Pricing and coordination in a dual-channel supply chain considering internet celebrity’s sales effort and personal influence

Abstract

The main purpose of this paper is to study the coordination, price and sales effort decisions of a dual channel supply chain under live streaming commerce mode. In nowadays’ e-commerce age, more and more people have interest in live streaming especially after the outbreak of COVID-19, but the research on live streaming supply chain is lacking. To fill this gap, a supply chain composed of a manufacturer and an internet celebrity is established, in which the demand is affected by the internet celebrity’s sales effort and personal influence. Considering different power structures of the supply chain, price and sales effort decisions are studied in four models: Nash, manufacturer dominant (MD), internet celebrity dominant (KD) and cooperative game models. Subsequently, the feasible region of bargaining game is discussed in terms to share the extra profits and coordinate the supply chain. The manufacturer and the internet celebrity can be coordinated through bargaining problem in the cooperation model, and the extra profits sharing ratio is depend on each other’s bargaining power. Numerical analysis is further provided to test the propositions and show the impacts of market share rate, internet celebrity’s commission rate and personal influence on supply chain’s performance.

1 Introduction

In the past years, the continuing development of cloud computing, big data, artificial intelligence, among many other latest technology advancements really subvert people’s lives. Automation and data analytics have become major forces to improve the efficiency of operations management [1]. In the end of 2019, the sudden outbreak of COVID-19 has an enormous strike on offline retail. More and more business companies launch well-prepared promotional activities online, and e-commerce live broadcast as a new business model has been booming. The total global e-commerce sales reached $4.28 trillion in 2020, a significant increase of 27.6% compared with that of last year [2]. A report on China’s live streaming industry revealed that the number of live streaming users in China has reached 635 million by the end of 2021, accounting for 61.5% of total internet users, including 464 million e-commerce live streaming users [3].

With the continuous development of logistics and network technology, live broadcasting as a commercial form also experiences constant updating and iteration, which becomes a necessary channel for sales. In live streaming rooms, the anchors act like a retailer to introduce the products in details and encourage people to buy, moreover, they can interact with people watching the broadcast immediately [4]. It has been proved that internet celebrity live broadcast can promote sales and bring popularity for the products [5, 6].

The high turnover brought by live streaming is closely related to the personal influence of e-commerce anchors. The anchors have more fans, the sales will be higher especially in some top streamer’s live broadcasting rooms. During the Double Eleven Shopping Festival in 2021, Jiaqi Li sold products collectively worth about $1.66 billion in 12 hours [1]. Many people including stars and business leaders come into the live streaming rooms with goods in these two years, such as Mingzhu Dong, Michael Yu, and so on. The professional and interactive features shown by the internet celebrities in the process of live broadcasting can significantly improve consumers’ purchase intention, which is depend on the sales effort they made [7-9]. Mingzhu Dong, chairman of Gree Electric Appliance, made the first live streaming in April 2020. The number of people online reached 4.31 million, but the sales were less than $33 thousand. On May 10th, Mingzhu Dong carried out the second livestream, however, the sales were more than $43.4 million within 3 hours. Different from the first live, Mingzhu Dong introduced the products in details, had a good interaction with audience, and the atmosphere in the live streaming room was better.

With the booming of the live streaming, the development of the live streaming supply chain is becoming more and more important. While the supply chain under live streaming mode is quite different from the traditional supply chain. In traditional supply chain, the consumers buy goods from the retailers or directly from the manufactures. In live streaming supply chain, the manufacturers promote the products by live broadcasting, and the internet celebrity can also be invited to join in the supply chain to sell the products. To expand the influence, the price in internet celebrity’s channel is decided by themselves [10, 11]. The internet celebrity’s service fee is varied according to different anchors, different products can get different commissions, anchors with a certain number of fans need to pay an additional service fee. Sales effort and personal influence of the internet celebrity have a great role in the supply chain under live streaming mode, but the participation of internet celebrities requires booth fee and commission rates at the same time. Therefore, the supply chain manager needs to develop an appropriate mode to cooperated with.

In practice, the manufacturer and the internet celebrity are two independent entities striving to maximize the interests for themselves. In order to fully realize the functions of live streaming supply chain, a dual-channel supply chain considering internet celebrity’s sales effort and personal influence is established, moreover, three decentralized decision-making models and a cooperation model are analyzed. The questions concerning this study are as follows:

How does the internet celebrity’s sales effort and personal influence affect the decisions of the members and the profits in the supply chain?

Who should dominant in the live streaming supply chain: the manufacturer or the internet celebrity?

How can this new kind of supply chain (dual-channel supply chain considering internet celebrity’s sales effort and personal influence) be coordinated?

The contributions of this paper are as follows: (i) motivated by the practical observation of internet celebrities bringing goods in the live streaming room, the impacts of internet celebrity’s sales effort and personal influence on the demand are considered. This appears new in the literature and may provide suggestions for supply chain managers to select internet celebrities. (ii) few researches consider influence spillover, combining the fact that more and more manufacturers sell products to customers directly through the live streaming room, we establish a dual-channel live streaming supply chain and consider the internet celebrity’s personal influence spillover between two sales channels. (iii) cooperation scenario is modelled that aims for achieving the overall profit maximum, subjecting to Pareto improvement for each member. In addition, the members’ attitude to risks are considered in the bargaining problem. These may help the manufacturer and the internet celebrity to improve the profits and cooperate in the long term.

The structure of this paper is as follows. In Section 2, the related literature is reviewed. In Section 3, model descriptions and assumptions are introduced. In Section 4, three decentralized models are established and the optimal decisions are obtained separately. In Section 5, the feasibility of cooperation and the solution of bargaining problem are analyzed. Section 6 provides numerical analysis. Finally, some conclusions and further insights are summarized in Section 7.

2 Related literature

This work is related to three streams of literature: live streaming mode, supply chain coordination and impacts of internet celebrities on demand.

2.1 Live streaming mode

Live streaming mode, as a new sales method, has attracted a lot attentions from scholars and businessmen. However, there is still a lack of theoretical research on live streaming mode as far as we know. The rapid development of e-commerce live streaming benefits from anchor interaction and user experience, and it was recognized and trusted by consumers in terms of innovation [12, 13]. E-commerce platform mode, transferring mode and live streaming platform mode by game theoretical method were studied as three common live streaming modes [14]. The results were found that each mode might be the best for the members, which was dependent on the basic net sales and the channel rate. The rational pricing decisions and excellent quality were the keys to profitability and impulse spending in the live e-commerce supply chain with online influencers [15].

Some scholars studied the influence of live stream mode on consumers through empirical research. Two empirical investigations (an online survey and a live streaming observations) were conducted, in which the information production behavior like broadcasting and information receiving behavior including watching and commenting on streaming media are analyzed [16]. It was found that the interaction between consumers and anchors would be the key to the development of e-commerce live streaming in the future [17]. Guided shopping dance would increase consumers’ purchase intention, and user experience was the core of live streaming. Meanwhile, the number of page views was found as the strongest predictor of live broadcast sales [18].

2.2 E-commerce supply chain coordination

Supply chain coordination have caught much attention in the literature. Previous studies have adequately discussed the supply chain coordination, such as wholesale price contact [19], revenue-sharing contact [20], cost-sharing contact [21], buyback contract [22-24], quantity discount contact [25], sales rebate contract [26] and so on.

In recent years, many coordination models and contracts were studied in which members sold products through e-commerce. The online-offline supply chain was established composed of a manufacture and a retailer considering corporate environmental responsibility and lateral inventory transshipment, where they proposed a revenue-sharing contract to realize the coordination [27]. A cost-sharing mechanism in the logistics service supply chain was explored, and the influences in members’ decisions were discussed [28]. The dynamic cost-sharing contract approach was proposed and utilized to coordinate the small-scale e-commerce supply chain [29]. The dual-channel reverse supply chain (DRSC) was built with a recycling center and a third-party recycler (TPR), and a revenue sharing contract was adopted to optimize the benefits of supply chain members in decentralized model [30].

A single contract cannot coordinate the e-commerce supply chain completely in some cases, some scholars designed joint contract and two-part tariff contract. The coordination problems in an offline channel and an online platform were investigated through wholesale price and cost-sharing contracts [31]. Cao et al. [32] investigated an online supply chain involving a manufacture and a retailer where they sold new and remanufactured products, and explored a two-part tariff contract to coordinate the supply chain when the return cost sharing contract is invalid. Considering the effect of product variety in a dual-channel supply chain in which the retailer has countervailing power, Wang et al. [33] found the two-part tariff pricing contracts could help the supply chain obtain the same profit in decentralized model as that in the centralized model. Both the improved revenue-sharing contract and the two-part tariff contract realized coordination for the omni-channel supply chain, and all members achieved Pareto improvement [34].

2.3 Impacts of internet celebrities on demand

Many previous research in terms to traditional supply chain studied that the demand was affected by retail price [35, 36], sales effort [9, 37] and quality [38, 39]. Time sensitivity, green degree and carbon reduction dependent demand were also considered in some literature about green supply chain [40, 41].

With the emergence of live streaming, research on internet celebrity selling products to consumers through live streaming mode was becoming more and more popular. Three contextual and environmental stimuli effects were investigated based on 300 valid survey data, and found the stimuli have direct effect on cognitive and emotional states, which offering insights that how live streaming stimuli influence consumers [42]. An online retailer selling products was considered through activities like live streaming platform during the selling season, and the extended demand was depended on the sales effort level [43]. Considering spillover effect on the other sales channel, Zhang et al. [11] and Zhang et al. [44] built dual-channel supply chains, and studied the impact of internet celebrities’ effort on the demand. The headmost anchors always built closed-loop systems for themselves [45]. However, the Mid-waist anchors need services and sales effort such like attracting fans, choosing product and building supply chain.

With the development of economy and information technology, live streaming, as a new e-commerce mode, more and more supply chain members engage in. Most of the related paper focused on the decisions in the supply chain under live streaming mode. The research in terms to coordination in live streaming supply chain remains in its infancy. However, the live streaming supply chain is quite different from the traditional supply chain, especially in the mode and characteristic of live streaming.

Motivated by the research on e-commerce supply chain coordination, this paper built a dual-channel live streaming supply chain composed of an internet celebrity and a manufacturer, discussed the price and sales effort decisions, and the coordination between the supply chain members is achieved. Inspired by the internet celebrity, another difference from current research is that our study considers the internet celebrity’s personal influence on demand. Moreover, the spillover effect on the demand is considered too, providing more insights for supply chain managers.

The closely related literature to compare this paper is listed in Table 1.

Table 1

Literature comparison

Literature	Coordi-nation	Demand influencing factor	Channel types		Decision variables
			Single	Dual	Live	Sales	Cross	Others
					price	effort	price
Liu et al. [46]		consumers’ regret, price, effort	√		√	√	×	quality
Peng et al. [47]		random rewards, price	√		√	×	×	quantity
Jiang and Cai [14]		impulsive consumption, price	√		√	×	×	wholesale price
Zhang et al. [11]		spillover effect, price		√	√	×	√	service level
Zhang et al. [43]		spillover effect, price, effort		√	√	√	√
Cao et al. [27]	√	environmental responsibility, price		√	√	×	√	wholesale price
Zhu and Liu [28]	√	price, effort	√		√	√	×	wholesale price
Rathnasiri et al. [29]	√	inventory, capacity	√		×	×	×	cost
This paper	√	personal influence/spillover effect		√	√	√	√

3 Model description and assumptions

3.1 Model description

This study investigates a dual-channel supply chain involving a manufacturer and an internet celebrity, in which the manufacturer can provides products directly to consumers through live streaming and it can also sell the products by the internet celebrity who can make the consumer feel more connected to the products in a short time. Usually, Internet celebrity with a little popularity and fans charge fees in the form of booth fee and commission. No matter how many products are sold, booth fee is fixed, and the commission is depended on the sales. Meanwhile, the manufacturer engages in live streaming to increase the sales.

The manufacturer determines the sale price in its own channel, and the internet celebrity determines the sale price and the sales effort for selling and marketing products in its own channel. In the internet celebrity’s live broadcast channel, the demand is assumed to be dependent on the price, the sales effort and the personal influence. In fact, through qualitative research and experiment research, the internet celebrity’s personal influence also has impacts on the consumer’s willingness to buy the products from the manufacturer. In the manufacturer’s channel, the demand is assumed to be dependent on the price and the internet celebrity’s personal influence spillover [10, 11]. The symbols and notations used are as follows in Table 2.

Table 2

Symbols and notations

Parameters
t	internet celebrity’s commission rate;
c	the unit manufacturing cost;
α	the market potential;
d	the market potential share of internet celebrity’s live broadcast channel;
b	the marketing cost coefficient of internet celebrity;
m	the personal influence of internet celebrity;
f	internet celebrity’s booth fee;
β	the responsiveness of demand to retail price;
γ	the responsiveness of demand to cross-price;
k	the responsiveness of demand to internet celebrity live mode;
h	the responsiveness of demand to internet celebrity’s personal influence spillover.
Decision variables
p m	the sale price in manufacturer’s live broadcast channel;
p r	the sale price in internet celebrity’s live broadcast channel;
e	internet celebrity’s sales effort;
Superscript
*	the optimal (equilibrium) solution;
c	the supply chain in the centralized model;
N	the supply chain in the Nash game model;
KD	the supply chain in the internet celebrity dominant (KD) decentralized model;
MD	the supply chain in the manufacturer dominant (MD) decentralized model;

3.2 Model assumptions

Assumption 1: p _m  > c > 0, which makes sure that the manufacturer can obtain profit from its own live broadcast channel. p _r  > c > 0, which make sure that the manufacturer can obtain profit from internet celebrity live broadcast channel.

Assumption 2: The function of the customer’s demand in manufacturer live broadcast channel is assumed to be D _m  (p _m ) = (1 - d) α - βp _m  + γp _r  + hm. The function of the customer’s demand in internet celebrity live broadcast channel is assumed to be D _r  (p _r , e) = dα - βp _r  + γp _m  + k (e + m). α, β, γ, h, m and k are all nonnegative and independent, 0 < d < 1, D₁ > 0, D₂ > 0, β > γ means the demand is more sensitive to the price in its channel than the other.

Assumption 3: The function of sales effort costs of the internet celebrity is $l\left(e\right)=\frac{1}{2}b{e}^2$ . The function is convex, it is obvious that l (0) = 0, l′ (e) = 2be > 0, l″ (e) = 2b > 0.

The profit functions of the supply chain as a whole system and two members are formulated as:

$\begin{array}{c}{\pi }_c\left(p_m,p_r,e\right)=\left(p_m-c\right)\left[\left(1-d\right)\alpha -\beta p_m+\gamma p_r+\mathit{hm}\right]\hfill \\ +\left(p_r-c\right)\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]-\frac{1}{2}b{e}^2\hfill \end{array}$(1)

$\begin{array}{c}{\pi }_m\left(p_m\right)=\left(p_m-c\right)\left[\left(1-d\right)\alpha -\beta p_m+\gamma p_r+\mathit{hm}\right]\hfill \\ +\left[\left(1-t\right)p_r-c\right]\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]-f\hfill \end{array}$(2)

$\begin{array}{c}{\pi }_k\left(p_r,e\right)=t{p}_r\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]\hfill \\ +f-\frac{1}{2}b{e}^2\hfill \end{array}$(3)

The manufacturer’s profit includes sales profit from the its live broadcast channel and the internet celebrity’s live broadcast channel. The internet celebrity’s profit involves commission and booth fee, the commission on the sales profit come from its own live broadcast channel. In addition, the booth fee is fixed which is depended on its reputation, experience, the product type and the cooperation mode.

3.3 Data and computational environment

The MATLAB R2021b was utilized to compute and analyze the models. The numerical solution and related graphs were obtained by this software.

4 Decision making structures

In this section, three non-cooperative models based on game theory are built to study the decisions in Nash game, KD game and MD game. For the sustainable development of network celebrity live broadcast in supply chain, a cooperative model is expanded at last.

4.1 Nash game

When the manufacturer and the internet celebrity make decisions separately and simultaneously with the same power. This is called a Nash game and the solution to this structure is the Nash equilibrium [48]. The decision problems can be solved as follows: $\begin{array}{cc}\hfill \{\begin{array}{c}\mathit{Max}{\pi }_m\left(p_m\right)=\left(p_m-c\right)\left[\left(1-d\right)\alpha -\beta p_m+\gamma p_r+\mathit{hm}\right]\hfill \\ +\left[\left(1-t\right)p_r-c\right]\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]-f\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \\ \mathit{Max}{\pi }_k\left(p_r,e\right)=t{p}_r\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]\hfill \\ +f-\frac{1}{2}b{e}^2\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \end{array}& \hfill \end{array}$

Proposition 1. When 2 b tβ - k²t² > 0, the Nash game has the following unique Nash equilibrium:

$p_r^{N*}=\frac{\begin{array}{c}\alpha b\gamma -bc{\gamma }^2+2\alpha b\beta d+b\beta c\gamma \hfill \\ -\alpha bd\gamma +2b\beta km+b\gamma hm\hfill \end{array}}{4b{\beta }^2-2b{\gamma }^2-2\beta k^{2}t+b{\gamma }^{2}t}$(4)

$e^{N*}=\frac{kt\left(\begin{array}{c}\alpha \gamma -c{\gamma }^2+2\alpha \beta d+\beta c\gamma \hfill \\ -\alpha d\gamma +2\beta km+\gamma hm\hfill \end{array}\right)}{4b{\beta }^2-2b{\gamma }^2-2\beta k^{2}t+b{\gamma }^{2}t}$(5)

$p_m^{N*}=\frac{\begin{array}{c}\hfill 2\alpha b\beta +2b{\beta }^{2}c-\alpha k^{2}t+2b\gamma km-\beta c{k}^{2}t\hfill \\ \hfill +\alpha d{k}^{2}t+c\gamma k^{2}t-h{k}^{2}mt-2\alpha b\beta d-2b\beta c\gamma \hfill \\ \hfill +2\alpha bd\gamma +2b\beta hm-\alpha bd\gamma t-b\gamma kmt\hfill \end{array}}{4b{\beta }^2-2b{\gamma }^2-2\beta k^{2}t+b{\gamma }^{2}t}$(6)

The proof of proposition 1 is shown in the Appendix.

The maximum expected profits of the manufacturer and the internet celebrity in Nash game model are expressed as follows.

${\pi }_m^{N\text{*}}=\left(p_m^{N\text{*}}-c\right)\left[\left(1-d\right)\alpha -\beta p_m^{N\text{*}}+\gamma p_r^{N\text{*}}+hm\right]+\left[\left(1-t\right)p_r^{N\text{*}}-c\right]\left[\begin{array}{c}d\alpha -\beta p_r^{N\text{*}}+\gamma p_m^{N\text{*}}\\ +k\left(e^{N\text{*}}+m\right)\end{array}\right]-f$(7)

$\pi k_m^{N\text{*}}=t{p}_r^{N\text{*}}\left[\begin{array}{c}d\alpha -\beta p_r^{N\text{*}}+\gamma p_m^{N\text{*}}\\ +k\left(e^{N\text{*}}+m\right)\end{array}\right]+f-\frac{1}{2}b{e}^{N\text{*}}{}^2$(8)

4.2 KD game

In this section, the decentralized problem is discussed as the KD Stackelberg game, in which the internet celebrity is the leader and the manufacturer is the follower. The manufacturer decides the price in its own channel first, and then the internet celebrity determines the price and sales effort in its own channel. The optimization model can be formulated as follows:

$\begin{array}{cc}\hfill \{\begin{array}{c}\mathit{Max}{\pi }_k\left(p_r,e,p_m^{\mathit{KD}*}\left(p_r\right)\right)\hfill \\ =t{p}_r\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]+f-\frac{1}{2}b{e}^2\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \\ \mathit{where}{p}_m^{\mathit{KD}*}\left(p_r\right)\mathit{are}\mathit{solved}\mathit{by}\mathit{the}\mathit{following}\mathit{problem}\hfill \\ \mathit{Max}{\pi }_m\left(p_m\right)=\left(p_m-c\right)\left[\left(1-d\right)\alpha -\beta p_m+\gamma p_r+\mathit{hm}\right]\hfill \\ +\left[\left(1-t\right)p_r-c\right]\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]-f\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \end{array}& \hfill \end{array}$

Proposition 2. In the KD model, when $2\mathit{bt}\left(\beta -\frac{{\gamma }^2\left(2-t\right)}{2\beta }\right)-k^2{t}^2>0$ , the Stackelberg equilibrium solution denoted as $\left(p_r^{\mathit{KD}*},e^{\mathit{KD}*},p_m^{\mathit{KD}*}\right)$ can be obtained:

$p_r^{\mathit{KD}*}=\frac{b\left(\begin{array}{c}\alpha \gamma -c{\gamma }^2+2\alpha \beta d+\beta c\gamma -\alpha d\gamma \hfill \\ +2\beta km+\gamma hm\hfill \end{array}\right)}{2\left(2b{\beta }^2-2b{\gamma }^2-\beta k^{2}t+b{\gamma }^{2}t\right)}$(9)

$e^{\mathit{KD}*}=\frac{kt\left(\begin{array}{c}\alpha \gamma -c{\gamma }^2+2\alpha \beta d+\beta c\gamma -\alpha d\gamma \hfill \\ +2\beta km+\gamma hm\hfill \end{array}\right)}{2\left(2b{\beta }^2-2b{\gamma }^2-\beta k^{2}t+b{\gamma }^{2}t\right)}$(10)

$\begin{array}{c}{p}_m^{\mathit{KD}*}=\frac{\beta c+hm-\alpha \left(d-1\right)-\gamma \left(c+\frac{b\left(t-1\right){\sigma }_1}{{\sigma }_2}\right)+\frac{b\gamma {\sigma }_1}{{\sigma }_2}}{2\beta }\hfill \\ \mathit{Where}\hfill \\ {\sigma }_1=\alpha \gamma -c{\gamma }^2+2\alpha \beta d+\beta c\gamma -\alpha d\gamma \hfill \\ +2\beta km+\gamma hm\hfill \\ {\sigma }_2=2\left(2b{\beta }^2-2b{\gamma }^2-\beta k^{2}t+b{\gamma }^{2}t\right)\hfill \end{array}$(11)

Proposition 2 is verified in the Appendix.

The maximum expected profits of the manufacturer and the internet celebrity in the KD model are expressed as follows.

$\begin{array}{l}{\pi }_m^{KD*}=\left(p_m^{KD\text{*}}-c\right)\left[\left(1-d\right)\alpha -\beta p_m^{KD\text{*}}+\gamma p_r^{KD\text{*}}+hm\right]\\ +\left[\left(1-t\right)p_r^{KD\text{*}}-c\right]\left[\begin{array}{c}d\alpha -\beta p_r^{KD\text{*}}+\gamma p_m^{KD\text{*}}\\ +k\left(e^{KD\text{*}}+m\right)\end{array}\right]-f\end{array}$(12)

$p{i}_k^{KD*}=t{p}_r^{KD\text{*}}\left[\begin{array}{c}d\alpha -\beta p_r^{KD\text{*}}+\gamma p_m^{KD\text{*}}\\ +k\left(e^{KD\text{*}}+m\right)\end{array}\right]+f-\frac{1}{2}b{e}^{KD\text{*}}{}^2$(13)

4.3 MD game

In this section, the decentralized problem is discussed as a MD Stackelberg game, in which the manufacturer is the leader and the internet celebrity is the follower. The internet celebrity determines the price and sales effort in its own channel first, and then the manufacturer decides the price in its own channel. The optimization model can be formulated as follows:

$\begin{array}{cc}\hfill \{\begin{array}{c}\mathit{Max}{\pi }_m\left(p_m,p_r^{\mathit{MD}*}\left(p_m\right),e^{\mathit{MD}*}\left(p_m\right)\right)\hfill \\ =\left(p_m-c\right)\left[\left(1-d\right)\alpha -\beta p_m+\gamma p_r+\mathit{hm}\right]\hfill \\ +\left[\left(1-t\right)p_r-c\right]\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]-f\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \\ \mathit{where}{p}_r^{\mathit{MD}*}\left(p_m\right)\mathit{and}{e}^{\mathit{MD}*}\left(p_m\right)\mathit{are}\mathit{solved}\hfill \\ \mathit{by}\mathit{the}\mathit{following}\mathit{problem}\hfill \\ \mathit{Max}{\pi }_k\left(p_r,e\right)=t{p}_r\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]\hfill \\ +f-\frac{1}{2}b{e}^2\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \end{array}& \hfill \end{array}$

Proposition 3. In the MD model, when $\frac{\beta \left(6b^2{\gamma }^2-2b^2{\gamma }^{2}t\right)-2b{\gamma }^2{k}^{2}t}{{\left(2b\beta -k^{2}t\right)}^2}-2\beta <0$ and 2 b tβ - k²t² > 0, the Stackelberg equilibrium solution denoted as $\left(p_r^{\mathit{MD}*},e^{\mathit{MD}*},p_m^{\mathit{MD}*}\right)$ can be obtained:

$p_r^{\mathit{MD}*}=\frac{\begin{array}{c}41\alpha 1b^{2}1{\beta }^2+41b^{2}1{\beta }^{3}1c+\alpha 1k^{4}1t^2-41\alpha 1b^{2}1{\beta }^{2}1d\hfill \\ -21b^{2}1\beta 1c1{\gamma }^2-21b^{2}1{\beta }^{2}1c1\gamma +41b^{2}1{\beta }^{2}1h1m+\beta 1c1k^{4}1t^2\hfill \\ -\alpha 1d1k^{4}1t^2+h1k^{4}1m1t^2-41b1{\beta }^{2}1c1k^{2}1t+b1c1{\gamma }^{2}1k^{2}1t\hfill \\ +41\alpha 1b^{2}1\beta 1d1\gamma +41b^{2}1\beta 1\gamma 1k1m-41\alpha 1b1\beta 1k^{2}1t-b1\gamma 1k^{3}1m1t\hfill \\ -21\alpha 1b^{2}1\beta 1d1\gamma 1t+41\alpha 1b1\beta 1d1k^{2}1t+b1\beta 1c1\gamma 1k^{2}1t\hfill \\ -\alpha 1b1d1\gamma 1k^{2}1t-21b^{2}1\beta 1\gamma 1k1m1t-41b1\beta 1h1k^{2}1m1t\hfill \end{array}}{21\left(\begin{array}{c}41b^{2}1{\beta }^3+b^{2}1\beta 1{\gamma }^{2}1t-31b^{2}1\beta 1{\gamma }^2-41b1{\beta }^{2}1k^{2}1t\hfill \\ +b1{\gamma }^{2}1k^{2}1t+\beta 1k^{4}1t^2\hfill \end{array}\right)}$(14)

$e^{\mathit{MD}*}=\frac{\begin{array}{c}\hfill -kt\left(\begin{array}{c}\alpha \gamma k^{2}t+2\beta k^{3}mt+2\alpha \beta d{k}^{2}t\hfill \\ +\beta c\gamma k^{2}t-\alpha d\gamma k^{2}t+\gamma h{k}^{2}mt\hfill \end{array}\right)\hfill \\ \hfill -bkt\left(\begin{array}{c}c{\gamma }^3-2\alpha \beta \gamma -4\alpha {\beta }^{2}d+\beta c{\gamma }^2\hfill \\ -2{\beta }^{2}c\gamma +\alpha d{\gamma }^2-4{\beta }^{2}km+{\gamma }^{2}km\hfill \\ +2\alpha \beta d\gamma -2\beta \gamma hm\hfill \end{array}\right)\hfill \end{array}}{2\left(\begin{array}{c}4b^2{\beta }^3+b^2\beta {\gamma }^{2}t-3b^2\beta {\gamma }^2-4b{\beta }^2{k}^{2}t\hfill \\ +b{\gamma }^2{k}^{2}t+\beta k^4{t}^2\hfill \end{array}\right)}$(15)

$p_m^{\mathit{MD}*}=\frac{\begin{array}{c}\hfill \begin{array}{c}b\left(4\alpha bd+2bc\gamma +4bkm\right){\beta }^2\hfill \\ -b\left(\begin{array}{c}2k^{3}mt-2\alpha b\gamma +bc{\gamma }^2+2\alpha d{k}^{2}t\hfill \\ +c\gamma k^{2}t+2\alpha bd\gamma -2b\gamma hm\hfill \end{array}\right)\beta \hfill \end{array}\hfill \\ \hfill -b\left(\begin{array}{c}bc{\gamma }^3+\alpha bd{\gamma }^2+b{\gamma }^{2}km\hfill \\ +\alpha \gamma k^{2}t-\alpha d\gamma k^{2}t+\gamma h{k}^{2}mt\hfill \end{array}\right)\hfill \end{array}}{2\left(\begin{array}{c}4b^2{\beta }^3+b^2\beta {\gamma }^{2}t-3b^2\beta {\gamma }^2-4b{\beta }^2{k}^{2}t\hfill \\ +b{\gamma }^2{k}^{2}t+\beta k^4{t}^2\hfill \end{array}\right)}$(16)

The proof of proposition 3 is shown in the Appendix.

The maximum expected profits of the manufacturer and the internet celebrity in the MD model are expressed as follows.

$\begin{array}{l}{\pi }_m^{MD*}=\left(p_m^{MD\text{*}}-c\right)\left[\left(1-d\right)\alpha -\beta p_m^{MD\text{*}}+\gamma p_r^{MD\text{*}}+hm\right]\\ +\left[\left(1-t\right)p_r^{MD\text{*}}-c\right]\left[\begin{array}{c}d\alpha -\beta p_r^{MD\text{*}}+\gamma p_m^{MD\text{*}}\\ +k\left(e^{MD\text{*}}+m\right)\end{array}\right]-f\end{array}$(17)

${\pi }_k^{MD*}=t{p}_r^{MD\text{*}}\left[\begin{array}{c}d\alpha -\beta p_r^{MD\text{*}}+\gamma p_m^{MD\text{*}}\\ +k\left(e^{MD\text{*}}+m\right)\end{array}\right]+f-\frac{1}{2}b{e}^{MD\text{*}}{}^2$(18)

5 Cooperation

Three non-cooperative games are discussed before. In this section, the manufacturer and internet celebrity are assumed to cooperate to derive the maximum profits of the whole supply chain. It is formulated as a centralized decision-making model. The optimization model can be formulated as follows:

$\begin{array}{cc}\hfill \{\begin{array}{c}\mathit{max}{\pi }_c\left(p_m,p_r,e\right)=\left(p_m-c\right)\left[\begin{array}{c}\left(1-d\right)\alpha -\beta p_m\hfill \\ +\gamma p_r+\mathit{hm}\hfill \end{array}\right]\hfill \\ +\left(p_r-c\right)\left[d\alpha -\beta p_r+\gamma p_m+k\left(e+m\right)\right]-\frac{1}{2}b{e}^2\hfill \\ s.t.p_m>c;p_r>c;e>0\hfill \end{array}& \hfill \end{array}$

Proposition 4. In the centralized model, when -2 b β² + β k² + 2 b γ² < 0, the optimal decisions and profits of the manufacturer and internet celebrity are given as follows:

$p_m^{C*}=\frac{\begin{array}{c}\hfill \begin{array}{c}2\alpha b\beta -\alpha k^2+2b{\beta }^{2}c-2bc{\gamma }^2-\beta c{k}^2\hfill \\ +\alpha d{k}^2-c\gamma k^2-h{k}^{2}m+2b\gamma km\hfill \end{array}\hfill \\ \hfill -2\alpha b\beta d+2\alpha bd\gamma +2b\beta hm\hfill \end{array}}{2\left(2b{\beta }^2-\beta k^2-2b{\gamma }^2\right)}$(19)

$p_r^{C*}=\frac{\begin{array}{c}\alpha b\gamma +b{\beta }^{2}c-bc{\gamma }^2-\beta c{k}^2+\alpha b\beta d\hfill \\ -\alpha bd\gamma +b\beta km+b\gamma hm\hfill \end{array}}{2b{\beta }^2-\beta k^2-2b{\gamma }^2}$(20)

$e^{C*}=\frac{k\left(\begin{array}{c}\alpha \gamma -{\beta }^{2}c+c{\gamma }^2+\alpha \beta d\hfill \\ -\alpha d\gamma +\beta km+\gamma hm\hfill \end{array}\right)}{2b{\beta }^2-\beta k^2-2b{\gamma }^2}$(21)

$\begin{array}{l}{\pi }_m^{C*}=\left(p_m^{C\text{*}}-c\right)\left[\left(1-d\right)\alpha -\beta p_m^{C\text{*}}+\gamma p_r^{C\text{*}}+hm\right]\\ +\left[\left(1-t\right)p_r^{C\text{*}}-c\right]\left[\begin{array}{c}d\alpha -\beta p_r^{C\text{*}}+\gamma p_m^{C\text{*}}\\ +k\left(e^{C\text{*}}+m\right)\end{array}\right]-f\end{array}$(22)

${\pi }_k^{C*}=t{p}_r^{C\text{*}}\left[\begin{array}{c}d\alpha -\beta p_r^{C\text{*}}+\gamma p_m^{C\text{*}}\\ +k\left(e^{C\text{*}}+m\right)\end{array}\right]+f-\frac{1}{2}b{e}^{C\text{*}}{}^2$(23)

The Hessian matrix of Equation (1) is $\left[\begin{array}{ccc}\hfill -2\beta \hfill & \hfill 2\gamma \hfill & \hfill 0\hfill \\ \hfill 2\gamma \hfill & \hfill -2\beta \hfill & \hfill k\hfill \\ \hfill 0\hfill & \hfill k\hfill & \hfill -b\hfill \end{array}\right]$ ; if -2 b β² + β k² + 2 b γ² < 0, the Hessian matrix is a negative define matrix. It ensures that π _c  (p _m , p _r , e) is a strictly concave function when -2 b β² + β k² + 2 b γ² < 0, and the supply chain’s profit has a unique solution when the manufacturer cooperates with the internet celebrity as a centralized system. The optimal decisions ( $p_m^{C*}$ , $p_r^{C*}$ , e^C*) obtain the maximum profits for the whole supply chain. The proof of proposition 4 is shown in the Appendix.

Proposition 5. When the parameters meet all the negative define conditions verified before, due to the double marginalization [49], ${\pi }_m^{N*}+{\pi }_k^{N*}={\pi }_c^{N*}<{\pi }_c^{C*}$ , ${\pi }_m^{\mathit{MD}*}+{\pi }_k^{\mathit{MD}*}={\pi }_c^{\mathit{MD}*}<{\pi }_c^{C*}$ , ${\pi }_m^{\mathit{KD}*}+{\pi }_k^{\mathit{KD}*}={\pi }_c^{\mathit{KD}*}<{\pi }_c^{C*}$ , which means the maximum profits of the entire supply chain system cannot be achieved in the decentralized models no matter which member dominates the game.

It is obviously that both the manufacturer and the internet celebrity want to participate in the cooperation only if their own profits are higher than that in non-cooperative game models. The maximum profits of the manufacturer and the internet celebrity are: $\begin{array}{cc}\hfill \begin{array}{c}{\pi }_m^{\mathit{max}}=\max \left({\pi }_m^{N*},{\pi }_m^{\mathit{KD}*},{\pi }_m^{\mathit{MD}*}\right)\hfill \\ {\pi }_k^{\mathit{max}}=\max \left({\pi }_k^{N*},{\pi }_k^{\mathit{KD}*},{\pi }_k^{\mathit{MD}*}\right).\hfill \end{array}& \hfill \end{array}$ The feasibility of the cooperative game is discussed as follows.

$\begin{array}{c}\Delta \pi m=\pi mC*-{\pi }_m^{\mathit{max}}=\left(p_m^{C*}-c\right)\left[\begin{array}{c}\left(1-d\right)\alpha -\beta p_m^{C*}\hfill \\ +\gamma p_r^{C*}+\mathit{hm}\hfill \end{array}\right]\hfill \\ +\left[\left(1-T\right)p_r^{C*}-c\right]\left[d\alpha -\beta p_r^{C*}+\gamma p_m^{C*}+k\left(e^{C*}+m\right)\right]\hfill \\ -F-{\pi }_m^{\mathit{max}}=-\mathit{GT}-F+A\hfill \end{array}$(24)

$\begin{array}{c}\Delta \pi k=\pi kC*-{\pi }_k^{\mathit{max}}={\mathit{Tp}}_r^{C*}\left[\begin{array}{c}d\alpha -\beta p_r^{C*}+\gamma p_m^{C*}\hfill \\ +k\left(e^{C*}+m\right)\hfill \end{array}\right]\hfill \\ +F-\frac{1}{2}b{e^{C*}}^2-{\pi }_k^{\mathit{max}}=\mathit{GT}+F-B\hfill \end{array}$(25) Where $\begin{array}{cc}\hfill \begin{array}{c}{D}_r^{C*}=d\alpha -\beta p_r^{C*}+\gamma p_m^{C*}+k\left(e^{C*}+m\right)\hfill \\ D_m^{C*}=\left(1-d\right)\alpha -\beta p_m^{C*}+\gamma p_r^{C*}+\mathit{hm}\hfill \\ A=\left(p_m^{C*}-c\right)D_m^{C*}+\left(p_r^{C*}-c\right)D_r^{C*}-{\pi }_m^{\mathit{max}}>0\hfill \\ B=\frac{1}{2}b{e^{C*}}^2+{\pi }_k^{\mathit{max}}>0\hfill \\ G=p_r^{C*}{D}_r^{C*}>0\hfill \end{array}& \hfill \end{array}$

The profits of the manufacturer and the internet celebrity are both dependent on F and T. When F₁ = - GT + A, Δπ _m  = 0 which means the extra profit gained by the internet celebrity is the highest, however, there is no motivation for the manufacturer to join this cooperation. When F₂ = - GT + B, Δπ _k  = 0 which means the extra profit gained by the manufacturer is the highest, but the internet celebrity won’t join in. Every pair (T, F) located between the line F₁ and F₂ is feasible. In this paper, Nash bargaining problem [50] is adopted to set the optimal solution (T^*, F^*). Suppose the utility functions of the manufacturer and the internet celebrity are: u _m  (T, F) = Δπ _m  (T, F)  ^{ξ ₁} , u _k  (T, F) = Δπ _k  (T, F)  ^{ξ ₂} . Where ξ₁ and ξ₂ represents the attitude of the manufacturer and the internet celebrity to risk.

Then, the optimal Nash bargaining solution (T^*, F^*) is obtained by solve the following problem. $\begin{array}{cc}\hfill \begin{array}{c}\mathit{max}N\left(T,F\right)=u_m\left(T,F\right)u_k\left(T,F\right)\hfill \\ ={\left[-\mathit{GT}-F+A\right]}^{{\xi }_1}{\left[\mathit{GT}+F-B\right]}^{{\xi }_2}\hfill \\ s.t.B-\mathit{GT}<F<A-\mathit{GT};0<T<1.\hfill \end{array}& \hfill \end{array}$

Proposition 6. In bargaining problem, the supply chain members can realize Pareto utility optimization by Nash bargaining equilibrium solution (T^*, F^*), $F^{*}=\frac{A{\xi }_2+B{\xi }_1}{{\xi }_1+{\xi }_2}-G{T}^{*}$ , ξ₁ and ξ₂ represent the members’ attitude to risk.

6 Numerical analysis

In this section, numerical analysis is applied with Matlab software to explain and confirm the propositions before. Furthermore, the influence of the main parameters on optimal decisions and profits is discussed, and the feasibility of the coordination is demonstrated. Referring to the case of previous study [10, 43], without loss of generality, internet celebrity’s booth fee f = 0, the other parameters are assumed in Table 3, which all meet the optimal conditions mentioned before.

Table 3

Numerical assumptions

Parameters	Value	Parameters	Value
t	0.35	β	1
c	10	k	0.3
α	500	γ	0.2
d	0.5	h	0.1
b	10	M	100

6.1 The supply chain’s performance in different models

Consider ξ₁ = ξ₂ = 1, the manufacturer and the internet celebrity are both risk-neutral. The coordination parameters, decisions and profits under different decision-making models are shown in Table 4. It shows that the total supply chain’s profits in the centralized model is greater than that in the decentralized models, while which in the coordination model can also achieve maximum as that in the centralized model. Compared to the decisions made in the centralized model, retail price in two sales channels and the internet celebrity’s sales effort decisions are all higher than that in the decentralized models. In the cooperation model, the manufacturer and the internet celebrity share the extra profits through bargaining problem, the profits of the manufacturer and the internet celebrity are both more than that in the decentralized models without coordination, which incentivizes the manufacturer and the internet celebrity to participate in the cooperation. Moreover, the members’ profits and optimal decisions in Nash model are all lower than that in other models.

Table 4

The optimal decisions and profits under different conditions

Decision -making model	Coordination parameters	p m	p r	e	π k	π m	π
KD model	— —	160.2	158.9	1.7	8531.0	34095.4	42626.4
MD model	— —	161.8	156.4	1.6	8551.3	33996.6	42547.8
Nash model	— —	159.8	156.2	1.6	8528.6	33992.4	42521.0
Centralized model	— —	169.7	178.7	5.1	— —	— —	43071.6
Cooperation model	F = 8891.9 - 24440T	169.7	178.7	5.1	8763.7	34307.9	43071.6

6.2 The influence of d on supply chain performance

In this section, d is assumed to change from 0.1 to 0.9, and the other parameters are the same as before. As shown in Fig. 1, the retail price decided by the manufacturer decreases with the increase of the internet celebrity’s potential market proportion. On the contrary, with the increase of the internet celebrity’s potential market proportion, the optimal price in the internet celebrity’s live broadcast channel is increased. When the market share of internet celebrity is greater than 0.476471, the optimal price of internet celebrity in the live streaming rooms is higher than that of the manufacturer. The expected profit of the supply chain in the centralized model is a smooth surface under the decisions of p _m and p _r , which confirms proposition 4, the expected profit function of the supply chain is concave and has the unique solution to achieve optimal. Thus, the manufacturer should consider the market share of the internet celebrity to improve the profit of the entire system, and then the extra profit can be reallocated which was proved in proposition 5.

Fig. 1

The influence of d on price and profit.

The impacts of internet celebrity’s potential market share on the optimal decisions are shown in Fig. 2. According to Fig. 2, for all values of d, the optimal prices and sales effort decisions in the centralized model are all more than which in the other decentralized models, which confirms proposition 5. When d is increased, the sales effort of internet celebrity is increased especially in the centralized model. It is implied that the internet celebrity tends to devote more sales effort when d is larger, which is in line with the common sense.

Fig. 2

The influence of d on decisions.

6.3 The influence of t and k on supply chain performance

Assume t is changed from 0.1 to 0.9, and the other parameters are the same as before. t is a profit transfer rate from the manufacturer to the internet celebrity, so it has no effects on the optimal decisions under the centralized model. The influence of the internet celebrity’s commission rate on the optimal decisions in the decentralized models is shown in Fig. 3. It is illustrated that the optimal price in the live broadcast channel of manufacturer is decreased with the increase of t, so does the price in the live broadcast channel of internet celebrity. In addition, $p_m^{C*}>p_m^{\mathit{MD}*}>p_m^{\mathit{KD}*}>p_m^{N*}$ , $p_r^{C*}>p_r^{\mathit{KD}*}>p_r^{\mathit{MD}*}>p_r^{N*}$ , the prices in three models confirms proposition 5. Moreover, the optimal price in the channel of manufacturer is declined quicker than that in the live streaming rooms of internet celebrity when t is increased, which is indicated that the manufacturer tends more to expand market sales through reducing the price. It can be also found that the sales effort is increased with the increase of t, which means the internet celebrity tend to devote more sales effort when the commission rate for them is higher.

Fig. 3

The influence of t on decisions.

The optimal profits of the members in the KD and MD models with the change of t and k are shown in Figs. 4 and 5. It can be seen from Fig. 4 that the profits of the manufacturer in the KD and MD models are both increased with the decrease of t and the increase of k, moreover, the increase of k has a greater impact on the profit of manufacturer when t is lower.

Fig. 4

The impacts of t and k on the manufacturer’s profit.

Fig. 5

The impacts of t and k on the internet celebrity’s profit.

Figure 5 demonstrates that the profit of internet celebrity is increased with the increase of t and the increase of k, meanwhile, the increase of k has a greater impact on the profit of internet celebrity when t is higher.

6.4 The feasible region of the cooperation

In this section, it is assumed that m is changed from 80 to 120, and the other parameters are the same as that in Table 1. As Fig. 6 shown, the personal influence of internet celebrity has positive impacts on the profit of supply chain. Therefore, it’s necessary to expand the personal influence of internet celebrity. It can be also found that the optimal profits in the centralized model is the highest, and which in Nash game model is the lowest, while the optimal profits in the MD and KD models are between those. The proposition 4 is verified by the above results. To remove the impact of double marginalization, the manufacturer and the internet celebrity should cooperate, and the feasible region of cooperation is presented in Fig. 7.

Fig. 6

The influence of m on the profit of the supply chain.

Fig. 7

The feasible region of the bargaining problem.

Equations (24) and (25) can be calculated as F₁ = 9104.33 - 24440 T, F₂ = 8679.38 - 24440 T, $F=\frac{9104.33{\xi }_2+8679.38{\xi }_1}{{\xi }_1+{\xi }_2}-24440T$ . It can be seen from Fig. 7, when F is closer to F₁, the allocation of the extra profit for the manufacturer will be lower, and which for the internet celebrity will be higher. On the contrary, the closer F approaches F₂, the lower the internet celebrity’s extra profit share, the higher the manufacturer’s extra profit share. Consider ξ₁ = ξ₂ = 1, the two members have the same bargaining power, F is in the middle of F₁ and F₂. When ξ₁ > ξ₂, F will be closer to F₂, the manufacturer has more bargaining power, the more extra profit gained. In the same way, the internet celebrity has more bargaining power when ξ₁ < ξ₂, and will obtain more extra profit.

7 Conclusions

With the rise of live e-commerce, more and more manufacturers engage in the live streaming rooms. Motivated by the operational practice of live streaming, we in this paper discuss optimal price and sales effort decisions of a dual-channel live streaming supply chain involving a manufacturer and an internet celebrity by building four models: Nash, MD, KD and cooperative game models. The impacts of the internet celebrity’s personal influence and sales effort on the demand are considered, and the spillover effect of the internet celebrity’s personal influence on the other channel is also taken in consideration. Moreover, the feasibility of coordination is verified and the numerical analysis is provided. The results show that, (1) the internet celebrity’s sales effort is positively related to the commission rate and market share, and the prices in two channels are negatively related to the internet celebrity’s commission rate. With the increase of the internet celebrity’s personal influence and sales effort, the profit of the supply chain increases. (2) the optimal price in the internet celebrity’s channel is not always lower than that in the manufacturer’s channel under the centralized model. (3) the optimal profits of the supply chain obtained in the decentralized decision-making model no matter who dominants are all lower than in the centralized decision-making model. (4) the manufacturer and the internet celebrity can be coordinated through bargaining problem in the cooperation model, and the total profit achieve maximum.

Based on the analysis in this paper, we discover some managerial insights.

When the internet celebrity’s potential market share of is low, the optimal price in the internet celebrity’s live streaming room should be lower than in the other sales channel to attract potential consumers. On the contrary, the optimal price in the internet celebrity’s broadcast channel should be higher when the potential market share of the internet celebrity is more.

In order to realize a long-term and scientific development, it’s necessary for the internet celebrity to expand the profit of the supply chain by improving the personal influence and the consumers’ responsiveness to live streaming mode.

The manufacturer can motivate the internet celebrity to devote more sales effort in live streaming room by promoting the internet celebrity’s commission rate and market share. The internet celebrity’s commission rate has negative effects on the prices without effects on the total profit, but the internet celebrity’s sales effort has positive impacts on the total profits. Thus, don’t just focus on the internet celebrity’s personal influence while ignoring the sales efforts they make.

The manufacturer and the internet celebrity should cooperate as an entire system to obtain the maximum profit of the supply chain. The supply chain members can be coordinated by the booth fee and the commission rate through bargaining problem. There is a feasible region for the manufacturer and the internet celebrity to bargain, and the bargaining result is depended on their attitude to risk.

There are several limitations of our work, this study only investigates the decisions and coordination in a dual-channel supply chain in which the manufacturer and the internet celebrity sell the products to consumers, but the impacts of live streaming platforms are ignored. In addition, the products will be return after received when the consumers are dissatisfied, but it hasn’t been taken into consideration. For further research, the return probability of products sold in the live streaming rooms could be considered in the profit functions. Furthermore, it is interesting in extending the research to the supply chain within multiple manufacturers and multiple internet celebrities. Moreover, further research could consider the transaction fees for live streaming platforms and the competition in different platforms.

Appendix

Proof of proposition 1.

In Nash game, the manufacturer and the internet celebrity have the same decision power. Taking the second derivative of Equation (2) with respect to p _m , $\frac{{\partial }^2{\pi }_m\left(p_m\right)}{\partial {p_m}^2}=-2\beta <0$ . It is obviously that the manufacturer has the unique solution to maximize its own profit. Taking the derivative of Equation (3), we can obtain the Hessian matrix $H_N=\left[\begin{array}{cc}\hfill -2\beta t\hfill & \hfill kt\hfill \\ \hfill kt\hfill & \hfill -b\hfill \end{array}\right]$ . When 2 b tβ - k²t² >0, the Hessian matrix of Equation (3) is negative define, which means the internet celebrity have the optimal solution.

To derive the Nash equilibrium solutions, taking the first derivative of Equation (3) with respect to p _r and e, taking first the derivative of Equation (2) with respect to p _m , we have:

$\begin{array}{cc}\hfill \begin{array}{c}\frac{\partial {\pi }_k\left(p_r,e\right)}{\partial p_r}=t\left(\alpha d-\beta p_r+\gamma p_m+k\left(e+m\right)\right)-\beta p_{r}t\hfill \\ \frac{\partial {\pi }_k\left(p_r,e\right)}{\partial e}=k{p}_{r}t-be\hfill \\ \frac{\partial {\pi }_m\left(p_m\right)}{\partial p_m}=hm-\beta p_m-\gamma \left(c+p_r\left(t-1\right)\right)+\gamma p_r\hfill \\ +\beta \left(c-p_m\right)-\alpha \left(d-1\right)\hfill \\ p_r^{N*},e^{N*}\mathrm{and}{p}_m^{N*}\mathrm{can}\mathrm{be}\mathrm{solved}\mathrm{by}\frac{\partial {\pi }_k\left(p_r,e\right)}{\partial p_r}=\frac{\partial {\pi }_k\left(p_r,e\right)}{\partial e}\hfill \\ =\frac{\partial {\pi }_m\left(p_m\right)}{\partial p_m}=0.\hfill \end{array}& \hfill \end{array}$

Proof of proposition 2.

To derive the KD Stackelberg equilibrium solutions, the manufacturer’s reaction function is derived as follows. $\begin{array}{cc}\hfill \begin{array}{c}\frac{d^2{\pi }_m\left(p_m\right)}{d{p_m}^2}=-2\beta <0\hfill \\ \frac{d{\pi }_m\left(p_m\right)}{d{p}_m}=hm-\beta p_m-\gamma \left(c+p_r\left(t-1\right)\right)+\gamma p_r\hfill \\ +\beta \left(c-p_m\right)-\alpha \left(d-1\right)=0\hfill \\ p_m^{\mathit{KD}*}\left(p_r\right)=\frac{\beta c-\gamma \left(c+p_r\left(t-1\right)\right)+hm+\gamma p_r-\alpha \left(d-1\right)}{2\beta }\hfill \end{array}& \hfill \end{array}$

Substituting $p_m^{\mathit{KD}*}\left(p_r\right)$ into Equation (3), taking the derivative with respect to p _r and e, we have: $H_{\mathit{KD}}=\left[\begin{array}{cc}\hfill -2t\left(\beta -\frac{\gamma \left(\gamma -\gamma \left(t-1\right)\right)}{2\beta }\right)\hfill & \hfill kt\hfill \\ \hfill kt\hfill & \hfill -b\hfill \end{array}\right]$ . When $2\mathit{bt}\left(\beta -\frac{{\gamma }^2\left(2-t\right)}{2\beta }\right)-k^2{t}^2$ >0, the Hessian matrix is a negative define matrix, which means (p _r , e) have a unique solution to maximize π _m  (p _r , e). $p_r^{\mathit{KD}*}$ and e^KD* can be solved by: $\begin{array}{cc}\hfill \begin{array}{c}\frac{\partial {\pi }_k\left(p_r,e\right)}{\partial p_r}=t\left(\alpha d-\beta p_r+k\left(e+m\right)+\frac{\gamma \left(\begin{array}{c}\beta c-\gamma \left(c+p_r\left(t-1\right)\right)\hfill \\ +hm+\gamma p_r-\alpha \left(d-1\right)\hfill \end{array}\right)}{2\beta }\right)\hfill \\ -p_{r}t\left(\beta -\frac{\gamma \left(\gamma -\gamma \left(t-1\right)\right)}{2\beta }\right)=0\hfill \\ \frac{\partial {\pi }_k\left(p_r,e\right)}{\partial e}=k{p}_{r}t-be=0\hfill \end{array}& \hfill \end{array}$

Substituting $p_r^{\mathit{KD}*}$ into $p_m^{\mathit{KD}*}\left(p_r\right)$ , $p_m^{\mathit{KD}*}$ is obtained.

Proof of proposition 3.

To derive the manufacturer dominant Stackelberg equilibrium solutions, the internet celebrity’s reaction function is first derived as follows.

$\begin{array}{cc}\hfill \begin{array}{c}\frac{d{\pi }_m\left(p_r,e\right)}{\mathit{dpr}}=t\left(\alpha d-\beta p_r+\gamma p_m+k\left(e+m\right)\right)-\beta p_{r}t\hfill \\ \frac{d{\pi }_m\left(p_r,e\right)}{d\mathrm{e}}=k{p}_{r}t-be\hfill \end{array}& \hfill \end{array}$

The Hessian matrix is $H_{\mathit{MD}}=\left[\begin{array}{cc}\hfill -2\beta t\hfill & \hfill kt\hfill \\ \hfill kt\hfill & \hfill -b\hfill \end{array}\right]$ . When 2 b tβ - k²t² >0, the Hessian matrix is negative define. $\begin{array}{cc}\hfill \begin{array}{c}{p}_r^{\mathit{MD}*}\left(p_m^{\mathit{MD}*}\right)=\frac{b\left(\alpha d+\gamma p_m+km\right)}{2b\beta -k^{2}t}\hfill \\ e^{\mathit{MD}*}\left(p_m^{\mathit{MD}*}\right)=\frac{kt\left(\alpha d+\gamma p_m+km\right)}{2b\beta -k^{2}t}\hfill \end{array}& \hfill \end{array}$

Substituting $p_r^{\mathit{MD}*}\left(p_m^{\mathit{MD}*}\right)$ and $e^{\mathit{MD}*}\left(p_m^{\mathit{MD}*}\right)$ into Equation (2), taking the second derivative with respect to p _m , we have $\frac{\beta \left(6b^2{\gamma }^2-2b^2{\gamma }^{2}t\right)-2b{\gamma }^2{k}^{2}t}{{\left(2b\beta -k^{2}t\right)}^2}-2\beta$ . When $\frac{\beta \left(6b^2{\gamma }^2-2b^2{\gamma }^{2}t\right)-2b{\gamma }^2{k}^{2}t}{{\left(2b\beta -k^{2}t\right)}^2}-2\beta <0$ , p _m have a unique solution to maximize . $p_m^{\mathit{MD}*}$ can be solved by . Substituting $p_m^{\mathit{MD}*}$ into $p_r^{\mathit{MD}*}\left(p_m^{\mathit{MD}*}\right)$ and $e^{\mathit{MD}*}\left(p_m^{\mathit{MD}*}\right)$ , $p_r^{\mathit{MD}*}$ and e^MD* are obtained.

Proof of proposition 4.

The first-order of Equation (1) with respect to p _m , p _r , e, respectively are as follows: $\begin{array}{cc}\hfill \begin{array}{c}\frac{\partial {\pi }_c\left(p_m,p_r,e\right)}{\partial p_m}=hm-\beta p_m+\gamma p_r+\beta \left(c-p_m\right)\hfill \\ -\gamma \left(c-p_r\right)-\alpha \left(d-1\right)\hfill \\ \frac{\partial {\pi }_c\left(p_m,p_r,e\right)}{\partial p_r}=\alpha d-\beta p_r+\gamma p_m+\beta \left(c-p_r\right)\hfill \\ -\gamma \left(c-p_m\right)+k\left(e+m\right)\hfill \\ \frac{\partial {\pi }_c\left(p_m,p_r,e\right)}{\partial e}=-be-k\left(c-p_r\right)\hfill \end{array}& \hfill \end{array}$

The Hessian matrix is $\begin{array}{cc}\hfill H_C=\left[\begin{array}{ccc}\hfill -2\beta \hfill & \hfill 2\gamma \hfill & \hfill 0\hfill \\ \hfill 2\gamma \hfill & \hfill -2\beta \hfill & \hfill k\hfill \\ \hfill 0\hfill & \hfill k\hfill & \hfill -b\hfill \end{array}\right]& \hfill \end{array}$

It can be found that |H₁| < 0, 4β² - 4γ² > 0. It is obviously that when -2 b β² + β k² + 2 b γ² < 0, 2b β - k² > 0, then |H₂| > 0 and |H₃| < 0. It’s proved that the Hessian matrix is a negative define matrix which means π _c  (p _m , p _r , e) is a jointly concave function, and the optimal solution is uniquely solved by

$\begin{array}{cc}\hfill \{\begin{array}{c}\hfill hm-\beta p_m+\gamma p_r+\beta \left(c-p_m\right)-\gamma \left(c-p_r\right)-\alpha \left(d-1\right)=0\hfill \\ \hfill \alpha d-\beta p_r+\gamma p_m+\beta \left(c-p_r\right)-\gamma \left(c-p_m\right)+k\left(e+m\right)=0\hfill \\ \hfill -be-k\left(c-p_r\right)=0\hfill \end{array}& \hfill \end{array}$

Proof of proposition 6.

Let the first-order conditions of N (T, F) with respect to T and F, $\begin{array}{cc}\hfill \begin{array}{c}\frac{\partial N\left(T,F\right)}{\partial T}=\frac{\partial N\left(T,F\right)}{\partial F}\hfill \\ ={\xi }_2{\left(A-F-GT\right)}^{{\xi }_1}{\left(F-B+GT\right)}^{{\xi }_2-1}\hfill \\ -{\xi }_1{\left(A-F-\mathit{GT}\right)}^{{\xi }_1-1}{\left(F-B+GT\right)}^{{\xi }_2}\hfill \end{array}& \hfill \end{array}$

It is obvious that $\frac{\partial N\left(T,F\right)}{\partial T}=\frac{\partial N\left(T,F\right)}{\partial F}=0$ when $F^{*}=\frac{A{\xi }_2+B{\xi }_1}{{\xi }_1+{\xi }_2}-G{T}^{*}$ , and the Nash bargaining model achieves optimal. Therefore, there is a line in parallel with F₁ and F₂ that maximize N (T, F), and the position of the line is depended on the attitude of the manufacturer and the internet celebrity to risk.