Strategic interplay between store brand strategy and selling mode choice

Abstract

With the development of e-commerce, online retailers has increasingly developed their own store brand to compete with the national brand. It is necessary for the manufacturers who sell national brand products through these online retailers to consider strategically when making the selling mode decisions. Motivated by such practice issue, we consider a theoretic model comprised of a national brand manufacturer and an online retailer. The retailer chooses to introduce an average store brand, a premium store brand or neither, and the manufacturer sells national brand products through the retailer under reselling or agency selling mode. Comparing and analyzing the six possible scenarios, we investigate the interaction between the retailer’s store brand strategy and the manufacturer selling mode selection. We find that the unit production cost and the quality of the national brand affect the retailer’s store brand strategy. Under the reselling format, the retailer prefers to introduce a premium store brand when the national brand cost is relatively low. While under the agency selling format, the retailer prefers to introduce a premium store brand when the cost is low but the quality is high. Interestingly, we further find that the manufacturer may benefit from the premium store brand introduction. Whether for the reselling or agency selling, there exists the “win–win” situation for the manufacturer and the retailer with a higher national brand quality. Overall, our results provide several management insights for the online retailer’s store brand introduction and the manufacturer’s selling mode selection.

1 Introduction

Retailers’ store brands, which has shown tremendous growth in traditional retailing over past years, are starting to develop in emerging online retailing. According to the 2021 International Private Label Yearbook of the Private Label Manufacturers Association (PLMA), retailers’ store brand has sustained their market share position above 30% in most European countries, and nearly half of the countries reached above 40% (PLMA 2021). Similar growth also can be seen across other areas, especially in the developing market of Asia-Pacific (Nielsen 2018). Among these store brands, an increasing number of them are introduced by the online-retailer. For example, Amazon has launched more than 70 store brands including 243,000 products up to now, and the sales of its clothing store brand has reached 9% of total sales in this category (Pulse 2020). However, the online-retailers appear to be differentiated on their introduction of store brand products. Specifically, some of the retailers introduce the average store brand, which are positioned as inferior substitutes and sold at lower prices to national brands. Retailers such as Suning.com and Alibaba Group aim to offer comparable quality but lower price to manufacturers’ national brands. The store brand JIWU, provided by the online retailer Suning.com, sells a variety of consumer product categories at a cheap price. One of the most typical products is an aroma lamp, which has similar product functions with the national brand MUJI but is priced at only 30% off it (Retailing 2019). The other retailers give up the low price competition and turn to the high-end market. They choose to introduce premium store brands that are highly personalized and improved in quality commitment and purchasing experience (Worldwide 2021). Examples of the premium store brand are Amazon’s High-end household goods brands Rivet, Stone & Beam and NetEase’s home brand Lifease. These differentiated store brand strategies may lead to different performance of retailers.

The rapid development of store brand business causes the national brand manufacturer’s concern. According to CNBC’s report, multiple sellers and brands are worried about Amazon’s entry into store brand business (CNBC 2018). These brands generally sell products through retailers in reselling mode or agency selling mode. Under a reselling mode, the retailer wholesales national brand products from the brand manufacturer and then decides the price for consumers. While under an agency selling mode (also called marketplace mode), the brand manufacturer determines retail price directly and shares revenue with the retailer based on a given percentage. Essentially, the two different selling modes attribute to different pricing power and collaborative relationship. The adopted mode also affects the store brand introduction decision and the performance of supply chain members. Another CNBC’s report points that the sellers and brands which rely on Amazon’s marketplace for sales could face bigger hurdles, as more store brands are launched by Amazon (CNBC 2019). Thus there is a need to consider the store brand strategy and selling mode selection jointly.

These observations give rise to several issues that are investigated in this paper. First, considering the national brand’s selling mode, which store brand strategy should the retailer adopt? And how should the retailer decide the selling price for the store brand introduced? Second, how does the retailer’s store brand strategy affect the manufacturer’s selling mode decision? And when should the manufacturer choose reselling format or agency selling format? Third, does the store brand introduction always hurt the national brand manufacturer? Do there exist a “win–win” situation for the manufacturer and the retailer?

To address these issues, we develop a game model in which a retailer has an incentive to introduce an average store brand or premium store brand and a manufacturer sells national brand through the retailer under a reselling or an agency selling mode. We analyze six scenarios based on different possible strategy combination, including no store brand under agency selling or reselling, an average store brand introduction under agency selling or reselling and a premium store brand introduction under agency selling or reselling respectively. Then we discuss which store brand strategy and selling mode should adopted by the two firms. We further compare the equilibrium scenario with the most profitable case to find whether exists the “win–win” situation. Our studies obtain several findings and provide managerial insights as follows.

First, under a reselling mode the platform retailer prefers to introduce an average store brand or premium store brand. We find that when the unit production cost of the national brand is relatively low, introducing an average store brand is optimal for the retailer because she can benefit from the increasing demand of the national brand and the average store brand selling. When the unit production cost of the national brand is relatively high, the retailer should introduce a premium store brand to grab the high-type consumer market. Interestingly, when the national brand quality is high but its production cost is low, the manufacturer will benefit from the retailer’s premium store brand introduction. Second, when the manufacturer adopts an agency selling mode, the retailer prefers to introduce a premium store brand or neither. Specifically, when the national brand quality is high, the retailer choose to introduce a premium brand with a lower national cost. When the national brand quality is low, the retailer prefers to introduce a premium store brand with a higher national cost. Third, comparing the two type of selling modes, we find that the manufacturer’s selection of the selling format changes with the consideration of the retailer’s store brand introduction. There exist three possible equilibrium cases, including premium storebrand introduction under reselling or agency selling mode and no store brand under agency selling mode. Besides, the retailer and the national brand manufacturer can achieve the “win–win” situation with the premium store brand introduction.

The rest of this paper is organized as follows. Section 2 reviews the related literature from three aspects. Then we build our main model in Sect. 3. Section 4 analyzes the equilibrium under different selling mode. Section 5 further discusses the selling modes decisions and the store brand decisions. Finally, we conclude this study and summarize the management insights.

2 Literature review

Our work is related to the two streams of literature on store brand introduction and agency selling or reselling mode choice. To highlight the contribution and positioning of this paper, we review the related literature of two streams in the following.

Studies have paid great attention to the selling mode selection in supply chain management, and most of them are concerned with the impact of different channel structure on the optimal sell mode decisions. Abhishek et al. (2016) build a model with two e-retailers to explore the conditions of which an agency selling format should be used instead of the reselling format. Tian et al. (2018) study a setting with two competitive suppliers and one intermediary to identify the optimal mode for the intermediary, and find that the suppliers’ competition and fulfillment cost moderate the choice for the intermediary to operate as a reseller, as a marketplace or a hybrid mode. Shen et al. (2019) study a supply chain structure comprised of a platform retailer, a traditional reseller and a manufacturer. They focus on the manufacturer’s channel choice between the retailers with different selling mode. Ha et al. (2021) consider a structure with one manufacturer and one online retailer, and investigate the manufacturer’s reselling and agency channel choice with incorporating the online retailer’s service effort. These studies mainly discuss the selection of selling mode from the perspective of channel structure, and some other studies consider many driving forces in selling format decision. For instance, Zhang and Zhang (2020) investigate how the reselling and agency selling modes affect the retailer’s information sharing strategies and the supplier’s channel entry decision. Chen et al. (2020) analyze a firm’s online mode choice in promotional period with the presence of retail competition. Although many existing literature explore the selling mode decision from different perspective, few studies discuss this problem from the manufacturer’s view in the context of the retailer’s store brand introduction.

There are also several studies on store brand introduction since the store brand shows considerable growth in the retail business practices. Earlier works are mainly interested in the impact of store brand on the performance of supply chain members. Mills (1995) shows a store brand introduction benefits the retailer at the expense of the national brand manufacturer, and it alleviates the double-marginalization problem in their distribution channel. Pauwels and Srinivasan (2004) empirically demonstrate that the store brand entry benefits the retailer and the premium-brand manufacturer, but harms the second-tier brand manufacturer. Ru et al. (2015) find that the manufacturer may benefit from the retailer’s store brand introduction as the interaction between them becomes vertical strategic independence. More recently, some studies explore the combinational effect of store brand introduction strategy and other supply chain strategies. For example, Shi and Geng (2021) demonstrate the impact of information sharing strategy on the retailer’s incentive to introduce a store brand. Jin et al. (2017) study the interaction between a retailer’s store brand decisions and a national brand manufacturer’s dual or single channel strategy. Li et al. (2018) investigate the strategic interaction between a retailer’s store brand introduction and a manufacturer’s online direct channel strategy. However none of these studies investigate the store brand decision with considering the different product quality strategy or store brand product line design. At this point, Zhang et al. (2021) analyze the retailer’s three store brand introduction strategies, including introducing a low-quality substitute product, a premium store brand or neither, with a consideration of the manufacturer’s encroachment. Other related studies (Chung and Lee 2017; Heese 2010)also explore the store brand introduction from the perspective of product-line design and store brand position, which are related to our study. Nevertheless, all of these research which pay attention to the high quality store brand strategy ignore the impact of different selling mode. While in practice, the retailer’s store brand may have different performance under different selling mode. Our study explore the retailer’s store brand strategy incorporating the national brand manufacturer’s selection of the selling format.

Few studies have noticed the interaction of store brand introduction and the different selling mode, which are most related to our research. Zhang and Hou (2022) examine the manufacturer’s decision of the two prevalent sale modes when the retailer introduce a private labels. They mainly focus on the environmental impact from the perspective of economic and sustainability, and they do not consider the high quality store brand strategy. Li et al. (2021) also investigate the effect of store brand introduction on the manufacturer. While they concentrate on how the different selling mode and the retailer’s investment moderate this effect, and they do not incorporate the selling format decision into the model. Taking these gaps into account, our paper contributes to the store brand field by considering the retailer’s high-quality store brand strategy and the selling mode decision jointly. In addition, we provide new insights that the retailer and the manufacturer can achieve the “win–win” situation with the premium store brand introduction under an agency selling mode.

3 Model

Consider a vertical supply chain that consists of a national brand manufacturer (he) and an online retailer (she). The manufacturer produces the national brand products and sells these products through a retailer. The retailer has an incentive to introduce a competing brand (an average store brand or a premium store brand) in addition to selling the national brand. The manufacturer can also contract with the online retailer to choose reselling mode or agency selling mode. We use the subscripts n, s, h to represent the national brand, average store brand and premium store brand, respectively. The selling price and wholesale price are denoted by p and w.

3.1 Product demand

Besides the national brand products, an average store brand or a premium store brand may be introduced to the market. We assume the quality levels of the average store brand and the national brand are $\gamma _{s}$, $\gamma _{n}$ respectively, where $\gamma _{s}\le \gamma _{n}<1$. This is consistent with our general observation that the national brand has a higher quality level compared to the average store brand, since it has competitive edges in components sourcing and product quality control. For example, a smart speaker from the cheaper store brand JIWU has poor performance compared to the national brand Huawei, according to the evaluation report of Shenzhen Institute of Consumption Quality (SICQ) (SICQ 2020). For simplicity, we further assume the quality level of the premium store brand is 1 to ensure that the premium store brand has a higher quality than the national brand. As the example mentioned in the Introduction section, Stone & Beam is an Amazon store brand dedicated to providing upscale furniture. A review from Insider Reviews says the quality of Amazon’s furniture is comparable to CB2 and West Elm. The tester comments that the pieces are sturdy, well-constructed, and easier to assemble (Kaplan 2022). The assumption is also common in related literature such as Ru et al. (2015), Shi and Geng (2021).

We follow Mills (1999) to characterize consumers’ purchasing behavior. There is a unit mass of consumers in the market with heterogeneous quality preference, each of whom buys at most one product. We assume the consumers’ valuation for product quality, $\theta$, is uniformly distributed over [0, 1]. Therefore, the consumer’s perceived value from a product with quality $\gamma$ can be obtained as $\theta \gamma$. This assumption reflects the fact that higher product quality gives consumers higher utility, and is widely used in store brand literature (Ru et al. 2015; Li et al. 2018; Zhang et al. 2021). Each consumer makes purchasing decision by maximizing their utility based on given product quality $\gamma$, retail price p, and their own valuation for the product $\theta$. Clearly, a consumer’s utility is $\gamma _{n}\theta -p_{n}$ if the nation brand product is bought, $\gamma _{s}\theta -p_{s}$ if the average store brand product is bought, and $\theta -p_{h}$ if the premium store brand product is bought.

Consumers make their purchasing decisions based on maximizing their utilities. If there is no store brand in the market, the consumers choose to buy the national brand if and only if their valuations satisfy γ_nθ − p_n ≥ 0. At this time, the demand for the national brand is given by $1-{p_{n}/\gamma_{n}}$. When the retailer introduces an average store brand to compete with the national brand, the consumers prefer to buy the national brand product if and only if their valuations satisfy $\gamma _{n}\theta -p_{n} \ge \gamma _{s}\theta -p_{s}$ and $\gamma _{n}\theta -p_{n} \ge 0$. So the marginal consumer who is indifferent between buying the national brand product and the average store brand product is located at $\frac{p_{n}-p_{s}}{\gamma _{n}-\gamma _{s}}$, and $\frac{p_s}{\gamma _{s}}$ is that between buying the average store brand product and nothing. Similarly, when the retailer introduces a premium store brand into the market, the consumers prefer to buy the premium store brand if and only if their valuations satisfy $\theta -p_{h}\ge \gamma _{n}\theta -p_{n}$ and $\theta -p_{h}\ge 0$. The indifferent consumer between buying the premium store brand band and the national brand is located at $\frac{p_{h}-p_{n}}{1-\gamma _{n}}$, and that between buying the national brand and nothing is located $\frac{p_{n}}{\gamma _{n}}$. Here, we assume that $\frac{p_{n}}{p_{h}}<\gamma _{n}<\frac{p_{n}}{p_{s}}\gamma _{s}$ to exclude the case wherein the store brand or the national brand is never bought. The main conclusions qualitatively hold when $\gamma _{n}\ge \frac{p_{n}}{p_{s}}\gamma _{s}$ or $\gamma _{n}\le \frac{p_{n}}{p_{h}}$. Consequently, the demands for the two different cases are as follows

$$\begin{aligned}&\left\{ \begin{array}{ll} D_{n1}=1-\frac{p_{n}-p_{s}}{\gamma _{n}-\gamma _{s}}, &{} \\ D_{s}=\frac{p_{n}-p_{s}}{\gamma _{n}-\gamma _{s}}-\frac{p_{s}}{\gamma _{s}}, &{} \\ \end{array} \right. \end{aligned}$$

(1)

$$\begin{aligned}&\left\{ \begin{array}{ll} D_{n2}=\frac{p_{h}-p_{n}}{1-\gamma _{n}}-\frac{p_{n}}{\gamma _{n}}, &{} \\ D_{h}=1-\frac{p_{h}-p_{n}}{1-\gamma _{n}}. &{} \\ \end{array} \right. \end{aligned}$$

(2)

3.2 Operating scenarios

The national brand manufacturer incurs a unit production cost c to produce the national brand products. Besides, the retailer also faces different production cost to introduce a store brand. We assume that the unit production cost of the average store brand and the premium store brand equals to c and $c_h$ ($c_h>c$), respectively. Generally, the high-quality products do better in terms of material supply, production process specification and quality control, which leads to higher marginal cost of the high-quality product. Therefore, we assume the premium store brand has a higher unit cost than the average store brand and the national brand. Similar assumption is commonly used in related studies, such as Ru et al. (2015). We further assume $c<\frac{\gamma _s}{2}$ to guarantee a positive marginal profit for the store brand product.

The interaction between the retailer and the manufacturer contains two stages, as shown in Fig. 1. The first stage is called strategy decision stage where the supply chain members make selling format decision and store brand decision. In this stage, the manufacturer first chooses a reselling agreement or an agency selling agreement. Then the retailer decides to introduce a premium store brand, average store brand, or nothing. In practice, the store brand and selling mode strategies are long-term decisions, and cannot be changed easily. Also, most famous national brand manufacturers enter in to the online platform as early as the development of e-commerce, such as Nike and Kate Spade. The second stage is called pricing stage where the retail prices of both the national brand and the store brand are determined. If the reselling format is adopted, the manufacturer sets the wholesale price $w_{n}$ for the national brand product. Then the retailer simultaneously sets the retail prices of the national brand product and the store brand product, if one has been introduced. If the agency selling format is adopted in previous stage, the manufacturer pays for an agency fee at a rate $\alpha$ of the sales revenue. Here, we assume the $\alpha$ is exogenous because the rate of agency fee is usually given by the online retailer based on product categories and is not adjusted frequently. For example, JD.com gives detailed standards of the agency fee proposition on its official website (JD.com 2021). If the standards need to be adjusted, JD.com will make a special announcement to explain. After that, the manufacturer sets the retail price for the national brand product. The retailer sets the retail price for the store brand product simultaneously if it has been introduced.

Fig. 1

Sequence of events

Given the retailer’s selling format decision and store brand decision, we consider six possible scenarios which are denoted by RA (introduce an average store brand with reselling mode), RP (introduce a premium store brand with reselling mode), RN (no store brand with reselling mode), AA (introduce an average store brand with agency selling mode), AP (introduce a premium store brand with agency selling mode), AN (no store brand with agency selling mode), respectively. In Sect. 4, we will derive and discuss the equilibrium in each case.

4 Equilibrium

In this section, we use backward induction to derive and analyze the equilibrium solutions under reselling format and agency selling format. All the equilibrium outcomes are provided in Appendix.

4.1 Reselling mode

Under the reselling format, the retailer may introduce an average store brand or a premium store brand or nothing. We start from the benchmark scenario RN, where the retailer only sells the national brand products to consumers under the reselling format. Based on the demand functions instructed, the retailer and manufacturer’s profit functions are

$$\begin{aligned} \pi _R^{RN} = \left( 1-\frac{p_n}{\gamma _n}\right) (p_n-w_n), \end{aligned}$$

(3)

$$\begin{aligned} \pi _M^{RN} = \left( 1-\frac{p_n}{\gamma _n}\right) (w_n-c). \end{aligned}$$

(4)

respectively. In the benchmark scenario RN, the manufacturer move first to determine the wholesale price $w_n$ and the retailer then decides the retail price.

Next, we consider the RA scenario in which the retailer introduce an average store brand under the reselling format. The retailer and manufacturer’s profit functions under this scenario are

$$\begin{aligned} \pi _R^{RA} = D_{n1}(p_n-w_n)+D_{s}(p_s-c), \end{aligned}$$

(5)

$$\begin{aligned} \pi _M^{RA} = D_{n1}(w_n-c), \end{aligned}$$

(6)

respectively. In this setting, the retailer sets retail prices for both the national brand products and the average store brand products after the manufacturer decides the wholesale price.

We further consider the RP scenario wherein the retailer introduces a high-end store brand under the reselling format. Therefore, the retailer and manufacturer’s profit functions under the RP scenario are

$$\begin{aligned} \pi _R^{RP} = D_{n2}(p_n-w_n)+D_{h}(p_h-c_h), \end{aligned}$$

(7)

$$\begin{aligned} \pi _M^{RP}= & {} D_{n2}(w_n-c), \end{aligned}$$

(8)

respectively. In this scenario, the manufacturer determines the wholesale price, and then the retailer sets retail prices for both the national brand and the premium store brand product simultaneously. Next, we compare the equilibrium solutions under reselling format to derive insights regarding how the store brand introduction strategies affects the equilibrium and which store brand should be introduced under reselling format. The main insights are summarized as follows.

Proposition 1

Under the reselling format, the store brand introduction lower the wholesale price and the selling price of the national brand, i.e., $w_n^{RN*}>w_n^{RA*}, w_n^{RN*}> w_n^{RP*}$, and $p_n^{RN*}>p_n^{RP*}$, $p_n^{RN*}>p_n^{RA*}$. Also, the average store brand introduction increases the demand of the national brand comparing with the case of no store brand introduction, i.e., $D_n^{RA*}>D_n^{RN*}$.

Proposition 1 shows the impact of store brand introduction on the selling price and demand of the national brand. Intuitively, when the retailer introduces an average or a premium store brand, the manufacturer would lower the wholesale price to ensure that the national brand selling price is not too high. As a result, the selling price of the national brand also decreases under the RA and RP scenarios. Moreover, when the retailer introduces a premium store brand, the demand of the national brand becomes more sensitive to the unit cost c and quality level $\gamma _n$. With the quality improvement or cost reduction, the demand of the national brand under RP scenario increases significantly. Hence, the demand of the national brand in RA scenario is higher than that in RN scenario due to the reduction of the selling price. But the demand of the national brand in RP scenario may be higher or lower than that in RN scenario, depending on the unit cost and product quality.

Proposition 2

Comparing the equilibrium profit under different store brand introduction strategic, we have (a) $\pi _R^{RN}<\pi _R^{RA}$, and $\pi _R^{RN}<\pi _R^{RP}$. (b) $\pi _R^{RA}<\pi _R^{RP}$ if $0<c<c_{RR}^*$, and $\pi _R^{RA} \ge \pi _R^{RP}$ if $c\ge c_{RR}^*$. (c) $\pi _M^{RN}>\pi _M^{RA}$, and $\pi _M^{RN}>\pi _M^{RP}$. (d) $\pi _M^{RA}<\pi _M^{RP}$ if $0<c<c_{RM}^*$, and $\pi _M^{RA}\ge \pi _M^{RP}$ if $c\ge c_{RM}^*$.

Proposition 2 shows the impacts of store brand introduction strategies on the retailer and the manufacturer’s profits under the reselling format. It indicates that the store brand introduction always benefits the retailer but hurts the manufacturer’s performance. This result is consistent with the industrial practice that more and more retailers have introduced their own store brand to compete with the national brand. Similar conclusion also can be seen in existing literature Li et al. (2018). We further observe that when the unit production cost is relatively low, the retailer has more incentive to introduce an average store brand. While when the unit production cost is high, the retailer introducing a premium store brand should be better off. Such cost effect is opposite for the manufacturer. This is because that if the retailer determines to introduce a premium store brand, the manufacturer has no price and quality advantages with a higher unit cost to compete with the premium store brand. Interestingly, we find that when the national brand cost is relatively low and has a smaller quality gap with the premium store brand, the manufacturer may benefit from the premium store brand introduction. This results suggests that if the manufacturer produces the national brand with higher efficiency (lower production cost and higher quality level), the retailer and the manufacturer may achieve the “win–win” result with the store brand introduction.

Fig. 2

The impact of store brand introduction on profits under reselling format

Figure 2 gives the supporting evidence to the retailer’s store brand introduction strategy. It shows that the retailer has higher performances to introduce either an average store brand or a premium store brand. We can see that both the manufacturer and the retailer’s profit decreases with the unit production cost except for the RA scenario. It suggests that when the average store brand is introduced, reducing production cost may be not enough effective for the manufacturer.

4.2 Agency selling mode

Besides the prevailing reselling format, the retailer has the option to operate as the marketplace by charging an agency fee for the manufacturer who plans to sell the national brand products. The key difference between these two selling format is who sets the retail price for product. Similarly, we begin with the benchmark scenario AN, where the retailer only service as the marketplace to earn money and the manufacturer moves to set selling prices for the national brand. The retailer and manufacturer’s profit functions in this scenario are

$$\begin{aligned} \pi _R^{AN } = \alpha \left( 1-\frac{p_n}{\gamma _{n}}\right) (p_n-c), \end{aligned}$$

(9)

$$\begin{aligned} \pi _M^{AN} = (1-\alpha )\left( 1-\frac{p_n}{\gamma _{n}}\right) (p_n-c), \end{aligned}$$

(10)

respectively. We move to analyze the scenario AA in which the retailer chooses to introduce an average store brand under the agency selling format. The retailer and manufacturer’s profit functions under the AA scenario are

$$\begin{aligned} \pi _R^{AA} = \alpha D_{n1}(p_n-c)+D_{s}(p_s-c), \end{aligned}$$

(11)

$$\begin{aligned} \pi _M^{AA} = (1-\alpha )D_{n1}(p_n-c), \end{aligned}$$

(12)

respectively. In this scenario, the retailer sets the selling price for the average store brand, and the manufacturer sets the selling price for the national brand product simultaneously.

Finally, we consider the scenario AP wherein the retailer has the option to introduces a premium store brand to compete with the national brand under the agency selling mode. The retailer and manufacturer’s profit functions under the AP scenario are

$$\begin{aligned} \pi _R^{AP} = \alpha D_{n2}(p_n-c)+D_{h}(p_h-c_h), \end{aligned}$$

(13)

$$\begin{aligned} \pi _M^{AP} = (1-\alpha )D_{n2}(p_n-c), \end{aligned}$$

(14)

respectively. In this setting, the manufacturer pays for a rate of commission $\alpha$. Then the retailer and manufacturer make the simultaneous pricing decision for the premium store brand and the national brand. We further compare the solutions in different case under the agency selling format and explore the impact of the production cost and the quality level in the following analysis.

Proposition 3

Under the agency selling format, the store brand introduction reduces the national brand selling price, i.e., $p_n^{AN*}>p_n^{AA*}$ and $p_n^{AN*}>p_n^{AP*}$. Moreover, the selling price $p_n^{AP*}$ is less sensitive to an increase in quality level than the selling prices $p_n^{AN*}$ and $p_n^{AA*}$.

Proposition 3 shows that introducing an average store brand or a premium store brand intensifies the price competition under agency selling format. Similar to the effect under reselling format, new product entering the market can corrode the national brand market share, which forces the manufacturer to price lower under the agency selling format. In addition, as the national brand quality increases, the selling prices of the national brand product under AN and AA scenario increase significantly but the selling price under AP scenario is almost unchanged. This implies that improving the product quality is not always efficient for the manufacturer to counter the store brand introduction under the agency selling agreement. When the retailer introduces an average store brand, improving the national brand quality is more productive for the manufacturer.

Proposition 4

Comparing the retailer and the manufacturer’s profits under AN, AA and AP scenario, we can find that (a)$\pi _R^{AN*}>\pi _R^{AA*}$ and $\pi _M^{AN*}>\pi _M^{AA*}$. (b)$\pi _R^{AN*}>\pi _R^{AP*}$ if $0<c<c_{ER}^{1*}$ or $c_{ER}^{2*}<c<\bar{c}$, $\pi _R^{AN*}\le \pi _R^{AP*}$ if $c_{ER}^{1*}\le c<\bar{c}$ or $0<c\le c_{ER}^{2*}$. (c)$\pi _M^{AN*}>\pi _M^{AP*}$ if $c_{EM}^{1*}\le 0$ or $0<c_{EM}^{1*}<c$; $\pi _M^{AN*}\le \pi _M^{AP*}$ if $0<c_{EM}^{1*}$ and $0<c\le c_{EM}^{1*}$.

Proposition 4 shows the retailer’s optimal store brand introduction strategy under agency selling format. Interestingly, introducing an average store brand is always not the optimal strategy for the retailer under agency selling format. Specifically, when the unit production cost of the national brand is very high but the quality level is relatively low, the retailer is better off with a premium store brand introduction. While when the unit production cost of the national brand is medium, the retailer prefers not to introduce any store brand. The underlying reason is that the agency fee is proportional to the national brand revenue and it accounts for a main part of the retailer’s profit under agency selling format. Figure 3 further illustrate the Proposition 4. As shown in these figures, the retailer’s profits under AN and AP scenarios decrease in production cost and quality level. However, the nature of the retailer’s performance regard to production cost is changed under AP scenario. If the national brand has a small quality gap with the premium store brand, the retailer’s profit decreases in the unit production cost. This is because that the negative effect of a cost reduction on the agency fee incomes dominates the positive effect on the premium store brand revenue. On the contrary, if the national brand has a big quality gap with the premium store brand, the retailer’s profit increase in production cost within a certain interval. The rising cost leads to an improvement on the premium store brand revenue, which covers the reduction of agency fee from the national brand.

Fig. 3

The impact of store brand introduction on retailer’s profit under agency selling format

5 Selling mode analysis

In this section, we continue to explore the manufacturer’s optimal selling format decision and the retailer’s preference based on previous analysis of store brand introduction strategic. As stated in above section, the retailer prefers to introduce an average store brand or a premium store brand under reselling format. While under agency selling format, the retailer is more likely to introduce a premium store brand or neither. Therefore, the following discussion focuses on RA RP AN and AP scenarios to further investigate the combined strategy on selling mode choice.

Proposition 5

The manufacturer may benefit from the premium store brand introduction under both the reselling format and agency selling format. Moreover, when the national brand quality is relatively low, the manufacturer prefers the agency selling mode, otherwise the manufacturer prefers a reselling mode.

Fig. 4

The most profitable scenario for the manufacturer

Comparing the manufacturer’s profit under different scenario, we further obtain the most profitable selling format for the manufacturer in Proposition 5. Intuitively, the store brand introduction, whether in a reselling mode or agency selling mode, hurts the national brand manufacturer due to product competition. When the quality level of the national brand is relatively low, an agency selling mode and no store brand introduction for the manufacturer is the most profitable. This is because the manufacturer has direct pricing power and there is no threat of store brand competition. However, when the quality level is relatively high, the manufacturer benefits more from the premium store brand introduction than the no store brand scenario under the agency selling agreement. Since a higher quality with a lower price makes the national brand becomes more cost-effective, the premium store brand increases the profit of the national brand manufacturer. Furthermore, when the quality level of the national brand is very high, the manufacturer’s preference turns to the reselling mode. In this situation, the price of the national brand begins to increases in the quality level, but the demand decreases in the quality level significantly. The positive effect of the direct pricing power is dominated by the negative effect of product competition. Figure 4 gives an evidence of the Proposition 5. We can see that when the retailer introduces a premium store brand, the manufacturer prefers the reselling mode with a higher quality level. Also, if the retailer introduces an average store brand, the manufacturer will be hurt either with a reselling mode or an agency selling mode. This conclusion gives the national brand manufacturers some inspirations. The manufacturers may change their selling modes according to the product quality of the store brand and market research on retailers’ store brand need to be strengthened.

Proposition 6

The reselling format is more profitable for the retailer in most cases. Under agency selling format, the retailer prefers to introduce a premium store brand only if the quality level of the national brand is relatively high.

Fig. 5

The most profitable scenario for the retailer

We continue to analyze the retailer’s preference for selling format considering the store brand introduction, which is summarized in Proposition 6. Combined with different selling mode to compare, we find no store brand introduction strategy under an agency selling can never be better than other store brand strategies. In other words, the retailer’s profit may be hurt when the manufacturer choose a agency selling. Figure 5 depicts the most profitable scenario for the retailer and further illustrate the Proposition 6. When the quality level of the national brand is low or medium, using the reselling agreement is optimal for both the average store brand strategy and the premium store brand strategy. When the national brand quality is relatively high, the retailer makes a trade-off between the reselling and agency selling mode to sell the premium store brand. In particular, if the national brand quality nears to the premium store brand quality, the retailer benefits more from the agency selling agreement with a premium store brand introduction. The reason lies in the fact that a higher national brand quality increases the retailer’s revenue of the platform agency fee which is the main part of the retailer’s profit. Although the agency selling mode lowers the price of premium store brand, the increasing revenue of the agency fee improves the retailer’s performance significantly. The boundary line for choosing a reselling mode or an agency selling mode depends on the magnitude of premium store brand cost ($c_h$). A higher unit production cost of the premium store brand enables the retailer to choose the agency selling mode at a lower level of the national brand quality.

Proposition 7

There exist three equilibrium scenarios for the interaction between the retailer’s store brand introduction and the manufacturer’s selling mode selection, including RP, RA and AP scenarios. The retailer and the manufacturer can achieve “win–win” situation under the AP and the RP scenarios.

Fig. 6

The equilibrium selling format decision

Fig. 7

Win–win situations for the manufacturer and retailer

We finally analyze the manufacturer’s equilibrium selling format decision in Proposition 7. Figures 6 and 7 gives a further illustration for Proposition 7. There exist three possible cases for store brand strategy and the selling mode selection. The first case is that the retailer does not introduce any store brand and the manufacturer choose an agency selling agreement. The location of this case ranges from a lower level of the national brand quality to a medium level. The second case is that the retailer introduce a premium store brand and the manufacturer choose an agency selling format. It occurs when the quality level of the national brand is medium or relatively high. Both the retailer and the manufacturer may benefit from this equilibrium to realize the most profitable scenario. The final case is that the retailer introduce a premium store brand and the manufacturer choose a reselling format. This equilibrium appears with two different conditions. When the national brand quality and the premium store brand cost is relatively low but the national brand cost is high, the national brand becomes less competitive compared to the premium store brand. In this situation, the manufacturer choose a reselling format to avoid direct competition with the premium store brand, which reduce the manufacturer’s losses. As a result, the retailer realizes the most profitable scenario while the manufacturer is hurt by the premium store brand introduction. The other condition is that When the national brand quality and the premium store brand cost is relatively high. The national brand has a higher cost performance in this situation and therefore the manufacturer prefers the reselling mode to charge a higher wholesale price. It’s obviously that the retailer’s different store brand introduction strategy changes the manufacturer’s preference for selling mode with the interaction. Recall the analysis in Proposition 5 and Proposition 6, we further find that the retailer and the manufacturer can achieve the “win–win” situation under the AP and the RP scenario. This provides the insight that if the retailer (like Amazon, JD.com mentioned above) wants to introduce a store brand to compete with the national brand, developing a higher quality product is a better choice for the retailer. Given a store brand introduction, the national brand manufacturer need to choose the selling agreement according to the product quality level and production cost. Furthermore, the manufacturers (like MUJI, Nike) should make efforts to decrease production cost of the national, which gives them larger decision range.

6 Conclusion

In this paper, we develop a game model which consists of a national brand manufacturer and an e-retailer who has an incentive to introduces an average store brand, a premium store brand or neither. We analyze six possible strategy combinations to investigate the interaction between the selling mode selection and store brand introduction strategy. We find that under the reselling mode, the retailer benefits from an average store brand or a premium store brand introduction while the manufacturer is hurt it. Under the agency selling mode, the retailer prefers to introduce a premium store brand or neither and the manufacturer may benefit from the premium store brand introduction. Further, we compare the two selling mode and obtain the three possible equilibrium of the store brand strategy and the selling mode decision. Interestingly, we find that the retailer and the manufacturer can reach the “win–win” situation with a premium store brand introduction under both the reselling and the agency selling mode.

Our findings generates valuable managerial insights regard of the store brand introduction strategy and the online selling mode selection. The details are summarized as follows. First, the retailer should take the selling mode of the national brand into consideration when introduce a store brand. If the retailer launches an average store brand under the agency selling mode, it will not be effective enough to compete with national brand advantage and the retailer’s performance will be hurt by the average brand introduction. Besides, the retailer should make efforts to develop higher quality store brand. This find explain the practice that some retailers, such as Lifease.com and Amazon, launch high-end store brand products which are developed with the highest standards and high-quality raw materials. Second, the national brand manufacturer should improve the product quality and lower the unit cost. Higher quality level may not only prevents the retailer to introduce the store brand under an agency selling mode, but also benefit the manufacturer even though the premium store brand is introduced. Moreover, the manufacturer needs to decide the selling agreement based on the magnitude of national brand quality and cost, and further takes the retailer’s potential reactions into consideration.

There are also several limitations in our study, which can be extensions in future research. We do not consider the hybrid mode of the reselling and agency selling in our theoretical mode. However, the national brand manufacturer may sell products through a reselling channel and a marketplace simultaneously in practice. Also, more and more e-retailers integrate the upstream supplier to provide high-end store brand products in the retailing industry. Future study can be extended to consider the supply chain integration and co-development.

Appendices

Appendix A

Under the scenario RN, the profit functions of the retailer and the manufacturer are given in Eqs. (3) and (4). By backward induction, we can obtain the optimal prices, product demand and profits of the manufacturer and the retailer as follows

$$\begin{aligned}&p_n^{RN*}=\frac{3\gamma _{n}+c}{4}, w_n^{RN*}=\frac{\gamma _{n}+c}{2},&\\&\pi _R^{RN*}=\frac{(c-\gamma _{n})^2}{16\gamma _{n}}, \pi _M^{RN*}=\frac{(c-\gamma _{n})^2}{8\gamma _{n}}.&\end{aligned}$$

Under the scenario RA, the profit functions of the retailer and the manufacturer are given in Eqs. (5) and (6). By backward induction, we can obtain the retailer and the manufacturer’s optimal decisions as follows

$$\begin{aligned}&p_n^{RA*}=\frac{3\gamma _{n}-\gamma _{s}+2c}{4}, p_s^{RA*}=\frac{c+\gamma _{s}}{2},&\\&w_n^{RA*}=\frac{2c+\gamma _{n}-\gamma _{s}}{2},&\\&\pi _R^{RA*}=\frac{4(c-\gamma _{s})^2+\gamma _{s}(\gamma _{n}-\gamma _{s})}{16\gamma _{s}}, \pi _M^{RA*}=\frac{\gamma _{n}-\gamma _{s}}{8}.&\end{aligned}$$

Under the scenario RP, the profit functions of the retailer and the manufacturer are given in Eqs. (7) and (8). Similarly, we can obtain the optimal prices and profits as follows

$$\begin{aligned}&p_n^{RP*}=\frac{2\gamma _{n}+c+c_h\gamma _{n}}{4}, p_h^{RP*}=\frac{c_h+1}{2},&\\&w_n^{RP*}=\frac{c_h\gamma _{n}+c}{2},&\\&\pi _R^{RP*}=\frac{(c-c_h\gamma _{n})^2+4\gamma _{n}(1-\gamma _{n})(1-c_h)^2}{16\gamma _{n}(1-\gamma _{n})},&\\&\pi _M^{RP*}=\frac{(c-c_h\gamma _{n})^2}{8\gamma _{n}(1-\gamma _{n})}.&\end{aligned}$$

Under the scenario AN, the profit functions of the retailer and the manufacturer are given in Eqs. (9) and (10). By backward induction, we can obtain the optimal prices and profits as follows

$$\begin{aligned}&p_n^{AN*}=\frac{c+\gamma _{n}}{2},&\\&\pi _R^{AN*}=\frac{\alpha (c-\gamma _{n})^2}{4\gamma _{n}}, \pi _M^{AN*}=\frac{(1-\alpha )(\gamma _{n}-c)^2}{4\gamma _{n}}.&\end{aligned}$$

Under the scenario AA, the profit functions of the retailer and the manufacturer are given in Eqs. (11) and (12). Similarly, we can obtain the optimal prices and profits of the retailer and the manufacturer’s as follows

$$\begin{aligned}&p_n^{AA*}=\frac{2\gamma _{n}^2-(2\gamma _{s}-3c)\gamma _{n}-\alpha c\gamma _{s}}{4\gamma _{n}-(1+\alpha )\gamma _{s}},&\\&p_s^{AA*}=\frac{(\gamma _{n}-\gamma _{s})(\alpha \gamma _{s}+\gamma _{s}+2c)+c\gamma _{s}(3-\alpha )}{4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s}},&\\&\pi _R^{AA*}=\frac{(\gamma _{n}-\gamma _{s})[(4\gamma _{n}-\alpha \gamma _{s})(\alpha \gamma _{n}\gamma _{s}-c\alpha \gamma _{s}-c\gamma _{s}+c^2)]}{\gamma _{s}(4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s})^2}\\&\qquad \quad \ +\frac{(\gamma _{n}-\gamma _{s})(\gamma _{n}\gamma _{s}^2)}{\gamma _{s}(4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s})^2},\\&\pi _M^{AA*}=\frac{(2\gamma _{n}-c)^2(\gamma _{n}-\gamma _{s})(1-\alpha )}{(4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s})^2}.&\end{aligned}$$

Under the scenario AP, the profit functions of the retailer and the manufacturer are given in Eqs. (13) and (14). Similarly, we can obtain the optimal prices and profits as follows

$$\begin{aligned}&p_n^{AP*}=\frac{c_h\gamma _n+\gamma _n+2c-\gamma _n^2-\alpha c\gamma _{n}}{4-(1+\alpha \gamma _{n})},&\\&p_h^{AP*}=\frac{2c_h+c+2-2\gamma _{n}-\alpha c}{4-(1+\alpha \gamma _{n})},&\\&\pi _R^{AP*}=\frac{(c-2c_h-2\gamma _{n}-\alpha c+c_h\gamma _{n}+\alpha c_h\gamma _{n}+2)\sigma _2}{(1-\gamma _{n})(4-\gamma _{n}-\alpha \gamma _{n})}&\\&\qquad \quad \ +\frac{\alpha \sigma _1^2}{\gamma _{n}(1-\gamma _{n})},&\\&\pi _M^{AP*}=\frac{(1-\alpha )\sigma _1^2}{\gamma _{n}(1-\gamma _{n})}.&\end{aligned}$$

Appendix B

1.1 B.1 Proof of Proposition 2

• (1) $\pi _R^{RA}-\pi _R^{RN}=\frac{4(c-\gamma _{s})^2+\gamma _{s}(\gamma _{n}-\gamma _{s})}{16\gamma _{s}}-\frac{(c-\gamma _{n})^2}{16\gamma _{n}}=\frac{4\gamma _{n}c_2-6c\gamma _{n}\gamma _s+3\gamma _n\gamma _s ^2-c^2\gamma _{s}}{16\gamma _{s}\gamma _n}.$ Since $4\gamma _{n}c_2-6c\gamma _{n}\gamma _s+3\gamma _n\gamma _s ^2-c^2\gamma _{s}=3\gamma _n(c-\gamma _{s})^2+c^2(\gamma _{n}-\gamma _{s})$, we can obtain that $\pi _R^{RA}-\pi _R^{RN}>0$.

• (2) $\pi _R^{RN}-\pi _R^{RP}=\frac{(c-\gamma _{n})^2}{16\gamma _{n}}-\frac{(c-c_h\gamma _{n})^2}{16\gamma _{n}(1-\gamma _{n})}-\frac{4\gamma _{n}(1-\gamma _{n})(1-c_h)^2}{16\gamma _{n}(1-\gamma _{n})}$. Since $c<c_h<1$, $\frac{(c-\gamma _{n})^2}{16\gamma _{n}}<\frac{(c-c_h\gamma _{n})^2}{16\gamma _{n}(1-\gamma _{n})}$. We can further obtain $\pi _R^{RN}<\pi _R^{RP}$.

• (3) Set $\Delta \pi =\pi _R^{RA}-\pi _R^{RP}=\frac{4(c-\gamma _{s})^2+\gamma _{s}(\gamma _{n}-\gamma _{s})}{16\gamma _{s}}-\frac{4\gamma _{n}(1-\gamma _{n})(1-c_h)^2}{16\gamma _{n}(1-\gamma _{n})}-\frac{4\gamma _{n}(1-\gamma _{n})(1-c_h)^2}{16\gamma _{n}(1-\gamma _{n})}$. c has two solutions $c_{RR1}=\frac{\gamma _{n}\gamma _{s}[(1-\gamma _{n})(4-\delta _1)-c_h]}{4\gamma _{n}(1-\gamma _{n})-\gamma _{s}}<0$ and $c_{RR2}=\frac{\gamma _{n}\gamma _{s}[4(1-\gamma _{n}-\delta _1(1+\gamma _{n}-c_h))]}{4\gamma _{n}(1-\gamma _{n})-\gamma _{s}}$ when $\Delta \pi =0$. $\frac{\partial \Delta \pi }{\partial c}\vert _{c=c_{RR2}}>0$, so $\Delta \pi$ is monotonically decreasing with respect to c under the constraint $\bar{c}>c>0$. Since $\Delta \pi \vert _{c=c_{RR2}}=0$, we can prove that $\pi _R^{RA}<\pi _R^{RP}$ when $0<c<c_{RR2}$ and $\pi _R^{RA}\ge \pi _R^{RP}$ when $c\ge c_{RR2}$. Set $c_{RR}^*=c_{RR2}$. (Note that $\delta _1=\frac{A}{\gamma _{n}\gamma _{s}(1-\gamma _{n})}$, where $A=-12c_h^2\gamma _{n}^2+16c_h^2\gamma _{n}-4c_h^2\gamma _{s}+32c_h\gamma _{n}^2-8c_h\gamma _{n}\gamma _{s}-32c_h\gamma _{n}+8c_h\gamma _{s}+4\gamma _{n}^3-4\gamma _{n}^2\gamma _{s}-20\gamma _{n}^2+5\gamma _{n}\gamma _{s}+16\gamma _{n}+3\gamma _{s}^2-4\gamma _{s}$)

• (4) $\pi _M^{RN}-\pi _M^{RA}=\frac{(c-\gamma _{n})^2}{8\gamma _{n}}-\frac{\gamma _{n}-\gamma _{s}}{8}=\frac{c(c-\gamma _{n})-\gamma _{n}(c-\gamma _{s})}{8\gamma _{n}}$. Since $\gamma _{n}>\gamma _{s}>c$, $\pi _M^{RN}>\pi _M^{RA}$.

• (5) $\pi _M^{RN}-\pi _M^{RP}=\frac{(c-\gamma _{n})^2}{8\gamma _{n}}-\frac{(c-c_h\gamma _{n})^2}{8\gamma _{n}(1-\gamma _{n})}=\frac{(c-\gamma _{n})^2(1-\gamma _{n})-(c-c_h\gamma _{n})^2}{8\gamma _{n}(1-\gamma _{n})}$. Since $(c-\gamma _{n})^2(1-\gamma _{n})-(c-c_h\gamma _{n})^2=\gamma _{n}^2-\gamma _{n}^3+2c\gamma ^2+2cc_h\gamma _{n}-2c\gamma _{n}-c^2\gamma _{n}>0$, we can prove that $\pi _M^{RN}>\pi _M^{RP}$.

• (6) $\pi _M^{RA}-\pi _M^{RP}=\frac{\gamma _{n}-\gamma _{s}}{8}-\frac{(c-c_h\gamma _{n})^2}{8\gamma _{n}(1-\gamma _{n})}=\frac{(\gamma _{n}-\gamma _{s})\gamma _{n}(1-\gamma _{n})-(c-c_h\gamma _{n})^2}{8\gamma _n(1-\gamma _{n})}$. When $(\gamma _{n}-\gamma _{s})\gamma _{n}(1-\gamma _{n})-(c-c_h\gamma _{n})^2<0$, that is $0<c\le c_{h}\gamma _{n}-\sqrt{\gamma _{n}(1-\gamma _{n})(\gamma _{n}-\gamma _{s})}$, $\pi _M^{RA}<\pi _M^{RP}$. When $c\ge c_{h}\gamma _{n}-\sqrt{\gamma _{n}(1-\gamma _{n})(\gamma _{n}-\gamma _{s})}$, $\pi _M^{RA}\ge \pi _M^{RP}$. Set $c_{RM}^*=c_{h}\gamma _{n}-\sqrt{\gamma _{n}(1-\gamma _{n})(\gamma _{n}-\gamma _{s})}$.

1.2 B.2 Proof of Proposition 4

• (1) Set $\Delta \pi =\pi _R^{AN*}-\pi _R^{AA*}=\frac{\alpha (c-\gamma _{n})^2}{4\gamma _{n}}-\frac{(\gamma _{n}-\gamma _{s})[(4\gamma _{n}-\alpha \gamma _{s})(\alpha \gamma _{n}\gamma _{s}-c\alpha \gamma _{s}-c\gamma _{s}+c^2)]}{\gamma _{s}(4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s})^2}+\frac{(\gamma _{n}-\gamma _{s})(\gamma _{n}\gamma _{s}^2)}{\gamma _{s}(4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s})^2}$. c has two solutions $c_{AN1}$ and $c_{AN2}$ when $\Delta \pi =0$. $c_{AN2}=\frac{\gamma _{n}\gamma _{s}\sigma _3}{\sigma _4}$, where $\sigma _3=8\gamma _{n}\gamma _{s}+8\alpha \gamma _{n}^2-\alpha \gamma _{s}^2-8\gamma _{n}^2+8\gamma _{n}\sqrt{\alpha \gamma _{s}(\gamma _{n}-\gamma _{s})}-2\gamma _{s}\sqrt{\alpha \gamma _{s}(\gamma _{n}-\gamma _{s})}+\alpha ^3\gamma _{s}^2-2\alpha \gamma _{s}\sqrt{\alpha \gamma _{s}(\gamma _{n}-\gamma _{s})}+2\alpha \gamma _{n}\gamma _{s}-6\alpha ^2\gamma _{n}\gamma _{s}$ and $\sigma _4=\alpha ^3\gamma _{s}^3-8\alpha ^2\gamma _{n}\gamma _{s}^2+2\alpha ^2\gamma _{s}^3+20\alpha \gamma _{n}^2\gamma _{s}-12\alpha \gamma _{n}\gamma _{s}^2+\alpha \gamma _{s}^3-16\gamma _{n}^3+16\gamma _{n}^2\gamma _{s}$. We can obtain that $c_{AN2}<0$ and $c_{AN1}>\bar{c}$. $\Delta \pi \vert _{c=0}$, so $\Delta \pi >0$ under the constrain $0<c<\bar{c}$. We can prove that $\pi _R^{AN*}>\pi _R^{AA*}$.

• (2) $\pi _M^{AN*}-\pi _M^{AA*}=\frac{(1-\alpha )(\gamma _{n}-c)^2}{4\gamma _{n}}-\frac{(2\gamma _{n}-c)^2(\gamma _{n}-\gamma _{s})(1-\alpha )}{(4\gamma _{n}-\gamma _{s}-\alpha \gamma _{s})^2}$. We solve $\pi _M^{AN*}-\pi _M^{AA*}=0$ and obtain two solutions, $c_{MAN1}$ and $c_{MAN2}$. $\bar{c}<c_{MAN1}<c_{MAN2}$. We can obtain that $\pi _M^{AN*}>\pi _M^{AA*}$.