Retailer service strategy on livestreaming platforms considering free riding behavior

Abstract

Livestreaming e-commerce is a significant and effective digital tool for retailers to boost sales during the COVID-19 outbreak. Services on livestreaming platforms may be provided by either the manufacturer or the retailer. As service free riding across products becomes increasingly prevalent, the key issues retailers face include selecting which product should be promoted and who should provide service on the livestreaming channel. Using a game-theoretic framework, we investigate a retailer’s optimal livestreaming service strategy that considers free riding between the retailer’s store brand and a manufacturer’s national brand. Our main findings are as follows. When service resources on the livestreaming channel are limited, (1) the retailer should not promote products with extremely low base demand, and (2) given that the national brand is promoted, if the two brands exhibit either very similar or significantly different features, the retailer should provide service for the national brand personally; otherwise, the retailer should delegate the service to the manufacturer. When livestreaming resources are unlimited, it may be unnecessary for the retailer to promote both brands if the store brand has a large base demand.

Introduction

The global outbreak of COVID-19 has posed great challenges to the retail industry. Customers may shift their purchases from offline to online due to public health concerns and “stay-at-home” orders, and some retail stores are closed temporarily due to COVID-19 lockdowns. In this pandemic world, digital technologies create new paths toward a retail revolution. Specifically, enabled by information technologies, augmented reality and artificial intelligence, livestreaming e-commerce has witnessed explosive growth during the COVID-19 outbreak. For example, JD.com launched its livestreaming platform to promote products and continues to enrich the live experience by adding 3D animations, augmented reality and enhanced personalization (Ruether, 2022). Amazon and YouTube introduced virtual try-on services in their livestreams, allowing consumers to try on clothes or makeup from their phones (Billington, 2020). Austin Li, one of China’s most influential live streamers, co-hosted livestreaming campaigns alongside a virtual idol called Tianyi Luo (Magloff, 2020). Nowadays, various retailers, including Walmart, Gree Electric, Suning, Amazon and JD.com, extensively conduct marketing activities through live video on livestreaming platforms such as Facebook Live and TikTok. Livestreaming e-commerce has proved to be an efficient digital tool for retailers to boost customer engagement and sales during the COVID-19 outbreak (Sun et al., 2019; Wongkitrungrueng & Assarut, 2020). According to Mckinsey, livestreaming-initiated sales could account for 10–20% of all e-commerce by 2026 (Mckinsey, 2021).

Livestreaming platforms provide customers with diverse services, including product demonstrations, “real-life” connections, virtual and interactive entertainment, immersive shopping experiences, live activities and so forth (Wang et al., 2022a; Wongkitrungrueng et al., 2020). These services help retailers boost sales by enhancing customer engagement and brand awareness (Kang et al., 2021; Sun et al., 2019; Wongkitrungrueng & Assarut, 2020). For a retailer with multiple brands of similar products (e.g., JD.com, Suning and Amazon), when one brand is promoted on the livestreaming platform, consumers improve their knowledge about similar products and develop trust in those products based on their common attributes (Tian et al., 2022; Zhang et al., 2019). Since the webpage of the product featured on a livestreaming channel always has artificial intelligence recommendation links to other products, customers can easily search for similar products after viewing a livestreaming. Gu and Tayi (2017) reported that 85% of respondents switch to buying a different but related product after inspecting one product in many product categories, including home appliances, furniture, apparel, shoes and accessories. This omnichannel shopping behavior unavoidably accelerates service free riding across products, which has been well-studied and documented in the context of information service and advertising (Carlton & Chevalier, 2008; Tian et al., 2022; Wu et al., 2004).

Service free riding across products significantly impacts retailers’ livestreaming service strategies. In reality, firms may not be able to provide livestreaming services for all products due to budgetary constraints or time limitations. For example, when firms engage Internet celebrities or influencers such as Austin Li and Viya to promote their products, most small- and medium-sized enterprises can only afford the fees charged by the streamers for a portion of products (Tao et al., 2022; Wang et al., 2022a). Meanwhile, the livestreaming shows hosted by these famous streamers usually last between 2 and 3 h, and these streamers cannot promote all products for firms in a short period (Wang et al., 2022a). Livestreaming resources can also be sufficient. For example, when firms use their own salespersons as the broadcasters, they can host livestreaming shows every day to promote all products. Additionally, digital technologies have stimulated the emergence of virtual anchors such as “Xiao Mei” on JD.com, and they can host 24-hour live streams to promote all products (Liu, 2022b). When a retailer sells both the manufacturer’s national brand and its own store brand, free riding will weaken the demand-enhancing effect of service on the livestreaming brand but generate a positive influence on the non-livestreaming brand. Thus, service competition between products changes. In addition, providing service for the store brand can mitigate double marginalization, while the marginal revenue of the store brand may be lower than that of the national brand. Therefore, retailers with limited resources face the key operational problem of choosing the right product to offer via livestreaming service. When resources are sufficient, providing services for both brands may lead to more revenue as well as a higher cost than promoting only one brand. Meanwhile, the former puts brands in direct competition for services, which may be more intense than the competition caused by free riding. Therefore, in such situations, it is worthwhile to explore whether, considering service free riding, all products should be promoted on livestreaming channels.

Livestreaming services for a national brand may be provided by either the retailer or the manufacturer. For example, JD.com hires its own product experts to recommend suppliers’ products via livestreaming (Linkshop, 2019), while the electronics manufacturer Gree sets up livestream events for its traditional distributors and recommends over 50 products from a professional vantage point (Liu, 2020). Additionally, Ha et al. (2021) point out that both retailers and large manufacturers may conduct marketing activities via live streams. Various studies have compared the two service modes, that is, retailer-provided service and manufacturer-provided service (Kolay, 2015; Kouvelis & Shi, 2020; Li et al., 2016). However, these studies don’t account for service free riding. When services are provided by different supply chain parties, the degrees of double marginalization and product competition will differ. Meanwhile, cross-product free riding also influences the competition between firms. On this basis, a debate arises regarding who should provide service for the national brand, considering service free riding from the store brand.

In this paper, we focus on addressing the following research questions:

1. Given limited livestreaming resources, which brand (the national brand or the store brand) should be promoted on the livestreaming channel, considering free riding?

2. When there are no constraints on livestreaming resources, should the retailer promote both brands in the presence of free riding?

3. Who should provide service for the national brand if it is promoted on the livestreaming channel?

To answer these questions, we construct a stylized model in which a retailer sells two competing products, namely, a manufacturer’s national brand and its own store brand. The two brands are differentiated in both horizontal features and base demands. When one brand is promoted on the retailer’s livestreaming channel, this service will have a spillover effect on the other brand owing to customer free riding behavior. We consider five service strategies for livestreaming channels: (1) retailer providing service for SB; (2) retailer providing service for NB; (3) manufacturer providing service for NB; (4) retailer providing services for both SB and NB and (5) retailer providing service for SB and manufacturer providing service for NB. By conducting comparative analyses across the first three service strategies, we derive the optimal livestreaming service strategy with limited resources. Then, we obtain the optimal service strategy with unlimited resources through further comparison with another two strategies. Finally, we conduct several numerical experiments to explore the influences of various parameters.

Our main findings are as follows. Given restricted livestreaming resources, when the store brand’s initial base demand is large enough (i.e., many customers prefer the store brand), the retailer should only promote the store brand to remove double marginalization on the livestreaming channel. When the base demand for the national brand is very high, providing livestreaming service for the national brand can lead to higher profits due to the high profit margin from the service. Moreover, the intensity of horizontal differentiation between brands plays a crucial role in the retailer’s selection of the service provider for the national brand. Specifically, if the two brands exhibit either little or significant feature differentiation, the retailer should personally provide livestreaming service for the national brand; otherwise, the retailer should delegate the service to the manufacturer to avoid high service costs. The reason for this lies in the trade-offs between the opposite effects of product substitutability on the two brands’ demands. Finally, given unlimited livestreaming resources, when the store brand’s base demand is enough large, the retailer should only promote the store brand to make full use of free riding; otherwise, the retailer should provide livestreaming services for both brands.

Our paper makes several contributions by combining service free riding, service provision modes and livestreaming service organically. First, we explore the impact of free riding on the retailer’s optimal service strategy. Second, we systematically answer the questions of which product provided with service and who to provide service on the livestreaming platform.

The remainder of our paper is organized as follows. Section 2 briefly summarizes the relevant literature. Section 3 presents our main model. Section 4 derives the equilibrium solutions for all service strategies. Section 5 compares strategies to determine the retailer’s optimal strategies with and without resource limitations. Section 6 conducts numerical experiments to explore the influences of some parameters and the robustness of our main findings. Section 7 presents conclusions and limitations. All proofs are given in the Appendix A.

Literature review

Our work is primarily related to three streams of research: research on livestreaming e-commerce, service free riding and service provision. In this section, we present a brief overview of each research area and discuss our contributions.

Livestreaming e-commerce

To thrive in the pandemic world, retailers are using digital technologies for various innovations. Among the different digital operations strategies, livestreaming e-commerce has gained great attention from researchers. A series of empirical studies have explored the relationships among the characteristics of livestreaming, customer motivations and behavioral intentions (Lin et al., 2022; Wang et al., 2022a). In addition, existing research investigates the effectiveness of livestreaming e-commerce in promoting consumer empowerment and boosting sales. For example, Kang et al. (2021) find that interactivity in livestreaming commerce has a curvilinear relationship with customer engagement behavior through tie strength. Wongkitrungrueng and Assarut (2020) verify that livestreaming has the potential to build customer engagement through two types of customer trust. Theoretically, some researchers have established models to explore the necessity for firms to introduce a livestreaming channel in various supply chain structures. For example, Pan et al. (2022) investigate the conditions under which a seller owning a traditional channel should add a livestream channel. Wang and Zhang (2022) build an analytical model to examine whether a manufacturer should adopt the influencer channel established by an e-tailer. They find that the influencer channel is beneficial to the e-tailer and consumers. However, whether a manufacturer benefits from it depends on the possibility of misfit elimination of the influencer channel and consumers’ hassle costs. Zhang et al. (2022) study whether and which part to introduce livestreaming services in the supply chain composed of a manufacturer and an e-commerce platform. In addition, Zhou et al. (2022) explore a manufacturer’s decision about using livestreaming as a new channel based on the social attributes of live streams. Different from them, we examine the optimal operation strategy of livestreaming service given that a livestreaming channel is employed. Regarding how to implement livestreaming service efficiently, Ha et al. (2021) analyze the optimal choice among three possible channel structures of an online intermediary that exerts service effort through live streams. Tao et al. (2022) study the financing scheme between a capital-constrained online retailer and an e-commerce platform in the context of livestreaming selling events. However, none of these studies explore the service strategy for multiple products on a livestreaming channel. Our work contributes to this body of literature in two ways. First, we account for the free riding of livestreaming service between a national brand and a store brand in a two-echelon supply chain. Second, we incorporate different service provision modes to thoroughly explore service strategies in the context of livestreaming e-commerce.

Service free riding

A substantial body of literature has studied the effect of free riding between the same product and the same type of channel (e.g., online or offline) (Pun et al., 2020; Shin, 2007; Wu et al., 2004). In the omnichannel world, cross-channel and cross-product free riding has become increasingly popular. With the development of advanced technologies, it is more convenient for customers to switch channels and products to comprehensively maximize the shopping experience. Most existing research examines the effect of free riding between online and offline channels selling the same product, such as showrooming and webrooming (Jing, 2018; Kuksov & Liao, 2018; Sun et al., 2022). Regarding free riding across products, Gu and Tayi (2017), Li et al. (2020) and Zhang et al. (2020b) analyze product assortment strategy considering free riding between two horizontally differentiated products. However, the above literature mainly considers the free riding of product fit information for reducing product returns. In contrast, we investigate free riding between a store brand and a national brand from the perspective of demand-enhancing services. Extant literature confirms the existence of service free riding across products. For example, Gallino and Moreno (2018) empirically find that virtual fitting rooms not only increase sales of the products displayed in the fitting rooms but also create spillovers for related products that are not available for virtual try-on. Gallino and Moreno provide a potential explanation for this effect: the virtual fitting room enhances customer loyalty and revisit behavior. Similarly, Zhang et al. (2019) verify that pop-up store visits increase the sales of both participating and nonparticipating retailers by strengthening customers’ trust and interest in similar products sold on the same platform. Moreover, Tian et al. (2022) analytically find that customers are more likely to engage in free riding across differentiated products than homogeneous products. We contribute to this area of literature by combining service free riding across products with different service modes to comprehensively examine service strategies on livestreaming channels.

Service provision

A large body of research has examined various service provision modes from the perspective of the identity of the service provider, such as retailer-provided service (Ranjbar et al., 2021; Tsay & Agrawal, 2000), manufacturer-provided service (Guan et al., 2020), third-party-provided service (Kouvelis & Shi, 2020), platform-provided service (Zhang et al., 2021), and a co-operative service mode (Liu et al., 2014; Zhang et al., 2020a). Furthermore, several papers have compared the above service modes to explore which party should perform the demand-enhancing service. For example, Li et al. (2016) compare four alternative service channels and consider different service providers in a supply chain. They demonstrate that a conflict in service channel selection may arise since both the retailer and the manufacturer prefer to either perform the service or hire the third party by themselves. As for the co-operative mode, Liu et al. (2014) study the efficacy of cost sharing by comparing co-operative advertising with advertising performed entirely by one party. Kouvelis and Shi (2020) address whether the manufacturer or the retailer should compensate the sales agent’s efforts under different supply contracts. Regarding the comparison between retailer-provided service and manufacturer-provided service, Kolay (2015) finds that a manufacturer delegating service to a retailer may reduce product base demand, channel members’ profits and consumer welfare. Bian et al. (2017) incorporate different power structures to identify when the manufacturer should outsource service to the retailer. Wang et al. (2022b) examine two competitive retailers’ choices between retailer-provided service and manufacturer-provided service. It is worth noting that the above papers compare different service provision modes without considering service free riding. In the context of livestreaming e-commerce, the prevalent free riding phenomenon exerts a significant impact on retailers’ service strategies. Thus, it is meaningful to further explore the choice of service mode while considering free riding. We not only examine which party should provide service but also address which product should be endowed with livestreaming service.

In summary, none of the above papers has simultaneously consider livestreaming service, service free riding, and service provision mode. To partially fill this research gap, we systematically combine the above research streams to explore the optimal livestreaming strategies considering the competition between a national brand and a store brand. Our study makes contributions by answering the following questions: impact of free riding, choice of product and choice of service provider.

Model

We consider a two-echelon supply chain in which a manufacturer (referred to as “he”) sells a national brand (henceforth, NB) through a retailer (referred to as “she”). The retailer also sells her own store brand (henceforth, SB), which is considered an acceptable substitute for NB. To promote sales in the pandemic world, the retailer introduces a livestreaming channel. The retailer must decide which product (NB, SB or both) should be promoted in the livestreaming channel. The service for SB is provided only by the retailer, while the service for NB may be provided by either the retailer or the manufacturer. Following Jerath and Zhang (2010), we assume that the retailer is more powerful than the manufacturer in negotiations over service provision because the retailer has a sufficiently large consumer market that the manufacturer seeks to tap into. As illustrated in Fig. 1, the retailer has five possible livestreaming service strategies: (1) Strategy S, in which the retailer provides service for SB; (2) Strategy R, in which the retailer provides service for NB; (3) Strategy M, in which the manufacturer provides service for NB; (4) Strategy SR, in which the retailer provides service for SB and NB; and (5) Strategy SM, in which the retailer provides service for SB and the manufacturer provides service for NB.

Fig. 1.

Retailer service strategies

Note that the resources in the livestreaming channel (e.g., livestreaming duration or capital budget) may be limited or unlimited. Under a situation of limited resources, the retailer can promote only one product and will choose from the first three strategies (Strategy S, Strategy R and Strategy M). Since the two products have some common attributes and are sold on the same platform, the non-livestreaming product can free ride on the service for the livestreaming product, enabled by customers’ switching behavior. In other words, the livestreaming product’s service has a positive spillover effect on the non-livestreaming product. When there are no resource constraints, the retailer can promote both brands in her livestreaming channel. Since both brands are provided with services, consumers do not need to switch between the two products. Thus, service free riding no longer exists in Strategy SR and Strategy SM.

We denote the wholesale price of NB as $\omega$, the service level as $e$ and the retail price as $p$. In addition, we use subscripts $s$ and $n$ and superscripts $S$, $R$, $M$, $\mathit{SR}$and $\mathit{SM}$ to indicate the corresponding products and strategies, respectively. Table 1 summarizes all the notations used throughout the paper.

Table 1.

Table of notation

Indexes	Definition
$S,R,M,SR,SM$	Superscripts, indexes for service strategies
$s,n$	Subscripts, indexes for SB and NB, respectively
Decision variables	Definition
$\omega$	Wholesale price for NB
$e$	Service level
$p$	Retail price
Other notations	Definition
$a$	SB’s base demand
$\theta$	Product substitutability rate
$\tau$	Degree of free riding
$D$	Product demand
$\mathrm{\Gamma }$	Indicator for service strategy (the value is 0 or 1)
${\pi }_r$, ${\pi }_m$	Profits for the retailer and the manufacturer, respectively

We use $a_i(i=s,n)$ to represent the initial base demand of product $i$ when it is the only available product, the price is set to zero and no service is performed. The difference between $a_s$ and $a_n$ reflects customers’ initial preference for SB and NB. Without a loss of generality, we normalize $a_n=1$ and set $a_s=a$. We do not impose any restrictions on the relative magnitudes of $a_s$ and $a_n$. Specifically, if $a>1$, SB’s initial base demand is larger than NB’s; otherwise, NB has a larger base demand than SB.

Since NB and SB have some common feature attributes, we use $\theta (0\le \theta <1)$ to denote the rate of substitutability or interchangeability between the two brands. More specifically, $\theta$ measures the degree of feature differentiation (Chanchoi & Coughlan, 2006). When $\theta =0$, the two products are fully differentiated and completely independent; as $\theta$ approaches 1, the products converge toward being maximally substitutable.

In Strategy S, the retailer only provides service level $e_s^S$ for SB. Following Liu et al. (2014), the new base demand of SB becomes $a+e_s^S$ under the demand-enhancing effect of service. In the presence of service free riding, $e_s^S$ will also trigger NB’s demand. We denote the degree of free riding from NB by $\tau$, which indicates that a fraction $\tau$ of customers who enjoy SB’s service in the livestreaming channel will further browse and consider NB in the sale channel. Moreover, enjoying one product’s service only enables consumers to learn about the common attributes between SB and NB, in the spirit of findings by Tian et al. (2022) and Gu and Tayi (2017). Obviously, the larger $\theta$ is, the more common attributes there are between the two products. Consistent with Jerath and Zhang (2010), the service for the livestreaming product will increase the base demand of the non-livestreaming product to a greater extent as $\theta$ increases. Thus, when NB is the only available product in the sale channel and its price is set to zero, $e_s^S$ will boost NB’s base demand to $1+\theta \tau e_s^S$ in the presence of free riding.

In Strategy R, the retailer provides service level $e_n^R$ for NB. Then, the new base demand of NB becomes $1+e_n^R$. To simplify our discussion, we consider a symmetric free riding degree in our main model. That is, the degree of free riding from SB is equal to $\tau$. In Sect. 6.2, we examine a general setting of asymmetric free riding degree and find that our main findings continue to hold. With the effect of free riding, the new base demand of SB becomes $a+\theta \tau e_n^R$. Similarly, in Strategy M, the new base demands of NB and SB become $1+e_n^M$ and $a+\theta \tau e_n^M$, respectively.

When both products are provided with services, service free riding no longer exists. In this case, providing service for one product only enhances this product’s base demand but does not affect the other product’s base demand. Therefore, the new initial base demands of SB and NB become:

$$a_s^{\prime }=a+{\mathrm{\Gamma }}^{\mathit{SR}}{e}_s^{\mathit{SR}}+{\mathrm{\Gamma }}^{\mathit{SM}}{e}_s^{\mathit{SM}},$$ 1

$$a_n^{\prime }=1+{\mathrm{\Gamma }}^{\mathit{SR}}{e}_n^{\mathit{SR}}+{\mathrm{\Gamma }}^{\mathit{SM}}{e}_n^{\mathit{SM}},$$ 2

where ${\mathrm{\Gamma }}^i=0$ or 1 is the indicator of Strategy $i$ $(i=S,R,M,SR,SM).$

We adopt the following utility function of a representative consumer to derive demand functions for the two brands. This function is widely used in the literature (Cai et al., 2012; Ingene & Parry, 2007):

$$U\equiv \sum _{i=s,n}\left(a_i,D_i,-,\frac{{D_i}^2}{2},-,p_i,D_i\right)-\theta D_s{D}_n,$$ 3

where $D_i$ is the demand for product $i$. We substitute the above new base demands into Eq. (3). By maximizing the utility function, we can obtain the demands for the two products in all strategies. Section 4 characterizes the demand functions in detail.

Following previous studies (Ha et al., 2021; Tsay & Agrawal, 2000), the corresponding service cost incurred is $k{e}^2$. This quadratic form conveys the increasing marginal service cost and diminishing returns. To enable a fair comparison between different service provision modes and for tractability, we assume $k=1$ for all service strategies. This assumption is consistent with the existing literature (Jerath & Zhang, 2010; Kolay, 2015; Liu et al., 2014). We also relax this assumption in Sect. 6.3 where firms differ in their efficiency in offering service. Intuitively, the result shows that strategies with high efficiency in service provision are easier to be optimal. Finally, for brevity, we normalize the operational and production costs for all firms to zero, a practice that has been widely adopted in the extant literature (Cai et al., 2012; Guan et al., 2020; Raj et al., 2020). Let ${\pi }_r$ and ${\pi }_m$ denote the profits of the retailer and the manufacturer, respectively. The profits are then calculated as follows:

$${\pi }_r=\left(p_n,-,\omega \right)D_n+p_s{D}_s-{\mathrm{\Gamma }}^S{e}_s^{S^2}-{\mathrm{\Gamma }}^R{e}_n^{R^2}-{\mathrm{\Gamma }}^{\mathit{SR}}\left(e_s^{S{R}^2},+,e_n^{S{R}^2}\right)-{\mathrm{\Gamma }}^{\mathit{SM}}{e}_s^{S{M}^2},$$ 4

$${\pi }_m=\omega D_n-{\mathrm{\Gamma }}^M{e}_n^{M^2}-{\mathrm{\Gamma }}^{\mathit{SM}}{e}_n^{S{M}^2}.$$ 5

In line with prior research (Iyer, 1998; Kuksov & Liao, 2018), the game proceeds as follows. In Stage 1, the retailer chooses the service strategy. In Stage 2, the manufacturer determines his wholesale price for NB. In Stage 3, the service provider(s) sets the service level(s). In Stage 4, the retailer decides on retail prices for both products. Appendix B considers another game sequence, that is, service providers set service levels in Stage 2, and the manufacturer determines his wholesale price for NB in Stage 3. Comparing the equilibrium results between the two game sequences, we find that both firms prefer the game sequence in our main model. The reason is as follow. When the wholesale price is determined before the service level(s), the manufacturer can reduce his wholesale price to stimulate the retailer’s service level, and thus double marginalization is further alleviated.

Equilibrium solutions

Based on the game sequence given in Sect. 3, we use backward induction to derive the optimal solutions for all service strategies. To simplify our discussion, we divide the five strategies into two categories: those that promote only one brand and those that promote both brands. The former includes Strategy S, Strategy R and Strategy M; free riding exists in these strategies. By comparing the retailer’s profits across these three strategies, we can obtain the optimal service strategy when livestreaming resources are limited. Strategy SR and Strategy SM belong to the group of strategies that promotes both brands and has no service free riding. They exist only when there is no constraint on livestreaming resources. The optimal service strategy without resource limitations can be obtained by comparing all five strategies. Next, we derive the equilibrium solutions for the two categories of strategies.

Strategies promoting only one brand (Strategy S, Strategy R and Strategy M)

We substitute the initial new base demands for Strategy S, Strategy R and Strategy M (given in Sect. 3) into the utility function in Eq. (3). Then, by maximizing the utility function, we obtain the following demands for the two products:

$$D_s=\frac{a-\theta -p_s+\theta p_n+(1-\tau {\theta }^2){\mathrm{\Gamma }}^S{e}_s^S-(1-\tau )\theta ({\mathrm{\Gamma }}^R{e}_n^R+{\mathrm{\Gamma }}^M{e}_n^M)}{1-{\theta }^2},$$ 6

$$D_n=\frac{1-\theta a-p_n+\theta p_s+(1-\tau {\theta }^2)({\mathrm{\Gamma }}^R{e}_n^R+{\mathrm{\Gamma }}^M{e}_n^M)-(1-\tau )\theta {\mathrm{\Gamma }}^S{e}_s^S}{1-{\theta }^2}.$$ 7

The above constructions have the following implications. First, the higher a product’s base demand, the more competitive it is against the other product. Second, in the absence of free riding (i.e., $\tau =0$), the service for the livestreaming product enhances its demand in the following two ways: by encroaching on the market of the non-livestreaming product and by attracting new potential customers. In the presence of free riding, the non-livestreaming product in turn cannibalizes the livestreaming product’s demand and attracts potential customers in the whole market through service free riding. Overall, the service exerts a net demand-enhancing effect on the livestreaming product, a net negative impact on the non-livestreaming product’s demand, and a positive effect on the aggregate demand for both products. As the free riding degree $\tau$ increases, the first two effects become weaker, while the last effect on the total demand becomes stronger. As product substitutability rate $\theta$ increases, the first two effects become stronger, while the last effect becomes weaker. This is because fierce competition will intensify the market cannibalization between products. Then, the service for the livestreaming product erodes more demand from the non-livestreaming product but gains few new potential consumers. Zhou et al. (2018) and Li et al. (2019) directly use linear demand functions to model service free riding between firms, and they only reflect that one firm’s service has a net demand-enhancing effect for both firms. Different from them, our demand functions derived from the utility function capture more practical influences of service and service free riding. That is, the service for one product encroaches on the market of the other product, service free riding mitigates this negative influence through customers’ switching purchase behavior, and service free riding enhances the total demand of both products by attracting new customers in the potential market.

By plugging the demand functions (i.e., Eqs. 6 and 7) into the profit functions (i.e., Eqs. 4 and 5), we can derive the equilibrium outcomes through backward induction. Lemma 1 summarizes the equilibrium results for the three service strategies.

Lemma 1

For Strategy S, Strategy R and Strategy M, the equilibrium wholesale prices, service levels, retail prices and profits are given in Table 2.

Table 2.

Equilibrium outcomes for Strategy S, Strategy R and Strategy M

	Strategy S	Strategy R	Strategy M
${\omega }^{i\ast }$	$\frac{3-\theta (4-\tau )a}{6}$	$\frac{4-{\theta }^2{\tau }^2-\theta (4-\tau )a}{2(4-{\theta }^2{\tau }^2)}$	$\frac{4(1-\theta a)(1-{\theta }^2)}{7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2)}$
$e_s^{i\ast }$	$\frac{(6-{\theta }^2(4+\tau +{\tau }^2\left)\right)a-3\theta (1-\tau )}{6(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	–	–
$e_n^{i\ast }$	–	$\frac{\theta \left(\right(7-4{\theta }^2)\tau +3{\theta }^2{\tau }^2-2{\theta }^2{\tau }^3-4)a}{2(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}+\frac{4+{\theta }^4{\tau }^3-{\theta }^2\tau (4+\tau )}{2(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{(1-\theta a)(1-{\theta }^2\tau )}{7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2)}$
$p_s^{i\ast }$	$\frac{(24-{\theta }^2(28-11\tau +7{\tau }^2\left)\right)a-3\theta (1-\tau )}{12(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{(24+{\theta }^4(4-\tau ){\tau }^2-{\theta }^2(32-12\tau +7{\tau }^2\left)\right)a}{4(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}+\frac{\theta \tau (4+{\theta }^4{\tau }^3-{\theta }^2\tau (4+\tau \left)\right)}{4(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{(7-{\theta }^2(8-\tau \left)\right)a+\theta \tau (1-{\theta }^2\tau )}{2(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)}$
$p_n^{i\ast }$	$\frac{27-3{\theta }^2(12-5\tau +2{\tau }^2)}{12(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}-\frac{\theta (12-9\tau -{\theta }^2(16-16\tau +5{\tau }^2-2{\tau }^3\left)\right)a}{12(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{\theta \left({\theta }^2\right(16-16\tau +9{\tau }^2-3{\tau }^3)-2(8-5\tau \left)\right)a}{4(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}+\frac{40-{\theta }^2(48-20\tau +22{\tau }^2)+{\theta }^4{\tau }^2(12-5\tau +3{\tau }^2)}{4(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{\theta \left({\theta }^2\right(4+\tau )-5)a+12-{\theta }^2(12-\tau )-{\theta }^4{\tau }^2}{2(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)}$
${\pi }_r^{i\ast }$	$\frac{(16-{\theta }^2(16-\tau (8-5\tau )\left)\right)a^2}{16(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}+\frac{3-2\theta (4-\tau )a}{16(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{3(16-{\theta }^2(16-(8-5\tau )\tau \left)\right)a^2}{16(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}+\frac{(4-{\theta }^2{\tau }^2{)}^2-2\theta (4-\tau \left)\right(4-{\theta }^2{\tau }^2)a}{16(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{(49-2{\theta }^2(48-7\tau )+{\theta }^4(48-16\tau +{\tau }^2\left)\right)a^2}{4(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2){)}^2}-\frac{2\theta (16-7\tau -{\theta }^4(8-\tau ){\tau }^2-2{\theta }^2(8-4\tau -3{\tau }^2\left)\right)a}{4(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2){)}^2}+\frac{16-{\theta }^2(16-{\tau }^2(1-{\theta }^2\tau {)}^2)}{4(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2){)}^2}$
${\pi }_m^{i\ast }$	$\frac{(3-\theta (4-\tau )a{)}^2}{24(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{(4-{\theta }^2{\tau }^2-\theta (4-\tau )a{)}^2}{8(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}$	$\frac{(1-\theta a{)}^2}{7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2)}$

Strategies promoting both brands (Strategy SR and Strategy SM)

Similarly, we plug the new initial base demands in Eqs. (1) and (2) into the utility function in Eq. (3). Then, maximizing the utility function gives the following demand functions for Strategy SR and Strategy SM:

$$D_s=\frac{a-\theta -p_s+\theta p_n+{\mathrm{\Gamma }}^{\mathit{SR}}(e_s^{\mathit{SR}}-\theta e_n^{\mathit{SR}})+{\mathrm{\Gamma }}^{\mathit{SM}}(e_s^{\mathit{SM}}-\theta e_n^{\mathit{SM}})}{1-{\theta }^2},$$ 8

$$D_n=\frac{1-a\theta -p_n+\theta p_s+{\mathrm{\Gamma }}^{\mathit{SR}}(e_n^{\mathit{SR}}-\theta e_s^{\mathit{SR}})+{\mathrm{\Gamma }}^{\mathit{SM}}(e_n^{\mathit{SM}}-\theta e_s^{\mathit{SM}})}{1-{\theta }^2}.$$ 9

From the above demand functions, it is easy to see that in addition to price, the two products also compete on service level when they are both promoted in the livestreaming channel. Moreover, as $\theta$ increases, the service competition becomes intensified.

Following the game sequence in Sect. 3, we use backward induction to derive the equilibrium outcomes for the two strategies, as shown in Lemma 2.

Lemma 2

For Strategy SR and Strategy SM, the equilibrium wholesale prices, service levels, retail prices and profits are given in Table 3.

Table 3.

Equilibrium outcomes for Strategy SR and Strategy SM

	Strategy SR	Strategy SM
${\omega }^{i\ast }$	$\frac{3-4\theta a}{6}$	$\frac{4(3-4a\theta ){(1-{\theta }^2)}^2}{21-46{\theta }^2+24{\theta }^4}$
$e_s^{i\ast }$	$\frac{\left(9,-,8,{\theta }^2\right)a-6\theta }{27-48{\theta }^2}$	$\frac{(7-10{\theta }^2+4{\theta }^4)a-4\theta +3{\theta }^3}{21-46{\theta }^2+24{\theta }^4}$
$e_n^{i\ast }$	$\frac{3-4a\theta }{18-32{\theta }^2}$	$\frac{(3-4a\theta )(1-{\theta }^2)}{21-46{\theta }^2+24{\theta }^4}$
$p_s^{i\ast }$	$\frac{2\left(9,-,14,{\theta }^2\right)a-3\theta }{27-48{\theta }^2}$	$\frac{28{(1-{\theta }^2)}^{2}a-4\theta +3{\theta }^3}{42-92{\theta }^2+48{\theta }^4}$
$p_n^{i\ast }$	$\frac{45-8\theta (9\theta +(3-4{\theta }^2\left)a\right)}{54-96{\theta }^2}$	$\frac{36-73{\theta }^2+36{\theta }^4-4\theta (5-9{\theta }^2+4{\theta }^4)a}{42-92{\theta }^2+48{\theta }^4}$
${\pi }_r^{i\ast }$	$\frac{4\left(3,-,4,{\theta }^2\right)a^2-8\theta a+3}{36-64{\theta }^2}$	$\frac{4(49-4{\theta }^2(32-29{\theta }^2+9{\theta }^4\left)\right)a^2}{4{(7-6{\theta }^2)}^2(3-4{\theta }^2)}-\frac{8\theta {(4-3{\theta }^2)}^{2}a-3{(4-3{\theta }^2)}^2}{4{(7-6{\theta }^2)}^2(3-4{\theta }^2)}$
${\pi }_m^{i\ast }$	$\frac{{(3-4a\theta )}^2}{54-96{\theta }^2}$	$\frac{{(3-4a\theta )}^2{(1-{\theta }^2)}^2}{{(3-4{\theta }^2)}^2(7-6{\theta }^2)}$

From the above equilibrium outcomes, we can easily observe that the service level for SB always increases with $a$ while that for NB always decreases with $a$. This means that in Strategy SR, the retailer will offer a higher level of service for the more profitable product (i.e., the product with larger base demand) to gain more profit. In Strategy SM, a firm will offer a higher level of service for its own brand when it has a base demand advantage over the other brand.

Optimal service strategy

In this section, we conduct comparative analyses across strategies to derive the retailer’s optimal strategy. Since livestreaming resources may be limited or unlimited, we discuss the optimal results under these two situations separately. Below, we first compare the retailer’s profits in Strategy S, Strategy R and Strategy M to derive the optimal strategy with resource limitations. Then, the profits in all five strategies are compared to obtain the optimal strategy when livestreaming resources are unlimited.

With resource limitation

All discussions in this subsection presume the common feasible area of Strategy S, Strategy R and Strategy M, as detailed in the Appendix A. As a first step toward understanding our managerial insights, we compare the optimal wholesale prices and service levels across the three strategies, as displayed in Lemma 3.

Lemma 3

The optimal wholesale prices and service levels in Strategy S, Strategy R and Strategy M have the following orders:

(1) ${\omega }^{M\ast }\ge {\omega }^{R\ast }>{\omega }^{S\ast }$if $a\le a_{\omega }^{\mathit{MR}}$ and ${\omega }^{R\ast }>{\omega }^{M\ast }>{\omega }^{S\ast }$ if $a>a_{\omega }^{\mathit{MR}}$.

(2) $e_n^{R\ast }>e_n^{M\ast }\ge e_s^{S\ast }$ if $a\le a_e^{\mathit{MS}}$, $e_n^{R\ast }\ge e_s^{S\ast }>e_n^{M\ast }$ if $a_e^{\mathit{MS}}<a\le a_e^{\mathit{RS}}$ and $e_s^{S\ast }>e_n^{R\ast }>e_n^{M\ast }$ if $a>a_e^{\mathit{RS}}$.

Simple intuition intimates that the manufacturer will lower his wholesale price to motivate a higher service level from the retailer in Strategy R but will charge a higher wholesale price to fully cover his service cost in Strategy M. However, the first part of Lemma 3 demonstrates that the wholesale price in Strategy R may be higher than that in Strategy M when SB’s base demand is sufficiently high. We explain the reason as follows. As $a$ increases, SB becomes more competitive, leading to a lower wholesale price of NB. In Strategy R, the reduction of wholesale price will stimulate a higher service level from the retailer. This in turn enhances the demand of NB and thus alleviates the reduction of wholesale price. In Strategy M, with a lower wholesale price, the manufacturer will offer a lower level of service, and hence NB’s demand may further reduce. Therefore, although the wholesale prices in Strategy R and Strategy M both decrease with $a$, the latter falls at a faster rate than the former. Then, when $a$ is higher than some threshold value, the wholesale price in Strategy R will be larger than that in Strategy M. Finally, because one product’s demand increases with its own service level but decreases with the competing product’s service level, it is straightforward that the wholesale price in Strategy S is the lowest among these three strategies.

Note that firms determine their service levels based on net returns to service. Specifically, the higher the marginal revenue generated by the service, the more service efforts a firm is willing to make. The retailer can gain revenue from both SB and NB, while the manufacturer can only obtain revenue from NB. Although the service for NB boosts NB’s demand and erodes SB’s demand, service free riding mitigates these influences. As discussed in the demand functions in Sect. 4.1, the livestreaming service can enhance the total demand for the two products. In Strategy R, the retailer’s service for NB boosts the total demand of SB and NB and the retailer gains revenue from both products. Meanwhile, the manufacturer will reduce his wholesale price to stimulate more service from the retailer, which significantly lessens double marginalization. In Strategy M, although the manufacturer’s service for NB boosts NB’s demand, service free riding from SB will weaken the demand-enhancing effect. In addition, the manufacturer can only obtain revenue from NB. Overall, the retailer’s marginal service revenue in Strategy R is higher than the manufacturer’s in Strategy M. Thus, as indicated in the second part of Lemma 3, the retailer’s service level for NB in Strategy R is always higher than the manufacturer’s in Strategy M.

The second part of Lemma 3 also shows that the service level for SB grows gradually higher than that for NB as SB’s base demand increases. As discussed above, the retailer collects revenue from both products, whereas the manufacturer only earns revenue from NB. When the retailer provides service for SB, the livestreaming channel is free of double marginalization. In addition, the larger $a$, the higher retail price of SB. Then, when the service level for SB rises by one unit, the increase in SB’s demand will generate more revenue for the retailer. Since the retailer can collect a higher marginal revenue from SB, she is willing to provide more service for SB. Therefore, as $a$ increases, the service level in Strategy S grows gradually higher than the service level in Strategy M and even higher compared to the service level in Strategy R.

Comparing the profits of supply chain members across the three strategies, we present the main results in the following propositions.

Proposition 1

Given limited livestreaming resources, there exist thresholds such that

1. when $a\ge max(a_r^{\mathit{RS}},a_r^{\mathit{MS}})$, Strategy S is the retailer’s optimal service strategy;

2. when $a<max(a_r^{\mathit{RS}},a_r^{\mathit{MS}})$,

   1. Strategy M is the retailer’s optimal service strategy if ${\theta }_1\le \theta \le {\theta }_2$, and
   2. Strategy R is the retailer’s optimal service strategy otherwise.

We use Fig. 2 to further illustrate Proposition 1. As shown in Fig. 2, when SB has a high base demand, the retailer should promote SB in the livestreaming channel; otherwise, it is reasonable for the retailer to promote NB. Regarding the service provider for NB, the retailer should delegate the service to the manufacturer if the product substitutability rate is intermediate; otherwise, the retailer itself should provide service for NB. These results imply that base demand differentiation and feature differentiation between products jointly determine the retailer’s optimal choice of livestreaming service strategy.

Fig. 2.

Retailer’s optimal service strategy with resource limitation given $\tau$= 0.5. (Color figure online)

As stated above, double marginalization is directly removed from the livestreaming channel in Strategy S. In addition, providing service for SB will boost the demand of SB but reduce the demand of NB. Therefore, with Strategy S, there is a trade-off between the gain from SB and the loss from NB. Lemma 3 shows that the service level in Strategy S is highest when $a$ is extremely large. Meanwhile, a large $a$ implies a high marginal revenue from SB. Overall, with the demand-enhancing effect of the high service level, the retailer will reap an extremely large revenue from SB. Although her revenue from NB may be relatively low, the total service revenue can far exceed the high service cost, eventually resulting in a high net profit. In contrast, if $a$ is relatively small, the service levels in both Strategy R and Strategy M will be higher than the service level in Strategy S, as shown in Lemma 3. In this situation, the other two strategies will outperform Strategy S because of the small profit margin and the low service level of SB.

Regarding the choice of service provider for NB, the retailer should balance her profit gain in Strategy M and the service cost in Strategy R. The discussion of demand functions in Sect. 4.1 shows that service for NB increases NB’s demand but reduces SB’s demand. Both of these influences become more pronounced as $\theta$ increases. Moreover, with a larger $\theta$, the service boosts the total demand for the two products to a lower extent. Therefore, when $\theta$ is very low, the mode of retailer-provided service not only mitigates double marginalization but also produces a high total revenue for the retailer. In contrast to Strategy M, in Strategy R, the retailer’s increased profit can fully compensate for the service cost incurred. When $\theta$ is in the medium range, the positive effect of service on the total demand is not very significant. At the same time, the relatively intense competition between products can alleviate double marginalization in Strategy M. In this case, delegating service provision to the manufacturer helps the retailer avoid the high service cost, thus generating a higher profit for the retailer. Finally, when $\theta$ is very large, the service has a strong demand-boosting effect on NB, in large part through cannibalizing SB’s market. Thus, in Strategy M, the manufacturer will offer a high level of service to NB. This change in service level further encroaches on the market for SB. In this case, the retailer will have a lower profit in Strategy M than in Strategy R.

Proposition 1 has the following practical suggestions. First, retailers should not promote products with low base demand (unpopular products) in their livestreaming channels. Conversely, they should promote the bestselling, popular products on livestreaming platforms. This suggestion supports the recent practice of Suning’s selection team, who screened out the most popular items for live broadcast. Second, regarding the service provider for NB, retailers should invite well-known manufacturers to provide service when there are a medium number of attribute differences between brands (e.g., smartphones and home appliances). In other situations, when brands belong to the categories with significant attribute differences (e.g., clothing, jewelry and beauty products) or slight attribute differences (e.g., digital accessories, office supplies and daily consumables), retailers should provide service for national brands.

Proposition 2 presents the result of comparing manufacturer’s profits across the three service strategies.

Proposition 2

Given limited livestreaming resources, the manufacturer’s profits have the following order: ${\pi }_m^{\text{R*}}>{\pi }_m^{\text{M*}}>{\pi }_m^{\text{S*}}$.

Although NB can free ride the service for SB, this indirect effect is far weaker than the direct demand-expanding effect of the service for NB. Thus, the manufacturer’s profit is lowest with Strategy S. Lemma 3 shows that the service level for NB in Strategy R is always higher than in Strategy M. Meanwhile, the manufacturer does not bear any service cost in Strategy R. Therefore, the manufacturer’s profit is highest with Strategy R. According to Proposition 1 and Proposition 2, we can observe a conflict between the retailer and the manufacturer’s choice of service strategies. Therefore, when the retailer requires the manufacturer to provide service for NB, she should take measures to encourage the manufacturer, such as service cost sharing.

Without resource limitation

In this subsection, we further analyze the situation of unlimited livestreaming resources. When there is no resource constraint, the retailer can promote both products on her livestreaming channel. In addition to the above three service strategies, we compare the other two strategies: Strategy SR and Strategy SM. Through a comparison of all five strategies, Proposition 3 reports the retailer’s optimal service strategy given unlimited livestreaming resources.

Proposition 3

Given unlimited livestreaming resources, when $a\ge a_r^{\mathit{SRS}},$ Strategy S is the retailer’s optimal service strategy; otherwise, Strategy SR is the retailer’s optimal service strategy.

Figure 3 vividly illustrates Proposition 3. As depicted in Fig. 3, if SB has a large base demand, the retailer should only provide livestreaming service for SB; otherwise, the retailer should offer livestreaming services for both products herself. The proof of Proposition 3 in the Appendix A shows that Strategy SR is always superior to Strategy SM, Strategy R and Strategy M. The underlying reasons are as follows. In Strategy SR, the retailer can control and adjust the service levels for both products to balance product competition. Specifically, the retailer will provide more service for the more profitable product, as discussed in Lemma 2. Given the service competition between products, in Strategy SR, the manufacturer will further reduce his wholesale price to stimulate the retailer’s service for NB, thus mitigating double marginalization. However, in Strategy SM, the retailer and manufacturer will compete on service to maximize their respective profits. This direct and intensified service competition, together with worse double marginalization, is harmful to the retailer. Regarding Strategy R and Strategy M, providing service for NB always cannibalizes the market of SB, which hurts the retailer. Overall, although the retailer may incur a high total service cost in Strategy SR, her improved revenue is sufficient to compensate for the increased service cost.

Fig. 3.

Retailer’s optimal service strategy without resource limitation, given $\tau$= 0.5. (Color figure online)

In Strategy S, the livestreaming channel is free of double marginalization, and the retailer will gain a high service revenue from SB when $a$ is very large. In Strategy SR, the product competition will intensify when the two products have comparable base demands. Thus, when $a$ is large, the retailer’s profit in Strategy SR will be lower than in Strategy S. When $a$ is very small, providing service for SB will not generate significant revenue. In this scenario, Strategy SR is best since the retailer can adjust her service levels to offer more (less) service to the more (less) profitable product.

Proposition 3 suggests that it is unnecessary for retailers to provide services for all products even though service resources are unlimited. Specifically, the retailer should only promote SB on the livestreaming channel when the base demand of SB is not very low (e.g., Amazon’s Fire TV and Google’s Chromecast); otherwise, the retailer should provide service for both products by herself.

Proposition 2 compares the manufacturer’s profits across Strategy S, Strategy R and Strategy M. On this basis, we further compare these profits with the manufacturer’s profits in Strategy SR and Strategy SM to shed light on the manufacturer’s preference when livestreaming resources are unlimited.

Proposition 4

Given unlimited livestreaming resources, Strategy SR is the manufacturer’s optimal service strategy if $a\le a_m^{\mathit{SRR}},$ and Strategy R is the manufacturer’s optimal service strategy otherwise.

Proposition 2 demonstrates that Strategy R is always the manufacturer’s first choice when given limited livestreaming resources. However, Proposition 4 shows that this is not always true when livestreaming resources are unlimited. Specifically, when SB’s base demand is very small, Strategy SR may outperform Strategy R for the manufacturer. As indicated in the proof of Proposition 4, ${\pi }_m^{SR\ast }$ is always larger than ${\pi }_m^{SM\ast }$. In Strategy SM, the two firms compete on service levels. This intensified competition, together with worse double marginalization, drives Strategy SR to outperform Strategy SM for the manufacturer. When a is very small, the retailer will offer more service for NB in Strategy SR. However, the retailer will reduce her service level for NB in Strategy R since SB with very low base demand will free ride on the service level for NB. Thus, the manufacturer’s profit in Strategy SR may be higher than in Strategy R if is very small. Both firms may have conflicts in the choice of optimal service strategy.

Numerical analyses

In this section, we conduct several numerical experiments to explore the influences of various parameters. In Sect. 6.1, we investigate the impact of free riding degree on the retailer’s optimal service strategy. In Sect. 6.2, we extend our main model to explore the robustness of our findings under an asymmetric degree of free riding. In Sect. 6.3, we relax the assumption of the same service cost coefficient to study the impact of service cost coefficient. These numerical experiments are conducted based on hundreds of parameter combinations spanning the common feasible space. For brevity, we only present some representative examples, and the characteristics of the optimal decision reflected in these examples are still consistent in a larger numerical space.

Impact of free riding degree

Propositions 1 and 3 demonstrate that the retailer’s optimal strategy mainly depends on a and θ, while the threshold values of the two parameters are affected by the free riding degree τ. Figure 4 vividly presents the result in Proposition 1. That is, Strategy S is optimal if a is high, Strategy M is optimal if a is low and θ is intermediate, and Strategy R is optimal otherwise. Meanwhile, Fig. 4 illustrates the impact of τ on the retailer’s optimal strategy when service resources are limited. Specifically, when τ is very low, Strategy M will never be optimal; as τ increases, the area for Strategy M being best enlarges. This means that Strategy M is much easier to outperform Strategy R and Strategy S as τ increases. As discussed in Eqs. (6) and (7), service free riding in Strategy M mitigates the encroachment of NB’s service on SB’s market. With a higher τ, the above influence becomes even stronger. Then, the retailer can not only gain a high revenue from NB but also reap a not low revenue from SB.

Fig. 4.

Impact of free riding degree (red for Strategy S, blue for Strategy R, and green for Strategy M). (Color figure online)

When service resources are sufficient, Proposition 3 shows that Strategy SR or Strategy S may be optimal. It is easy to know that the retailer’s profit in Strategy SR is independent of $\tau$ and the one in Strategy S increases with $\tau$. Therefore, as $\tau$ increases, Strategy S is much easier to outperform Strategy SR. In other words, the region for Strategy S (Strategy SR) dominance enlarges (shrinks) as $\tau$ increases. The reason is that free riding in Strategy S enhances the total demand of both brands. Thus, a higher $\tau$ leads to a higher profit for the retailer in Strategy S.

Asymmetric degree of free riding

The degree of free riding may differ from one product’s service to another. We denote the degree of free riding from NB in Strategy S by ${\tau }^{\prime }$ to distinguish it from $\tau$ in Strategy R and Strategy M. Correspondingly, the equilibrium outcomes in Strategy S change when replacing $\tau$ with ${\tau }^{\prime }$. The equilibrium outcomes in Strategy R and Strategy M remain the same. Thus, ${\pi }_r^{M\text{*}}-{\pi }_r^{R\text{*}}$ stays unchanged. In view of analytical difficulties, we numerically compare ${\pi }_r^{R\text{*}}$and ${\pi }_r^{M\text{*}}$ with ${\pi }_r^{S\text{*}}$. Here, we present some of those results in Fig. 5.

Fig. 5.

Retailer’s profits under an asymmetric degree of free riding (red for Strategy S, blue for Strategy R, and green for Strategy M). (Color figure online)

Figure 5 demonstrates that our main results remain qualitatively unchanged. That is, when $a$ is very large, providing service for SB (i.e., Strategy S) is the optimal strategy. When $a$ is relatively small, Strategy M is best if $\theta$ is intermediate, and Strategy R is optimal otherwise. Moreover, as $\tau$ increases, the area for Strategy M expands, given a fixed ${\tau }^{\prime }$. Recall that NB’s service level will cannibalize SB’s market. With a high free riding degree from SB, the retailer in Strategy R may provide a high service level for NB, while the manufacturer in Strategy M may not offer an extremely high service level even though $\theta$ is very large. Thus, it is much easier for Strategy M to outperform Strategy R for the retailer because of the manufacturer’s low level of service. Finally, since there is no free riding in Strategy SR and Strategy SM, the asymmetric free riding degree likewise does not alter our main results when livestreaming resources are unlimited.

Different service cost coefficients

Here, we consider that the retailer and the manufacturer are different in terms of their efficiency in offering services. In other words, the service cost coefficients for the two firms are distinct. In particular, we assume that the service cost coefficient for the retailer is equal to 1 and the one for the manufacturer $k$ is more general. $k$ may be higher or lower than 1. When $k>1$, the retailer provides services more efficiently. Under this assumption, only the equilibrium outcomes in Strategy M and Strategy SM have a change.

Figure 6 illustrates the impact of service cost coefficient when resources are limited. When $k$ is relatively small, the optimal strategy will switch from Strategy R to Strategy M as compared to the main model. In addition, combining Figs. 4 and 6, we can easily observe that as $k$ increases, the region for Strategy M dominance shrinks gradually. These results are intuitive because large $k$ means the manufacturer’s inefficiency in offering service. Finally, with a high $a$, Strategy S is always best, which is in line with our main result.

Fig. 6.

Impact of service cost coefficient with resource limitation (red for Strategy S, blue for Strategy R, and green for Strategy M). (Color figure online)

Figure 7 presents the impact of service cost coefficient when resources are unlimited. As illustrated in Fig. 7, when $k$ is very small, Strategy SM may be optimal in addition to Strategy S and Strategy SR in the main model. As $k$ increases, the zone for Strategy SM being optimal gradually shrinks. It finally vanishes when $k$ rises to 1. This is also intuitive because the increase in $k$ boosts the manufacturer’s service cost and hence worsens double marginalization. Additionally, Strategy M is never optimal, while Strategy SM may be best even when $k$ and $a$ are both low. The reason is that a lower $k$ drives the manufacturer to offer more service for NB in Strategy M. This will encroach on more markets of SB. Thus, the retailer prefers Strategy SM by offering service to SB. Finally, in line with the main model. Strategy S is still best when $a$ is very large.

Fig. 7.

Impact of service cost coefficient without resource limitation (red for Strategy S, blue for Strategy R, green for Strategy M, orange for Strategy SR and brown for Strategy SM). (Color figure online)

Conclusion

To thrive in the pandemic world, various retailers have turned to a digital platform—livestreaming e-commerce. Despite the growing popularity and potential benefits of livestreaming platforms, retailers face great challenges regarding how to effectively implement this service strategy. Considering service free riding between NB and SB, we develop a game-theoretic model to explore a retailer’s optimal livestreaming service strategy. First, we conduct comparative analyses across three service strategies: the retailer providing service for SB, the retailer providing service for NB and the manufacturer providing service for NB. We use these analyses to derive the optimal strategy with limited livestreaming resources. We find that the disparity in base demand and feature differentiation between products jointly determine the retailer’s optimal choice. Specifically, the retailer should display the product with large base demand in the livestreaming channel due to the high service revenue. Regarding the choice of service provider for NB, it seems more profitable for the retailer to control the service level when the product substitutability rate is either extremely high or low. Otherwise, delegating service provision to the manufacturer is more beneficial. We then examine a scenario with unlimited livestreaming resources. By comparing an additional two strategies, we find that when the base demand of SB is large, the retailer should provide service only for SB. Otherwise, the retailer should provide service for both products to control service levels.

With two crucial levers, namely, base demand disparity and feature differentiation, our study yields several managerial implications. First, our results justify retailers’ recent practice of promoting bestselling products in livestreaming channels. For example, Suning set up a selection team with the code name “MK-1” to choose suitable products for its live broadcast room. This team mainly screens out the bestselling and popular offerings by studying market hot spots (Liu, 2022a). Second, our results offer managerial insights into the appropriate service provider for NB when resources are limited. We recommend that retailers invite famous manufacturers to provide service when there is a medium level of attribute differences between the store brand and national brands (e.g., for smartphones and home appliances). For products with substantial attribute differences (e.g., clothing, jewelry and beauty products) or slight attribute differences (e.g., digital accessories, office supplies and daily consumables), retailers should provide service themselves. This finding also supports Suning’s practices. Suning has launched a store, “Qu guang guang,” that combines livestreaming with offline shopping. In this store, Suning sets up special livestreaming areas for large brands and invites them to engage in livestreaming events (Xu, 2020). Finally, in the absence of resource limitations, we find that retailers should provide service for all products in most cases unless customers prefer SB.

There are several limitations to our study. First, our paper only focuses on two opposite service provision modes: manufacturer-provided service and retailer-provided service. In reality, a situation may arise in which both parties simultaneously provide service for the same product. Thus, it is worthwhile to examine other service modes, such as the co-operative mode. Second, we have limited our discussion to the competition between a store brand and a national brand. Since multiple manufacturers may compete in a service war, one interesting extension of our work would be to allow for horizontal competition between national brands. Third, this study considers market situations in which the retailer adopts a conventional reselling format. As the agency selling format is increasingly popular (e.g., Taobao and eBay), future research could extend our model to consider online marketplaces, which would yield further insights. Finally, our results show that there may exist conflicts between channel members regarding the choice of service strategy. Therefore, it is useful to design possible coordination mechanisms, such as cost-sharing or revenue-sharing contracts, to improve channel members’ profits and supply chain performance.

Appendix A

Proof of Lemma 1 We first derive the retailer’s best retail price responses. Specifically, given $\omega$ and $e$, the Hessian matrix of ${\pi }_r$ with respect to $(p_s,p_n)$ is $\left[\begin{array}{cc}\frac{2}{{\theta }^2-1}& \frac{2\theta }{1-{\theta }^2}\\ \frac{2\theta }{1-{\theta }^2}& \frac{2}{{\theta }^2-1}\end{array}\right]$ in all service strategies, which is negative definite. Thus, ${\pi }_r$ is jointly concave in $p_s$ and $p_n$. From the first-order conditions (FOCs), we obtain $p_s(\omega ,e)=\frac{a+{\mathrm{\Gamma }}^S{e}_s^S+\tau \theta ({\mathrm{\Gamma }}^R{e}_n^R+{\mathrm{\Gamma }}^M{e}_n^M)}{2}$ and $p_n(\omega ,e)=\frac{1+\omega +{\mathrm{\Gamma }}^R{e}_n^R+{\mathrm{\Gamma }}^M{e}_n^M+\tau \theta {\mathrm{\Gamma }}^S{e}_s^S}{2}$.

Then, we substitute the above response functions into the profit functions in Eqs. (4) and (5). The retailer continues to maximize her profit by setting the service level in Strategy S and Strategy R. Given $\omega$, ${\pi }_r$ is concave in $e$ when $\frac{{\partial }^2{\pi }_r}{\partial {e_s^S}^2}=\frac{{\partial }^2{\pi }_r}{\partial {e_n^R}^2}=\frac{3-{\theta }^2(4-2\tau +{\tau }^2)}{-2(1-{\theta }^2)}\le 0$. According to the FOCs, we have $e_s^S\left(\omega \right)=\frac{(\omega -1)\theta (1-\tau )+a(1-{\theta }^2\tau )}{3-{\theta }^2(4-2\tau +{\tau }^2)}$ and $e_n^R\left(\omega \right)=\frac{(1-{\theta }^2\tau )(1-\omega )-\theta \left(1,-,\tau \right)a}{3-{\theta }^2(4-2\tau +{\tau }^2)}$. In Strategy M, anticipating the retailer’s reactions, the manufacturer will determine his service level by maximizing his profit function. Since $\frac{{\partial }^2{\pi }_m}{\partial {e_n^M}^2}=-2<0$, ${\pi }_m$ is concave in $e_n^M$. With the FOC, we have $e_n^M\left(\omega \right)=\frac{(1-{\theta }^2\tau )\omega }{4(1-{\theta }^2)}$.

Finally, we substitute all response functions into the manufacturer’s profit function. The manufacturer will determine his optimal wholesale price by maximizing the profit function. Given the optimal wholesale price, all equilibrium outcomes in Table 2 are derived. Moreover, we can determine the feasible areas for each strategy to ensure that the equilibrium outcomes are nonnegative: $\theta <\sqrt{\frac{3}{4-2\tau +{\tau }^2}}$ and $\frac{3\theta (4-\tau )}{24-{\theta }^2(16-\tau (8-7\tau \left)\right)}\le a\le \frac{3}{\theta (4-\tau )}$ in Strategy S; $\theta <\sqrt{\frac{3}{4-2\tau +{\tau }^2}}$ and $\frac{\theta (4-\tau )(4-{\theta }^2{\tau }^2)}{24-{\theta }^2(16-\tau (8-7\tau \left)\right)}\le a\le \frac{4-{\theta }^2{\tau }^2}{\theta (4-\tau )}$ in Strategy R; and $\theta <\frac{\sqrt{2\sqrt{4-2\tau \left(1,-,\tau \right)}+\tau -4}}{\tau }$ and $\frac{\theta (4-\tau +{\theta }^2{\tau }^2)}{7-{\theta }^2(4-\tau )}\le a\le \frac{1}{\theta }$ in Strategy M.

Proof of Lemma 2 The solution procedure is similar to Lemma 1. For brevity, we omit the details. The feasible areas are $\theta <0.75$ and $\frac{6\theta }{9-8{\theta }^2}\le a\le \frac{3}{4\theta }$ for Strategy SR and $\theta <0.866$ and $\frac{4\theta -3{\theta }^3}{7-10{\theta }^2+4{\theta }^4}\le a\le \frac{3}{4\theta }$ for Strategy SM.

Proof of Lemma 3 We conduct comparison analyses in the common feasible domain for the three strategies, that is, $\theta <\sqrt{\frac{3}{4-2\tau +{\tau }^2}}$ and $\underset{\_}{a}=\text{max}\left(\frac{\theta (4-\tau )(4-{\theta }^2{\tau }^2)}{24-{\theta }^2(16-\tau (8-7\tau \left)\right)},,,\frac{\theta (4-\tau +{\theta }^2{\tau }^2)}{7-{\theta }^2(4-\tau )}\right)\le a\le \frac{3}{\theta (4-\tau )}=\overline{a}$.

1. The differences in optimal wholesale prices are as follows:

   $${\omega }^{R\text{*}}-{\omega }^{S\text{*}}=\frac{a\theta (4-\tau )(1-{\theta }^2{\tau }^2)}{6(4-{\theta }^2{\tau }^2)}>0,$$

   $${\omega }^{M\ast }-{\omega }^{S\ast }=\frac{3(1-{\theta }^2\tau {)}^2+\theta (4-7\tau -{\theta }^4(4-\tau ){\tau }^2+2{\theta }^2(4-8\tau +{\tau }^2))a}{6(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)}>0,$$

   $$\Delta {\omega }^{\mathit{MR}}={\omega }^{M\ast }-{\omega }^{R\ast }=\frac{\left(1,-,{\theta }^2,\tau \right)\left(\left(2,-,\theta ,\tau \right),\left(2,+,\theta ,\tau \right),\left(1,-,{\theta }^2,\tau \right),-,\theta ,\left(4,+,7,\tau ,-,12,{\theta }^2,\tau ,+,{\theta }^2,{\tau }^2\right),a\right)}{2\left(4,-,{\theta }^2,{\tau }^2\right)\left(7,-,{\theta }^2,\left(8,-,2,\tau ,+,{\theta }^2,{\tau }^2\right)\right)}.$$

   It is clear that $\Delta {\omega }^{\mathit{MR}}$ is decreasing in $a$. Letting $\Delta {\omega }^{\mathit{MR}}=0$ yields $a_{\omega }^{\mathit{MR}}=\frac{(2-\theta \tau )(2+\theta \tau )(1-{\theta }^2\tau )}{\theta (4+7\tau -12{\theta }^2\tau +{\theta }^2{\tau }^2)}$. Therefore, for $a\in [\underset{\_}{a},\overline{a}]$, we have ${\omega }^{M\text{*}}\ge {\omega }^{R\text{*}}>{\omega }^{S\text{*}}$if $a\le a_{\omega }^{\mathit{MR}}$ and ${\omega }^{R\text{*}}>{\omega }^{M\text{*}}>{\omega }^{S\text{*}}$ if $a>a_{\omega }^{\mathit{MR}}.$

2. The differences in the optimal service levels are as follows:

   $$\begin{array}{cc}\hfill \Delta e^{\mathit{RS}}& =e_n^{R\ast }-e_s^{S\ast }\hfill \\ \hfill & =\frac{(1-\theta \tau )\left(3\right(1+\theta \left)\right(4-{\theta }^2{\tau }^2)-(24+3\theta (4+\tau )-{\theta }^3\tau (4+\tau +{\tau }^2)-{\theta }^2(16-8\tau +7{\tau }^2)\left)a\right)}{6(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)},\hfill \end{array}$$

   $$\begin{array}{cc}& \Delta e^{\mathit{MS}}=e_n^{M\ast }-e_s^{S\ast }\hfill \\ \hfill & =\frac{3(6+\theta (7-2{\theta }^2(4-\tau )(1-\tau )-7\tau -{\theta }^4(1-\tau ){\tau }^2+2{\theta }^3\tau (4-(2-\tau \left)\tau \right)-2\theta (4+\tau +{\tau }^2)\left)\right)}{6(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}\hfill \\ \hfill & -\frac{42+\theta (18+\theta ((5-7\tau )\tau -76+6{\theta }^3\tau (4-(2-\tau \left)\tau \right)-6\theta (4+\tau +{\tau }^2)+{\theta }^4{\tau }^2(4+\tau +{\tau }^2)+2{\theta }^2(16-{\tau }^3)\left)\right)}{6(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}a,\hfill \end{array}$$

   $$\begin{array}{cc}\hfill \Delta e^{\mathit{MR}}& =e_n^{M\ast }-e_n^{R\ast }=\frac{(2-\theta \tau )(2+\theta \tau )(1-{\theta }^2\tau )\left({\theta }^2,\tau \right(2-(2-{\theta }^2)\tau )-1)}{2(4-{\theta }^2{\tau }^2)(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}\hfill \\ \hfill & +\frac{\theta (4-(49-100{\theta }^2+64{\theta }^4)\tau -3{\theta }^2(7-12{\theta }^2){\tau }^2+{\theta }^2(14-25{\theta }^2+4{\theta }^4){\tau }^3+{\theta }^4(2-{\theta }^2\left){\tau }^4\right)}{2(4-{\theta }^2{\tau }^2)(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}.\hfill \end{array}$$

All above differences are monotonic functions of $a$. Moreover, their first-order derivatives with respect to $a$ involve only two parameters (i.e., $0\le \tau \le 1$ and $0\le \theta <\frac{\sqrt{3}}{\sqrt{4-(2-\tau )\tau }}$). Following Jerath and Zhang (2010) and Yang et al., (2019), we resort to a graphic solution via contour plotting to find the signs of the first-order derivatives. We can offer any of the dozens of contour plots used in our paper, but we omit them here to focus on the exposition. By contour plotting in terms of $\theta$ and $\tau$, we find both $\Delta e^{\mathit{RS}}$ and $\Delta e^{\mathit{MS}}$ are decreasing in $a$. Letting $\Delta e^{\mathit{RS}}=0$ and $\Delta e^{\mathit{MS}}=0$, we have (1) $\Delta e^{\mathit{RS}}\ge 0$ if $a\le a_e^{\mathit{RS}}$ and $\Delta e^{\mathit{RS}}<0$ if $a>a_e^{\mathit{RS}}$, and (2) $\Delta e^{\mathit{MS}}\ge 0$ if $a\le a_e^{\mathit{MS}}$ and $\Delta e^{\mathit{MS}}<0$ if $a>a_e^{\mathit{MS}}$, where $a_e^{\mathit{RS}}=\frac{3(1+\theta )(4-{\theta }^2{\tau }^2)}{24+3\theta (4+\tau )-{\theta }^3\tau (4+\tau +{\tau }^2)-{\theta }^2(16-8\tau +7{\tau }^2)}$ and $a_e^{\mathit{MS}}=\frac{3(6+\theta (7-2{\theta }^2(4-\tau )(1-\tau )-7\tau -{\theta }^4(1-\tau ){\tau }^2+2{\theta }^3\tau (4-(2-\tau \left)\tau \right)-2\theta (4+\tau +{\tau }^2)\left)\right)}{42+\theta (18+\theta ((5-7\tau )\tau -76+6{\theta }^3\tau (4-(2-\tau \left)\tau \right)-6\theta (4+\tau +{\tau }^2)+{\theta }^4{\tau }^2(4+\tau +{\tau }^2)+2{\theta }^2(16-{\tau }^3)\left)\right)}$.

Because $\Delta e^{\mathit{MR}}\left(a,=,0\right)=\frac{-\left(2,-,\theta ,\tau \right)\left(2,+,\theta ,\tau \right)\left(1,-,{\theta }^2,\tau \right)\left(1,-,{\theta }^2,\tau ,\left(2,-,\left(2,-,{\theta }^2\right),\tau \right)\right)}{2\left(4,-,{\theta }^2,{\tau }^2\right)\left(7,-,{\theta }^2,\left(8,-,2,\tau ,+,{\theta }^2,{\tau }^2\right)\right)\left(3,-,{\theta }^2,\left(4,-,2,\tau ,+,{\tau }^2\right)\right)}<0$ and $\Delta e^{\mathit{MR}}\left(a,=,\overline{a}\right)=\frac{4+(143-348{\theta }^2+192{\theta }^4)\tau +(103{\theta }^2-92{\theta }^4){\tau }^2+{\theta }^2(-49+51{\theta }^2+4{\theta }^4){\tau }^3}{-2(4-\tau )(4-{\theta }^2{\tau }^2)(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}+\frac{-{\theta }^4(9+5{\theta }^2){\tau }^4+(2{\theta }^4+9{\theta }^6-4{\theta }^8){\tau }^5+{\theta }^6(-2+{\theta }^2){\tau }^6}{-2(4-\tau )(4-{\theta }^2{\tau }^2)(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}<0$, we have $e_n^{M\text{*}}<e_n^{R\text{*}}$.

In summary, for $a\in [\underset{\_}{a},\overline{a}]$, we have $e_n^{R\text{*}}>e_n^{M\text{*}}\ge e_s^{S\text{*}}$ if $a\le a_e^{\mathit{MS}}$, $e_n^{R\text{*}}\ge e_s^{S\text{*}}>e_n^{M\text{*}}$ if $a_e^{\mathit{MS}}<a\le a_e^{\mathit{RS}}$ and $e_s^{S\text{*}}>e_n^{R\text{*}}>e_n^{M\text{*}}$ if $a>a_e^{\mathit{RS}}$.

Proof of Proposition 1 (1) The difference between ${\pi }_r^{R\text{*}}$ and ${\pi }_r^{S\text{*}}$ is

$$\Delta {\pi }_r^{\mathit{RS}}={\pi }_r^{R\text{*}}-{\pi }_r^{S\text{*}}=\frac{(1-{\theta }^2{\tau }^2)(4-{\theta }^2{\tau }^2+a^2(-16+{\theta }^2(16-8\tau +5{\tau }^2)\left)\right)}{16(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}.$$

Clearly, $\Delta {\pi }_r^{\mathit{RS}}$ is concave in $a$. Letting $\Delta {\pi }_r^{\mathit{RS}}=0$ yields $a=\pm \frac{\sqrt{4-{\theta }^2{\tau }^2}}{\sqrt{16-{\theta }^2\left(16,-,8,\tau ,+,5,{\tau }^2\right)}}$. We then have ${\pi }_r^{R\text{*}}\le {\pi }_r^{S\text{*}}$ if $a\ge a_r^{\mathit{RS}}$ and ${\pi }_r^{R\text{*}}>{\pi }_r^{S\text{*}}$ if $a<a_r^{\mathit{RS}}$ for $a\in [\underset{\_}{a},\overline{a}]$, where $a_r^{\mathit{RS}}=\frac{\sqrt{4-{\theta }^2{\tau }^2}}{\sqrt{16-{\theta }^2\left(16,-,8,\tau ,+,5,{\tau }^2\right)}}$.

(2) The difference between ${\pi }_r^{M\text{*}}$ and ${\pi }_r^{S\text{*}}$ is

$$\Delta {\pi }_r^{\mathit{MS}}={\pi }_r^{M\text{*}}-{\pi }_r^{S\text{*}}=\frac{A_1{a}^2+B_{1}a+C_1}{16{(7+2{\theta }^2(-4+\tau )-{\theta }^4{\tau }^2)}^2(3-{\theta }^2(4-2\tau +{\tau }^2))},$$

where $A_1=-196-{\theta }^2\left(-,640,+,280,\tau ,-,49,{\tau }^2\right)-{\theta }^{10}{\tau }^4\left(-,16,+,8,\tau ,-,5,{\tau }^2\right)-{\theta }^4\left(704,-,672,\tau ,+,116,{\tau }^2,-,84,{\tau }^3\right)-4{\theta }^8{\tau }^2\left(-,64,+,48,\tau ,-,24,{\tau }^2,+,5,{\tau }^3\right)-2{\theta }^6\left(-,128,+,192,\tau ,+\right)\left(88,{\tau }^2,-,28,{\tau }^3,+,27,{\tau }^4\right),B_1=-2\theta \left(\left(-,35,+,112,{\theta }^2,-,64,{\theta }^4\right),\tau ,-,4,+,4,{\theta }^2,\left(16,{\theta }^4,-,5,-,14,{\theta }^2\right),{\tau }^2\right)\left(+,\left(28,{\theta }^2,+,18,{\theta }^4,-,48,{\theta }^6\right),{\tau }^3,-,4,{\theta }^4,\left(6,-,9,{\theta }^2,+,{\theta }^4\right),{\tau }^4,+,{\theta }^6,\left({\theta }^2,-,4\right),{\tau }^5\right),$ and $C_1=45-{\theta }^8{\tau }^4(19-8\tau +4{\tau }^2)-4{\theta }^2(28-11\tau +13{\tau }^2)-4{\theta }^6{\tau }^2(12-11\tau +{\tau }^2-2{\tau }^3)-2{\theta }^4(-32+16\tau -39{\tau }^2+8{\tau }^3+2{\tau }^4)$. A contour plot verifies $A_1<0$ in the entire feasible region. Thus, $\Delta {\pi }_r^{\mathit{MS}}$ is concave in $a$.

Letting $\Delta {\pi }_r^{\mathit{MS}}=0$ yields $a=\frac{{4{\theta }^3\tau \left(28,-,5,\tau ,+,7,{\tau }^2\right)-\theta }^9\left(4,-,\tau \right){\tau }^4-\theta \left(4,+,35,\tau \right)+4{\theta }^7{\tau }^2\left(16,-,12,\tau ,+,9,{\tau }^2,-,{\tau }^3\right)-2{\theta }^5\tau \left(32,+,28,\tau ,-,9,{\tau }^2,+,12,{\tau }^3\right)}{A_1}\pm \frac{2(7-2{\theta }^2(4-\tau )-{\theta }^4{\tau }^2)\sqrt{b_1}}{A_1}$, where $b_1=(-3+{\theta }^2(4+(-2+\tau )\tau \left)\right)(-15-2{\theta }^6{\tau }^3(12+\tau +5{\tau }^2)+{\theta }^8{\tau }^4(20+\tau (-8+5\tau ))+2{\theta }^2(16+\tau (-9+7\tau ))+{\theta }^4(-16+\tau (16+\tau (-11+\tau (12+5\tau )\left)\right)\left)\right)$. The contour plot uniquely illustrates that the smaller root is lower than $\underset{\_}{a}$. We then have ${\pi }_r^{M\text{*}}\le {\pi }_r^{S\text{*}}$ if $a\ge a_r^{\mathit{MS}}$ and ${\pi }_r^{M\text{*}}>{\pi }_r^{S\text{*}}$ if $a<a_r^{\mathit{MS}}$ for $a\in [\underset{\_}{a},\overline{a}]$, where $a_r^{\mathit{MS}}=\frac{{4{\theta }^3\tau \left(28,-,5,\tau ,+,7,{\tau }^2\right)-\theta }^9\left(4,-,\tau \right){\tau }^4-\theta \left(4,+,35,\tau \right)+4{\theta }^7{\tau }^2\left(16,-,12,\tau ,+,9,{\tau }^2,-,{\tau }^3\right)-2{\theta }^5\tau \left(32,+,28,\tau ,-,9,{\tau }^2,+,12,{\tau }^3\right)}{A_1}-\frac{2(7-2{\theta }^2(4-\tau )-{\theta }^4{\tau }^2)\sqrt{b_1}}{A_1}.$

(3) The difference between ${\pi }_r^{M\text{*}}$ and ${\pi }_r^{R\text{*}}$ is

$$\Delta {\pi }_r^{\mathit{MR}}={\pi }_r^{M\text{*}}-{\pi }_r^{R\text{*}}=\frac{A_2{a}^2+B_{2}a+C_2}{16(4-{\theta }^2{\tau }^2){(7+2{\theta }^2(-4+\tau )-{\theta }^4{\tau }^2)}^2(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)},$$

where $A_2={\theta }^2(-16-8(35-104{\theta }^2+64{\theta }^4)\tau +(-637+2096{\theta }^2-2976{\theta }^4+1536{\theta }^6){\tau }^2-4{\theta }^2(91-356{\theta }^2+304{\theta }^4){\tau }^3+2{\theta }^2(98-337{\theta }^2+312{\theta }^4+24{\theta }^6){\tau }^4-4{\theta }^4(-14+33{\theta }^2+6{\theta }^4){\tau }^5+{\theta }^6(4+15{\theta }^2\left){\tau }^6\right)$, $B_2=2\theta (16+4(35-112{\theta }^2+64{\theta }^4)\tau +4{\theta }^2(19+56{\theta }^2-64{\theta }^4){\tau }^2+{\theta }^2(-147+40{\theta }^2+128{\theta }^4){\tau }^3+4{\theta }^4(19-50{\theta }^2+20{\theta }^4){\tau }^4+(28{\theta }^4+34{\theta }^6-52{\theta }^8){\tau }^5-4{\theta }^6(6-9{\theta }^2+{\theta }^4){\tau }^6+{\theta }^8(-4+{\theta }^2\left){\tau }^7\right)$ and $C_2=-{(1-{\theta }^2\tau )}^2(16+8{\theta }^2(-4+\tau )\tau +{\theta }^8{\tau }^6-2{\theta }^6{\tau }^4(4-3\tau +2{\tau }^2)+{\theta }^4{\tau }^2(16-16\tau +13{\tau }^2\left)\right))$. Similarly, the contour plot shows $A_2<0$ in the entire feasible region, which means that $\Delta {\pi }_r^{\mathit{MR}}$ is concave in $a$.

Letting $\Delta {\pi }_r^{\mathit{MR}}=0$ yields $a=\frac{-\theta (2-\theta \tau )(2+\theta \tau )(1-{\theta }^2\tau )(4+\tau (35-{\theta }^6(4-\tau ){\tau }^2-{\theta }^2(108-(55-28\tau \left)\tau \right)+{\theta }^4(64+\tau ((37-4\tau )\tau -52\left)\right)\left)\right)}{A_2}\pm \frac{4\theta \tau (1-{\theta }^2\tau )(7-{\theta }^2(8-\tau (2-{\theta }^2\tau )\left)\right)\sqrt{b_2}}{A_2}$, where $b_2=(2-\theta \tau )(2+\theta \tau )(-1+2\theta +(-2+\theta \left)\theta ,\tau \right)(-1-2\theta +\theta (2+\theta \left)\tau \right)(3-{\theta }^2(4+(-2+\tau )\tau \left)\right)$. We denote the smaller and larger roots as $a_r^{MR1}$ and $a_r^{MR2}$, respectively. Then, for $a\in [\underset{\_}{a},\overline{a}]$, when $a_r^{MR1}\le a\le a_r^{MR2}$, we have ${\pi }_r^{M\text{*}}\ge {\pi }_r^{R\text{*}}$; otherwise, we have ${\pi }_r^{M\text{*}}<{\pi }_r^{R\text{*}}$. In addition, through contour plots, we find that for $\theta =a_r^{MR{1}^{-1}}(a,\tau )$, there exist two roots, that is, ${\theta }_{MR1}^1$>${\theta }_{MR1}^2$. Meanwhile, for $\theta =a_r^{MR{2}^{-1}}(a,\tau )$, there exists only one root ${\theta }_{MR2}$. Because $a_r^{MR1}$ and $a_r^{MR2}$ are the two roots for $\Delta {\pi }_r^{\mathit{MR}}=0$ and $a_r^{MR1}<a_r^{MR2}$, ${\theta }_{MR1}^2$ and ${\theta }_{MR2}$ cannot exist simultaneously in the common feasible area. We denote ${\theta }_{MR1}^1$ by ${\theta }_1$ and ${\theta }_{MR1}^2$ or ${\theta }_{MR2}$ by ${\theta }_2$. Then, the condition $a_r^{MR1}\le a\le a_r^{MR2}$ can be transformed into ${\theta }_1\le \theta \le {\theta }_2$. Therefore, we have ${\pi }_r^{M\text{*}}\ge {\pi }_r^{R\text{*}}$ if ${\theta }_1\le \theta \le {\theta }_2$ and ${\pi }_r^{M\text{*}}<{\pi }_r^{R\text{*}}$ if $\theta <{\theta }_1$$\text{or}\theta$ $>{\theta }_2$.

In summary, for $a\in [\underset{\_}{a},\overline{a}]$, when $a\ge \text{max}(a_r^{\mathit{RS}},a_r^{\mathit{MS}})$, Strategy S is the retailer’s optimal service strategy. When $a<\text{max}(a_r^{\mathit{RS}},a_r^{\mathit{MS}})$, Strategy M is the optimal service strategy if ${\theta }_1\le \theta \le {\theta }_2$, and Strategy R the optimal service strategy otherwise.

Proof of Proposition 2 Taking the differences, we have

$${{\pi }_m^R}^{\text{*}}-{{\pi }_m^S}^{\text{*}}=\frac{(1-{\theta }^2{\tau }^2)E_1}{24(4-{\theta }^2{\tau }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)},$$

$${{\pi }_m^M}^{\text{*}}-{{\pi }_m^S}^{\text{*}}=\frac{E_2}{24(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)},$$

$${{\pi }_m^M}^{\text{*}}-{{\pi }_m^R}^{\text{*}}=\frac{(1-{\theta }^2\tau )E_3}{8(4-{\theta }^2{\tau }^2)(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)},$$

where $E_1=3(4-{\theta }^2{\tau }^2)-a^2{\theta }^2(4-\tau {)}^2$,$E_2={\theta }^2\left(-,40,+,56,\tau ,-,7,{\tau }^2,+,{\theta }^4,{\left(-,4,+,\tau \right)}^2,{\tau }^2,-,2,{\theta }^2\right)\left(\left(-,16,+,24,\tau ,+,{\tau }^3\right)\right)a^2+6\theta (4-7\tau +(6{\theta }^2-4{\theta }^4){\tau }^2+{\theta }^4{\tau }^3)a+3(3+3{\theta }^4{\tau }^2-2{\theta }^2(4-5\tau +4{\tau }^2\left)\right)$, and $E_3=-{\theta }^2\left(16,+,8,\left(-,7,+,6,{\theta }^2\right),\tau ,+,\left(7,-,24,{\theta }^2\right),{\tau }^2,+,9,{\theta }^2,{\tau }^3\right)a^2+2\theta \left(16,+,4,\left(-,7,+,4,{\theta }^2\right),\tau ,-,8,{\theta }^2,{\tau }^2,+,\left(7,{\theta }^2,-,4,{\theta }^4\right),{\tau }^3,+,{\theta }^4,{\tau }^4\right)a-16+16{\theta }^2\tau +{\theta }^4{\tau }^4-{\theta }^6{\tau }^5$.

Clearly, $E_1$ is a concave function of $a$. Because $E_1(a=0)=3(4-{\theta }^2{\tau }^2)>0$ and $E_1(a=\overline{a})=3(1-{\theta }^2{\tau }^2)>0$, we have $E_1>0$ in the common feasible domain, which indicates that ${{\pi }_m^R}^{\text{*}}>{{\pi }_m^S}^{\text{*}}$.

We denote $\frac{{\partial }^2{E}_2}{\partial a^2}=2{\theta }^2\left(-,40,+,56,\tau ,-,7,{\tau }^2,+,{\theta }^4,{\left(-,4,+,\tau \right)}^2,{\tau }^2,-,2,{\theta }^2,\left(-,16,+,24,\tau ,+,{\tau }^3\right)\right)$ as $F_1$. Letting $E_2=0$ yields $a_{E2}^1=\frac{-3\theta \left(1,-,{\theta }^2,\tau \right)\left(4,-,\left(7,-,{\theta }^2,\left(4,-,\tau \right)\right),\tau \right)+2\theta (1-\tau )\sqrt{6(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}}{F_1}$ and $a_{E2}^2=\frac{-3\theta (1-{\theta }^2\tau )(4-(7-{\theta }^2(4-\tau )\left)\tau \right)-2\theta (1-\tau )\sqrt{6(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}}{F_1}$. When $F_1<0$, $E_2$ is concave in $a$ and $a_{E2}^1<a_{E2}^2$. Via contour plotting, we obtain $a_{E2}^1<\underset{\_}{a}$ and $a_{E2}^2>\overline{a}$. Thus, we have $E_2>0$. When $F_1>0$, $E_2$ is convex in $a$ and $a_{E2}^1>a_{E2}^2$. In this case, contour plots show that $a_{E2}^1>\overline{a}$ and $a_{E2}^2>\overline{a}$. Overall, we have $E_2>0$ in the entire feasible domain, which means that ${{\pi }_m^M}^{\text{*}}>{{\pi }_m^S}^{\text{*}}$. For a similar reason, we have ${{\pi }_m^M}^{\text{*}}>{{\pi }_m^R}^{\text{*}}$. In summary, we have ${{\pi }_m^R}^{\text{*}}>{{\pi }_m^M}^{\text{*}}>{{\pi }_m^S}^{\text{*}}$ for $a\in [\underset{\_}{a},\overline{a}]$.

Proof of Proposition 3 We conduct the profit comparisons in the common feasible domain for the five strategies, that is, $\theta <0.75$ and ${\underset{\_}{a}}_1=\frac{6\theta }{9-8{\theta }^2}\le a\le \frac{3}{4\theta }={\overline{a}}_1$. Taking the differences, we have

$${\pi }_r^{SR\ast }-{\pi }_r^{SM\ast }=\frac{{\left(3,-,4,a,\theta \right)}^2\left(1,+,8,{\theta }^2,-,7,{\theta }^4\right)}{4{\left(7,-,6,{\theta }^2\right)}^2\left(27,-,84,{\theta }^2,+,64,{\theta }^4\right)}\ge 0,$$

$${\pi }_r^{SR\ast }-{\pi }_r^{R\ast }=\frac{H_1{a}^2+H_{2}a+H_3}{16(4-{\theta }^2{\tau }^2)(9-16{\theta }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)},$$

$${\pi }_r^{SR\ast }-{\pi }_r^{M\ast }=\frac{I_1{a}^2+I_{2}a+I_3}{4(9-16{\theta }^2){(7-{\theta }^2(8-2\tau +{\theta }^2{\tau }^2\left)\right)}^2},$$

$${\pi }_r^{SR\text{*}}-{\pi }_r^{S\text{*}}=\frac{{\theta }^2\left(16,+,24,\tau ,-,\left(3,+,16,{\theta }^2\right),{\tau }^2\right)a^2-2\theta \left(12,+,\left(9,+,16,{\theta }^2\right),\tau ,-,16,{\theta }^2,{\tau }^2\right)a+9+24{\theta }^2\tau -12{\theta }^2{\tau }^2}{16\left(9,-,16,{\theta }^2\right)\left(3,-,{\theta }^2,\left(4,-,2,\tau ,+,{\tau }^2\right)\right)},$$

where $H_1=144-64{\theta }^6{\tau }^2\left(4,-,2,\tau ,+,{\tau }^2\right)-3{\theta }^2\left(112,-,56,\tau ,+,67,{\tau }^2\right)+16{\theta }^4(16-8\tau +25{\tau }^2-6{\tau }^3+3{\tau }^4)$, $H_2=-2\theta (48+4(9+16{\theta }^2)\tau -76{\theta }^2{\tau }^2-{\theta }^2(9+16{\theta }^2){\tau }^3+16{\theta }^4{\tau }^4)$, $H_3={\theta }^2(64+96\tau -4\left(3,+,20,{\theta }^2\right){\tau }^2-24{\theta }^2{\tau }^3+{\theta }^2(3+16{\theta }^2\left){\tau }^4\right)$, $I_1=147-30{\theta }^2(16-7\tau )-16{\theta }^{10}{\tau }^4-4{\theta }^8{\tau }^2(8+\tau )(8-3\tau )+{\theta }^4(592-\tau (464+129\tau \left)\right)-16{\theta }^6(16-\tau (16+\tau (23-3\tau )\left)\right)$, $I_2=-2\theta (52+63\tau +{\theta }^2(-48+\tau (-72+\tau (-54-64{\theta }^4+{\theta }^2(128-9\tau )+4{\theta }^6{\tau }^2\left)\right)\left)\right)$, and $I_3=3+19{\theta }^8{\tau }^4+{\theta }^2\left(64,+,84,\tau ,-,9,{\tau }^2\right)+{\theta }^6{\tau }^2\left(48,-,\tau ,\left(44,+,9,\tau \right)\right)-2{\theta }^4(32+\tau (48+(7-9\tau )\tau \left)\right)$.

By contour plotting, we find that $H_1>0$ and $I_1>0$, which means that ${\pi }_r^{SR\text{*}}-{\pi }_r^{R\text{*}}$ and ${\pi }_r^{SR\text{*}}-{\pi }_r^{M\text{*}}$ are convex in $a$. Moreover, contour plots show that ${H_2}^2-4H_1{H}_3<0$ and ${I_2}^2-4I_1{I}_3<0$. Thus, there exists no root for ${\pi }_r^{SR\text{*}}-{\pi }_r^{R\text{*}}=0$ and for ${\pi }_r^{SR\text{*}}-{\pi }_r^{M\text{*}}=0$. Finally, we have ${\pi }_r^{SR\text{*}}>{\pi }_r^{R\text{*}}$ and ${\pi }_r^{SR\text{*}}>{\pi }_r^{M\text{*}}$ in the common feasible domain.

As we can see, ${\pi }_r^{SR\text{*}}-{\pi }_r^{S\text{*}}$ is a convex function of $a$. Letting ${\pi }_r^{SR\text{*}}={\pi }_r^{S\text{*}}$, we have $a_r^{\mathit{SRS}}=\frac{16{\theta }^2\left(1,-,\tau \right)\tau +3\left(4,+,3,\tau \right)-2\tau \sqrt{(9-16{\theta }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}}{\theta \left(16,+,24,\tau ,-,\left(3,+,16,{\theta }^2\right),{\tau }^2\right)}$ and $a_r^{SRS1}=\frac{16{\theta }^2\left(1,-,\tau \right)\tau +3\left(4,+,3,\tau \right)+2\tau \sqrt{(9-16{\theta }^2)(3-{\theta }^2(4-2\tau +{\tau }^2\left)\right)}}{\theta \left(16,+,24,\tau ,-,\left(3,+,16,{\theta }^2\right),{\tau }^2\right)}$. Since $a_r^{SRS1}>{\overline{a}}_1$, we have ${\pi }_r^{SR\text{*}}\le {\pi }_r^{S\text{*}}$ when $a\ge a_r^{\mathit{SRS}}$ and ${\pi }_r^{SR\text{*}}>{\pi }_r^{S\text{*}}$ otherwise.

In summary, for $\theta <0.75$ and $a\in [{\underset{\_}{a}}_1,{\overline{a}}_1]$, Strategy S is the retailer’s optimal service strategy if $a\ge a_r^{\mathit{SRS}}$, and Strategy SR is optimal otherwise.

Proof of Proposition 4 Taking the differences, we have

$${\pi }_m^{SR\ast }-{\pi }_m^{SM\ast }=\frac{{\left(3,-,4,a,\theta \right)}^2\left(9,-,18,{\theta }^2,+,10,{\theta }^4\right)}{6{\left(3,-,4,{\theta }^2\right)}^2\left(63,+,96,{\theta }^4,-,166,{\theta }^2\right)}\ge 0,$$

$${\pi }_m^{SR\ast }-{\pi }_m^{R\ast }=\frac{a\left(\sqrt{2},\theta ,\left(4,-,\tau \right),J_1,-,16,\theta ,J_2,J_3\right)+\sqrt{2}{\theta }^2{\tau }^2{J}_1-4\sqrt{2}{J}_1+12J_2{J}_3}{4J_1{J}_2{J}_3},$$

where $J_1=\sqrt{54-96{\theta }^2},J_2=\sqrt{4-{\theta }^2{\tau }^2},$and$J_3=\sqrt{3-{\theta }^2(4-(2-\tau \left)\tau \right)}$. One can see that ${\pi }_m^{SR\text{*}}-{\pi }_m^{R\text{*}}$ is decreasing in $a$. Letting ${\pi }_m^{SR\text{*}}={\pi }_m^{R\text{*}}$ yields $a_m^{\mathit{SRR}}=\frac{12J_2{J}_3-\sqrt{2}(4-{\theta }^2{\tau }^2)J_1}{\theta (16J_2{J}_3-\sqrt{2}(4-\tau \left)J_1\right)}$. Combing the results in Proposition 2, we see that ${\pi }_m^{SR\text{*}}$ is the highest if $a\le a_m^{\mathit{SRR}}$, and otherwise ${\pi }_m^{R\text{*}}$ is the highest in the common feasible domain.

Appendix B

Consider that the service provider sets the service level first, then the manufacturer determines the wholesale price, and finally the retailer determines the retail prices. Similar to the main model, we use backward induction to solve the equilibrium outcomes. To avoid lengthiness, we omit the solution process. Table 4 gives the equilibrium profits of the two firms for each strategy, where we add the superscript 1 to denote this game sequence. It is worth noting that the equilibrium outcomes in Strategy M do not change under this new game sequence.

Table 4.

Equilibrium outcomes in the case of different game sequence

	${\pi }_r^{i1\ast }$	${\pi }_m^{i1\ast }$
Strategy S	$\frac{3-2a\theta (4-\mathrm{\tau })+a^2(16-{\theta }^2(12+{\mathrm{\tau }}^2\left)\right)}{48-4{\theta }^2(13-(2-\tau \left)\tau \right)}$	$\frac{2(1-{\theta }^2){(3-a\theta (4-\tau \left)\right)}^2}{{(12-{\theta }^2(13-(2-\tau )\tau \left)\right)}^2}$
Strategy R	$\frac{4+3a^2(5-4{\theta }^2)+2a\theta (-4+\tau )-{\theta }^2{\tau }^2}{4(15-{\theta }^2(16-\tau (2-(4-3{\theta }^2\left)\tau \right)\left)\right)}$	$\frac{2(1-{\theta }^2){(4-a\theta (4-\tau )-{\theta }^2{\tau }^2)}^2}{{(15-{\theta }^2(16-\tau (2-(4-3{\theta }^2\left)\tau \right)\left)\right)}^2}$
Strategy SR	$\frac{3-8a\theta -3a^2(5-4{\theta }^2)}{45-52{\theta }^2}$	$\frac{32{(3-4a\theta )}^2(1-{\theta }^2)}{{(45-52{\theta }^2)}^2}$
Strategy SM	$\frac{36-39{\theta }^2-8a\theta (12-13{\theta }^2)}{{(21-26{\theta }^2)}^2}+\frac{3a^2(49-100{\theta }^2+52{\theta }^4)}{{(21-26{\theta }^2)}^2}$	$\frac{{(3-4a\theta )}^2(7-8{\theta }^2)}{{(21-26{\theta }^2)}^2}$

The profit comparisons between different game sequences are as follows:

$${\pi }_r^{S\text{*}}-{\pi }_r^{S1\text{*}}=\frac{{\theta }^2{\left(3,-,a,\theta ,\left(4,-,\tau \right)\right)}^2{\left(1,-,\tau \right)}^2}{16\left(3,-,{\theta }^2,\left(4,-,\left(2,-,\tau \right),\tau \right)\right)(12-{\theta }^2(13-(2-\tau )\tau \left)\right)}\ge 0,$$

$${\pi }_m^{S\text{*}}-{\pi }_m^{S1\text{*}}=\frac{{\theta }^2{\left(3,-,a,\theta ,\left(4,-,\tau \right)\right)}^2{\left(1,-,\tau \right)}^2\left(24,-,{\theta }^2,\left(23,-,\left(2,-,\tau \right),\tau \right)\right)}{24\left(3,-,{\theta }^2,\left(4,-,\left(2,-,\tau \right),\tau \right)\right){\left(12,-,{\theta }^2,\left(13,-,\left(2,-,\tau \right),\tau \right)\right)}^2}\ge 0,$$

$${\pi }_r^{R\text{*}}-{\pi }_r^{R1\text{*}}=\frac{3{(1-{\theta }^2\tau )}^2{(4-a\theta (4-\tau )-{\theta }^2{\tau }^2)}^2}{16(4-{\theta }^2{\tau }^2)\left(3,-,{\theta }^2,\left(4,-,\left(2,-,\tau \right),\tau \right)\right)(15-{\theta }^2(16-\tau (2-(4-3{\theta }^2\left)\tau \right)\left)\right)}\ge 0,$$

$${\pi }_m^{R\text{*}}-{\pi }_m^{R1\text{*}}=\frac{{(1-{\theta }^2\tau )}^2{(4-a\theta (4-\tau )-{\theta }^2{\tau }^2)}^2(33+9{\theta }^4{\tau }^2-2{\theta }^2(16+\tau +4{\tau }^2\left)\right)}{8(4-{\theta }^2{\tau }^2)\left(3,-,{\theta }^2,\left(4,-,\left(2,-,\tau \right),\tau \right)\right){(15-{\theta }^2(16-\tau (2-(4-3{\theta }^2\left)\tau \right)\left)\right)}^2}\ge 0,$$

$${\pi }_r^{SR\text{*}}-{\pi }_r^{SR1\text{*}}=\frac{{(3-4a\theta )}^2(3+4{\theta }^2)}{4(-9+16{\theta }^2)(-45+52{\theta }^2)}\ge 0,$$

$${\pi }_m^{SR\text{*}}-{\pi }_m^{SR1\text{*}}=\frac{{(3-4a\theta )}^2(3+4{\theta }^2)(-99+92{\theta }^2)}{6{(45-52{\theta }^2)}^2(-9+16{\theta }^2)}\ge 0,$$

$${\pi }_r^{SM\text{*}}-{\pi }_r^{SM1\text{*}}=\frac{{\theta }^2{(-3+4a\theta )}^2(-364+1205{\theta }^2-1316{\theta }^4+468{\theta }^6)}{4(-3+4{\theta }^2){(147-308{\theta }^2+156{\theta }^4)}^2}\ge 0,$$

$${\pi }_m^{SM\text{*}}-{\pi }_m^{SM1\text{*}}=\frac{{\theta }^2{(3-4a\theta )}^2(-84+{\theta }^2(267+92{\theta }^2(-3+{\theta }^2)\left)\right)}{(-7+6{\theta }^2){(63-162{\theta }^2+104{\theta }^4)}^2}\ge 0.$$

In conclusion, both firms prefer that the wholesale price is determined before the service level(s).

Declarations

Conflict of interest

Ping Xie declares that she has no conflict of interest. Ruixia Shi declares that she has no conflict of interest. Di Xu declares that he has no conflict of interest.

Human and animal rights statement

This article does not contain any studies with human participants or animals performed by any of the authors.