. Figures 8(b)–8(d) show the sensitivity to initial values of system (33) when p1=15.09969,p2=13.4153,andB=0.68792. In Figure 8(b), blue curve is the retail price p1 with the initial valuesp1=15.09969, and red curve is the retail price p1 with the initial valuesp1=15.09968. We can see from the figure that when the retail price p1 initial value changed 0.00001, the system tends to be significantly different after about 20 cycles iteration. Similarly, Figures 8(c)–8(d) give the case when the retail price p2 initial value changed 0.0001 and the advertising investment B changed 0.00001. It can be seen that a small change of the initial value will cause quite different results in the end.Figure 8
Chaos attractor and the sensitivity to initial value.
(a)
(b)
(c)
(d)


## 4.3.3. The Influence of Parameters on the Profits

Figures9(a)–9(d) give the influence of βii=1,2,3 on the long-term average profit of the manufacturer (black curve) and the retailer (blue curve). As shown, with the increase of the βii=1,2,3, the profit of the manufacturer and the retailer is stable at first, and when the system enters into the bifurcation region, the retailer’s profits begin to decrease. But, the result is opposite for the manufacturer. Therefore, we conclude that in the RS model, the higher adjustment speed of the price is adverse to the leader retailer, but it is favorable to the follower manufacturer.Figures9(e)–9(j) show the influence of the cost c0, channel competition coefficient γ, and brand competition coefficient η on the long-term average profit in the system stable region (β1=0.001,β2=0.01,andβ3=0.1) and the bifurcation region (β1=0.01,β2=0.01,andβ3=0.1), respectively. From Figures 9(e) and 9(f) , we can see that both profits are in the stable region and the bifurcation region separately. So, in the RS model, the manufacturers can improve their profit through controlling the cost of the direct channel. From Figures 9(g) and 9(h), we can see that channel competition coefficient γ has obvious effects on profits. In the stable region, with the increase of γ, both manufacturer’s profit and retailer’s profit are rising; the profit of the manufacturer rise significantly greater than that of the retailer; when γ>0.005, the profit of the manufacturer and the profit of the retailer are the same; when γ>0.005, the profit of the manufacturer exceed the profit of the retailer. In the bifurcation region with the increase of γ, the manufacturer’s profit is still increasing while the retailer’s profit decreases rapidly. It shows that in the RS model, the manufacturer who sets up a direct sales channel is more advantageous than the retailer when the market is sensitive to channel competition. Figures 9(i) and 9(j) give the influence of the brand competition coefficient η on the profit. We can see from the figure, in the stable region, with the increase of η, both manufacturer’s and retailer's profit will increase, but when the system enters into the bifurcation region, the manufacturer’s profit is still increasing while the retailer’s profit decreases rapidly. So, in a word, controlling the cost and increasing the channel competitive strength and brand competition strength will make the manufacturer increase its profit in the game dominated by the retailer.Figure 9
The influence of parameters on the profits.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)


## 5. Comparison of the Two Models and Analysis

In this section, we compare the prices and profits of the two models in the stable region and in the bifurcation region, respectively. The results are shown in Table1.Table 1
Units for magnetic properties.
MS
MS
RS
RS
α
1
=
0.01
,
α
2
=
0.01
α
1
=
0.05
,
α
2
=
0.01
β
1
=
0.001
,
β
2
=
0.01
, β3=0.1
β
1
=
0.009
,
β
2
=
0.01
,
β
3
=
0.1
w
11.406
10.4333
8.7782
9.7763
p
0
13.3228
13.1668
13.3215
13.3111
p
1
15.0705
14.5760
15.0997
14.0365
p
2
13.4116
13.3964
13.4153
12.7192
B
0.6868
0.5634
0.6879
0.3506
π
m
141.4914
120.1906
112.289
192.2792
π
r
122.3231
143.2053
150.898
31.6133
π
263.8145
263.3959
263.187
223.8925Table1 shows the prices and the profits of the two models in different regions.In Table1, we compare different variables of the two models in the stable region and bifurcation region, respectively. It can be seen from Table 1, in the stable region, the wholesale price and direct selling price is higher in the MS model than that in the RS model. The total profit of the supply chain in the MS model is also more than that in the RS model. In the RS model, the lower wholesale price makes the retailer gain more profit. In the bifurcation region, with the increase of the competitive level, both in the MS model and the RS model, the retailer has the lower NB product price and SB product price. In addition, we can see that the game leader will have advantages in the stability region, but once enters into the bifurcation region or the chaos region, the game followers will adjust their price to follow the leader’s decision so that they can make a profit.


## 6. Conclusions

This paper researches two noncooperative dynamic pricing strategies used in a supply chain. We establish dynamic Stackelberg game models in which either the manufacturer or the retailer acts as a leader. In these two models, the manufacturer sells a national-brand (NB) product to an independent retailer or directly to consumers through a direct channel. The retailers sell a store-brand (SB) product when they sell the NB product coming from the manufacturer. So, there is competition both in different channels and in products with different brands. We consider the two Stackelberg game: either the manufacturer acts as a leader or the retailer acts as a leader. We use dynamic theory to analyze the complexity of the model, such as bifurcation and chaos. The results indicate that in the MS model, with the increase of the adjustment speed, both the wholesale price and the direct selling price are higher which in turn incur larger profits for the manufacturer, but it gets worse when the adjustment speed increases. In the RS model, with the increase of the adjustment speed, the increase of the adjustment speed is unfavorable to the retailer. By controlling the total cost of the direct channel and increasing channel competition strength and brand competitive strength, the manufacturers can increase their profits in the game dominated by the retailer. In addition, the stable region within the system will become smaller when the market is sensitive to the channel competition, brand competition, and advertising indifference. The research will serve as a good guidance for the supply chain managers who wish to achieve the best business results.



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*Source: 4802360-2019-12-05.xml*