Manufacturer’s optimal distribution strategy in the platform supply chain: Bundling or add-on?

Abstract

This study delves into the realm of distribution strategies employed in retail markets, particularly focusing on the widely utilized bundle-and-add-on strategy. Three distinct distribution strategies are examined: bundled-by-the-base-manufacturer (BBBM), bundled-by-the-platform, and the add-on approach within a platform supply chain context. Through comprehensive analysis, this paper investigates the optimal distribution strategy for manufacturers. Significantly, our research reveals that in cases where only bundling is feasible, base manufacturers can reap benefits from a self-bundling strategy when both the platform’s commission rate and the marginal cost of the bundled product are low. Additionally, the platform stands to gain from this approach when the commission rate is low, and the marginal cost of the bundled product is either moderate or very high. Notably, win–win scenarios can emerge for both manufacturers and platforms through specific bundling or platform bundling strategies under specific conditions. Furthermore, the study demonstrates that the price of bundles under the add-on strategy will be lower, attracting more consumers, especially when the commission rate is low. Intriguingly, when the marginal cost of the base product is low, the add-on strategy consistently emerges as the optimal choice; as costs escalate, BBBM might be optimal, especially when the commission rate is low, showcasing a nuanced understanding of the interplay between costs and strategic choices in distribution. The major contribution of this paper lies in its nuanced analysis of these strategies and their implications for both manufacturers and platforms in the retail landscape.

1 Introduction

Unlike bundling, where two or more individual goods are sold in a single package, add-on sales separate the base product from the add-on product. Firms can impose price discrimination on consumers because different consumers value the base product from the add-on product differently. Taking Apple’s earphones (i.e., iPod) as an example, some consumers who prefer the quality or brand of iPods will buy the earphones even if they do not buy Apple’s mobile phones. This phenomenon is also common in the hotel industry. For example, some hotels have begun to sell their rooms and breakfast services separately. In this way, the hotel can attract more consumers who want to enjoy breakfast. Thus, firms face a dilemma when they must give up extra profits from selling add-on products in a bundle. In the retail market, some firms adopt bundling, while others adopt add-on sales. For example, car dealers offer additional extended warranty programs to customers who buy new cars. The hotel sells rooms and charges additional services (Yin et al., 2021). In the example above, consumers can buy basic products and add-ons separately depending on the firm’s sales policy. When the firm adopts an add-on strategy, the firm can impose price discrimination on consumers. However, the firm also needs to face the potential risk that consumers may value the base product more than that of the add-on, and thus these consumers may refuse to purchase the add-on. Moreover, when the firm adopts a bundle strategy, that is, consumers have to buy the base product and the add-on together, it is intuitive that both the add-on manufacturer and the base manufacturer can profit from such a strategy. However, the firms cannot impose price discrimination on consumers to earn a higher profit. Thus, it is critical for firms to choose an optimal pricing strategy for the base product and its add-on.

Another phenomenon worth paying attention to is the increasing participation of e-commerce platforms in bundling. For example, e-commerce platforms such as JD.com sell home appliances but also offer insurance and sell smartphones, protective cases, membranes, and even communication network services. However, some e-commerce platforms explicitly oppose bundling. For example, Kuaishou, an emerging Chinese social e-commerce platform, recently banned “bundling” and announced the termination of third-party sellers who violated rules.¹ This clearly illustrates that different participants (platforms, base manufacturers, and add-on manufacturers) have different incentives for bundling or add-on strategies under the platform supply chain model. Specifically, when the bundle is provided by the platform, the base manufacturer and add-on manufacturer need to wholesale the products to the platform first, and then the platform sets the price for bundles. In this case, the bundle suffers a triple markup, and the double marginalization problem becomes more significant. However, when the base manufacturer provides the bundle, that is, the add-on manufacturer wholesales its product to the base manufacturer at first, and then the base manufacturer makes the pricing decision on the bundle. In this case, the bundle only suffers from double markup, but the base manufacturer transfers the commission to the bundle’s price. Thus, the commission rate charged by the platform plays a critical role in the decision of bundle strategy.

The above motivation urges us to study a firm’s bundling or add-on strategy in the platform supply chain context. The online shopping market has experienced incredible growth in recent years, for example, from 108 million online shoppers in 2009 to 782.41 million in 2020 (Marketing China, 2022). The record growth of e-commerce platforms has become even more significant in the wake of the COVID-19 pandemic. It is estimated that firms can increase sales by 20 percent by selling Chinese e-commerce platforms (Marketing China, 2022). Therefore, it is critical to investigate how firms make product operational decisions on e-commerce platforms. The platform-operating model has two key features associated with bundled or add-on sales. First, bundling can be performed by different entities, either by the base product manufacturer or platform. For example, JD.com sells mobile phones and accessories (such as protective cases) in its self-run stores. By contrast, mobile phone manufacturers (such as Apple) sell mobile phone accessories in their third-party stores. However, the manufacturer uses a bundle or sells on the platform separately, and they have to pay the platform a percentage of the commission. Our study captures these two key features by considering a specific structural and pricing approach to contribute to the literature on bundling and platform supply chains.

Based on the above discussion, in the platform supply chain considering bundling and add-on selling respectively, the platform can adopt the following strategies: when bundling is considered, (1) the manufacturer will implement bundling, and sell the two products at the bundling price, and the platform will charge the commission according to the proportion of sales (“Case M”); (2) The platform itself implements bundling, that is, the platform provides bundling product and sets the bundling price (“Case E”); When bundling is not considered (i.e., add-on selling), (3) the manufacturers of the two products sell their products directly through the platform respectively, consumers buy one or two products at a price set by the two manufacturers respectively, and the platform charges commission according to the percentage of sales (“Case A”). The following issues may arise in this case: (i) Who is better to undertake a bundling strategy between the platform and the manufacturer? (ii) How do the platform and manufacturer decide on sales models and pricing strategies when considering add-on strategy? and (iii) what is the strategic impact of bundling or add-on strategy on the platform, manufacturer, and supply chain?

To solve these problems, we developed a game-theoretical model consisting of two manufacturers selling base products and add-on products through a common platform respectively. As a benchmark, we first analyze the bundling model of a firm’s decision-making, and then investigate how firms make decisions under the add-on strategy. Finally, we analyze two extensions of correlated valuations to check the robustness of our main model.

Accordingly, the main contributions of this research are given as follows:

• This study challenges previous findings by revealing that the platform's commission rate significantly influences optimal pricing strategies in a platform supply chain, emphasizing the need to consider platform fees in pricing decisions.

• This study explores how different entities within the supply chain are affected by bundling strategies, offering valuable insights into pricing decisions and profits for supply chain participants, a factor often overlooked in prior studies.

• Contrary to earlier beliefs, the study finds that the add-on pricing strategy is optimal only when the base product’s cost is low. This nuanced understanding guides base manufacturers in choosing suitable strategies for selling base products and add-ons in a platform supply chain context.

• This study not only contributes theoretical insights but also provides actionable recommendations, helping businesses make informed decisions about their pricing and distribution strategies in platform supply chains.

The remainder of this paper is structured as follows. Section 2 reviews literature relevant to our study. Section 3 proposes the game-theoretical model and presents some assumptions. Section 4 considers three possible distribution strategies and derives equilibrium outcomes. Section 5 explores the firm’s optimal distribution strategy. Section 6 presents some numerical experiments to reveal the impact of relevant factors on the profitability of all firms under different distribution strategies. Section 7 presents extensions to verify the robustness of our basic model. Section 8 presents the conclusions and future research directions of this study.

2 Literature review

This study aims to explore the base manufacturer’s optimal product distribution strategy in a platform supply chain. Thus, our work is related to three streams of literature: (i) the selling mode in the platform supply chain, (ii) the bundling strategy, and (iii) the add-on strategy. This section reviews related studies and describes the research gaps.

2.1 Selling mode in the platform supply chain

The tremendous growth of e-commerce enforces the development of platform supply chains, and thus, an increasing number of scholars are paying attention to the firm’s choice of platform selling mode. Hagiu and Wright (2015) found that firms prefer the reselling model when market activities lead to spillover effects between products and network effects lead to negative results. Liu et al. (2020) studied the platform’s optimal selling mode when data-driven marketing (DDM) is considered. The results show that as DDM efficiency increases, the platform is more likely to choose the reselling model. Niu et al. (2021) investigated the interactions between firms’ strategies for introducing Internet-based installments and the platform’s selling mode in a competing market environment. They found that when incremental demand is moderate, the platform prefers to provide services, while the supplier will refuse to provide such services. Wang et al. (2021) investigated a platform retailer’s optimal selling strategy when consumer returns are considered. The results show that when the sales efficiency of the channels is different or close enough, the platform is more inclined to adopt the reselling strategy than the agency strategy. Xu and Choi (2021) analyzed the influence of the selling mode on firms’ strategy of adopting blockchain technology when cap-and-trade regulations are considered. They showed that the platform in the agency mode is always better off from blockchain technology, whereas the platform in the reselling mode might be worse off from such technology. Li et al. (2021) revealed that the manufacturer will be more willing to choose the agency mode after the platform introduces a private brand. Tian et al. (2018) and Shi et al. (2021) analyzed the platform’s choice of selling mode facing upstream or downstream competition, they showed that the platform’s choice depends on the commission rate and competition between two suppliers. In the closed-loop supply chain context, Ma and Hu (2022) found that only when the commission rate is very low or high, the supply chain members can reach a stable platform sales partnership. In the context of the platform supply chain, Zhang et al. (2022) found that the selection of a sales model is highly dependent on the inherent attributes of the product, and both the commission rate and slotting fee are also important moderators. In contrast to the above studies, this study focuses on the firm’s choice of selling mode in the platform supply chain, which has rarely been studied in previous literature.

2.2 Bundling strategy

In this subsection, we focus on literature related to the bundling strategy. Some studies explore firms’ optimal product-bundling strategies in the monopoly market. Giri et al. (2013) studied the impact of the bundling strategy on supply chain operations when reservation prices follow discrete and uniform continuous distributions. They found that the retailer can be better off when the manufacturer adopts a pure bundling strategy and the product’s cost is low. Banciu and Ødegaard (2016) investigated the optimal bundling strategy with dependent valuations of the bundle components, they found that the seller can benefit from bundles when the product’s marginal cost is relatively low. In a multi-echelon supply chain context, Eghbali-Zarch et al. (2019) found that bundling a basic good with a premium can benefit from the bundling strategy when the market size, product cost, and amount of required raw materials are small. Banciu et al. (2010) explored a seller’s bundling strategy with vertically differentiated products and limited resources. The results showed that the optimal strategy is affected by the resources’ absolute and relative availabilities and the product’s quality. Honhon and Pan (2017) examined a firm offering vertically differentiated products and analyze its optimal pricing decisions under different bundling strategies. Their results showed that adopting a bundling strategy can improve profits by increasing sales.

Moreover, some scholars focus on firms’ bundling strategies in the duopoly market. Lin et al. (2020) focus on two competing e-commerce platforms and study their optimal pricing and product-bundling strategies. The result shows that the mixed product-bundling strategy is the equilibrium. Jena and Ghadge (2020) studied the interactions between product bundling and advertising strategies in a competitive environment with different power structures. They find that the supply chain can obtain the highest profit from the manufacturer’s bundling strategy, which significantly impacts advertising efforts. Chen et al. (2019, 2020) study how product interrelatedness affects firms’ choice of bundling strategies. They find that the manufacturer benefits from a partially mixed bundling strategy when the products are substitutable with relatively high substitutability. They also found that the manufacturer prefers a pure bundling strategy when the products are complementary. Giri et al. (2017) found that adopting the bundling strategy is always optimal in a supply chain consisting of two suppliers that produce two complementary products. In contrast to previous studies that consider the influencing mechanism of the bundling strategy on firms’ profits, our study is more concerned with who is more appropriate to provide the bundling strategy, that is, BBBM or BBP. Meanwhile, when the base manufacturer adopts different bundling strategies, the supply chain members adjust their pricing decisions, accordingly influencing firms’ profits. However, the literature rarely examined the bundling strategy with different undertakers on all participants’ pricing decisions and profits.

2.3 Add-on strategy

As this study aims to reveal the differences between bundling and add-on strategies, we also review studies on the add-on strategy. Yin et al. (2021) investigated two product selling strategies, add-on, and bundling, and further explored the retailer’s optimal strategy. Their results showed that when firms are integrated, the retailer prefers the add-on strategy if the base product cost is low and the add-on cost is high, and the retailer prefers the add-on strategy if firms are independent. Shulman and Geng (2013) explored firms’ add-on pricing decisions when products are vertically differentiated and partial consumers are unaware of the initial purchase add-on fees. They found that, when quality asymmetry exists only in the underlying goods and consumers are bounded rationally, a high-quality firm will be worse off under the add-on strategy, while a low-quality firm will be better off under such a strategy. Geng and Shulman (2015) investigated the impact of cost-saving and heterogeneous consumer price sensitivity on a firm’s add-on pricing decision. Their results showed that when consumers are heterogeneous in price sensitivity, the positive effect of the add-on strategy caused by cost savings may diminish. Thus, the firm will suffer from the add-on strategy. Geng et al. (2018) investigated the interactions between add-on pricing and an online platform’s choice of selling mode. The results showed that the firm prefers the bundling strategy under the reseller mode and the separate selling strategy under the agency mode. Lin (2017) considered firms’ add-on strategies with vertically differentiated products in different industries. The results indicated that a high-quality firm is more likely to sell the add-on as an alternative choice, but a low-quality firm prefers to sell the add-on to all consumers when the add-on’s cost-to-value ratio is very low. Song and Li (2018) studied a firm’s dynamic pricing decision for basic services when the add-on is sold separately. They found that the unbundling value increases the consumer’s degree of dependence on the subscription price of the basic service and the add-on service.

Moreover, our study is similar to Yin et al. (2021) and Geng et al. (2018). However, there are some differences between our study and the above literature. First, Yin et al. (2021) studied the retailer’s optimal pricing strategy in different scenarios that differ by whether the add-on manufacturer or base manufacturer is independent of the retailer or not. However, we focus on the manufacturer’s optimal pricing strategy: bundling or add-on. Second, if the base manufacturer can benefit from the bundling strategy, we further explore which bundling strategy—BBP or BBBM—will be the optimal one, which was not considered in Yin et al.’s (2021) study. Third, although our research uses the same demand of models as Yin et al. (2021), the firms’ profit functions in our study are different. Yin et al. (2021) studied the retailer’s optimal pricing strategy in a traditional supply chain, but we explored the firm’s strategies in a platform supply chain. Thus, the firm’s pricing decision and optimal selling strategy are influenced by the features of the platform, such as the commission rate charged by the platform, which has not been studied in the literature. Fourth, in contrast to Geng et al. (2018), our study focuses on the base manufacturer’s optimal pricing strategy in a platform supply chain. However, Geng et al. (2018) investigated the interactions between the firm’s pricing strategy (i.e., bundling or add-on pricing), and also the distribution contract between the platform and firm (i.e., wholesale or agency). Further, we examine the impacts of different undertakers of the bundling strategy on firms’ profitability.

Given the research gap in Table 1, the main novelty of this study is examining the influence of different bundling strategy undertakers on pricing decisions and profits for supply chain participants, and also analyzing the optimal selling strategy of base manufacturers when considering both add-on and base product distribution separately.

Table 1 Summarizes the most relevant research in the literature and compares the topic, research focus, and goal of the case study

3 Problem statement

In this section, we propose a game-theoretical model and introduce our assumptions. Table 2 lists the main definitions and notations used for reference. All proofs are provided in the “Appendix”.

Table 2 Summary of notations

3.1 Firms

We consider a two-tier platform supply chain that includes a base manufacturer (M1) who produces the base product, an add-on manufacturer (M2) who produces the add-on, and an e-commerce platform (E). For example, manufacturers sell home appliances on e-commerce platforms such as JD.com, while third-party providers serve protection plans. Note that the add-on only becomes valuable if the consumer buys the base product (e.g., home appliance dealers sell extended warranty programs to consumers who have bought new home appliances). The base manufacturer can choose between two product sales modes, that is, bundling and add-on selling. Furthermore, when the bundling mode is chosen, the base manufacturer can choose to bundle by itself or dictate the platform to bundle. This assumption is plausible because the base manufacturer, such as Apple, Samsung, Nike, and Adidas, generally has more power to decide which selling mode to use.

Based on the above analysis, we consider three cases, as shown in Fig. 1. In Case E, both products are BBP which sells the bundled product in its own marketplace. In Case M, both products are bundled by either of the manufacturer. Without loss of generality and consistency with common situations, we consider only the case with both products BBBM. After bundling, the base manufacturer sells the bundled product to consumers through the platform, while the platform charges a commission rate on the base manufacturer. In Case A, two manufacturers sell their products independently through the platform, and each pays a commission rate to the platform to gain access rights. Manufacturer M1 produces the product at a marginal cost $c_{1} \in \left[ {0,1} \right]$ and manufacturer M2 produces the add-on at marginal cost $c_{2} \in \left[ {0,1} \right]$.

Fig. 1

Supply chain structures

3.2 Consumers

Consumers are heterogeneous in their valuations of both products. The total market size was normalized to 1. We denote a consumer’s valuation of product $i$ by $v_{i}$ ($i = 1,2$), and $i = 1$ signifies the base product, and $i = 2$ signifies the add-on. Similar to that in prior literature (e.g., Prasad et al., 2010; Yin et al., 2021), we assume both $v_{1}$ and $v_{2}$ are uniformly and independently distributed from 0 to 1. When a bundled product is offered (i.e., bundling strategy) for a consumer with valuation ($v_{1}$, $v_{2}$), they can purchase the product and obtains the utility $u_{b} = v_{1} + v_{2} - p_{b}$ or leave the market with zero utility, where $p_{b}$ is the bundled product’s price. As the base product and the add-on are offered separately (i.e., add-on strategy) with retail prices of $p_{1}$ and $p_{2}$ respectively, a consumer can obtain either utility $u_{a} = v_{1} - p_{1}$ if they purchase the base product only or the utility $u_{a} = v_{1} + v_{2} - p_{1} - p_{2}$ if they purchase both products or leave the market with zero utility. The consumer makes a purchase decision by obtaining the maximum utility from the market-offered product(s). Note that the customer who consumes an add-on should satisfy two conditions: (i) they have bought the base product, and (ii) the add-on provides them with positive utility.

Based on the consumer’s decisions, we can derive the demand for a product(s) between bundling and add-on strategies. When the bundling strategy is offered and the price $p_{b}$ is set, the demand for bundle $d_{b}$ can be derived by $u_{b} \ge 0$ as $d_{b} = \frac{1}{2}\left( {2 - p_{b}^{2} } \right)$ if $p_{b} \le 1$ and $d_{b} = \frac{1}{2}\left( {2 - p_{b} } \right)^{2}$ otherwise (Fig. 2 illustrates this). When the add-on strategy is offered, consumers can buy the base product and add-on in the platform’s marketplace. A consumer will buy the base product first if they can obtain utility of $u_{a} = v_{1} - p_{1} > 0$. Subsequently, the consumer considers whether to buy the add-on. A consumer will buy if she finds that the add-on gives them an additional non-negative utility (i.e., $v_{2} - p_{2} > 0$). Therefore, we can derive the base product’s demand $d_{1} = 1 - p_{1}$ and the add-on’s demand $d_{2} = \left( {1 - p_{1} } \right)\left( {1 - p_{2} } \right)$.

Fig. 2

Demands under bundling strategy

3.3 Firm and consumer decisions

Based on the above discussion, it is intuitive that the possible selection of bundling or add-on is driven by the base product manufacturer’s distribution offering and the platform’s rate structure (i.e., the commission rate $f$). In each case, the interaction between manufacturers and the platform is depicted in Fig. 1. The stages of the game for each case are given as below:

Stage 1: The product distribution strategy. The base manufacturer decides on the product distribution strategy, that is, the bundling strategy or add-on strategy.

Stage 2: The bundling strategy. If the manufacturer chooses the bundle-selling strategy, it must decide by itself or dictate the platform to bundle.

Stage 3: The operational decisions. Firms make pricing decisions in different cases as follows:

• (1) In Case E, the base manufacturer adopts a bundling strategy, and the platform acts as a reseller to sell both products as a bundle. In this case, manufacturers first simultaneously set wholesale prices $w_{1}$ and $w_{2}$ on the platform, which sets the bundle’s price $p_{b}$ for in turn.

• (2) In Case M, that is, the base manufacturer adopts a bundling strategy, and the platform acts as a marketplace for the manufacturers by charging a commission rate $f \in \left[ {0,1} \right]$. The add-on manufacturer first sets $w_{2}$ to the base manufacturer, which, in turn, sets the retail price $p_{b}$ for the bundle.

• (3) In Case A, that is, the base manufacturer adopts an add-on strategy, and the platform acts as a marketplace for the manufacturers by charging a commission rate $f$. In this case, assuming that both manufacturers accept the rate structure, then both set the retail prices $p_{1}$ and $p_{2}$ simultaneously.

Note that we treat commission rate $f$ as exogenously determined to focus on understanding the bundling problem. This assumption is also prevailed in previous studies such as Geng et al. (2018), Shen et al. (2019), and Zhen and Xu (2021). In reality, the commission rate is largely predetermined through industrial observations. Since both products generally belong to the same or similar product categories, we further assume that the platform charges them the same commission rate $f$.

4 The different selling strategies

In this section, we first provide the optimal solutions for each case, and then compare outcomes in these cases to reveal the impact of the bundling strategy on firms’ decisions and profits. Finally, we explore the base manufacturer or platform’s optimal bundling strategy.

4.1 Bundled by the platform strategy (Case E)

We first consider Case E, in which the base manufacturer adopts the bundling strategy of the platform. In this case, the platform resells both products as a bundle. The platform sells the bundle in its marketplace and chooses $p_{b}$ to maximize its profit.

$$ \pi_{e}^{E} = \left( {p_{b} - w_{1} - w_{2} } \right)d_{b} $$

(1)

Both manufacturers face the platform and make independent decisions. Manufacturer $i$ sets $w_{i}$ to maximize profit:

$$ \pi_{{m_{i} }}^{E} = \left( {w_{i} - c_{i} } \right)d_{b} $$

(2)

We can formulate the profit functions of the platform and manufacturers, and solve the game by backward induction. The firms’ optimal decisions under Case E (which we denote by superscript “E”) can be summarized in the lemma as below.

Lemma 1

Under Case E, the firms’ optimal pricing and realized demands are as follows: $w_{i}^{E*} = \frac{{\left( {2 + 3c_{i} - c_{3 - i} } \right)}}{4}$, $p_{b}^{E*} = \frac{{\left( {4 + c_{1} + c_{2} } \right)}}{3}$, $d_{b}^{E*} = \frac{{\left( {2 - c_{1} - c_{2} } \right)^{2} }}{18}$; the firms’ optimal profits are as follows: $\pi_{e}^{E*} = \frac{{\left( {2 - c_{1} - c_{2} } \right)^{3} }}{108}$ and $\pi_{{m_{i} }}^{E*} = \frac{{\left( {2 - c_{1} - c_{2} } \right)^{3} }}{72}$. See all the proofs in the “Appendix”.

From Lemma 1, we observe that the firms’ pricing decisions are mainly affected by the product’s marginal cost. Specifically, as the base product (add-on)’s cost increases, the base (add-on) manufacturer sets a higher wholesale price. However, as the add-on (base product)’s cost increases, the base (add-on) manufacturer decreases its wholesale price. Moreover, as the base product’s or add-on’s cost increases, the platform is more likely to set a higher retail price for the bundle, and thus product demand will decrease. In addition, as the product’s marginal cost increases, the manufacturers’ and platforms’ profits decrease. This also indicates that lowering the product’s cost is an effective way for manufacturers to earn greater profits.

4.2 Bundled by the base manufacturer strategy (Case M)

We now consider Case M, where the base manufacturer adopts a bundling strategy. In this case, the platform acts as a marketplace for manufacturers by charging commission rates. The base product manufacturer sells the bundle through the platform’s marketplace and chooses $p_{b}$ to maximize its profit.

$$ \pi_{m1}^{M} = \left( {\left( {1 - f} \right)p_{b} - w_{2} - c_{1} } \right)d_{b} $$

(3)

The add-on manufacturer acts as a supplier for the base product manufacturers. The add-on manufacturer chooses $w_{2}$ to maximize profit:

$$ \pi_{m2}^{M} = \left( {w_{2} - c_{2} } \right)d_{b} $$

(4)

The platform gains profit from charging the base product manufacturer by $\pi_{e}^{M} = fp_{b} d_{b}$. We can formulate the profit functions of the platform and manufacturers and solve the game as above subsection. The firms’ optimal decisions in Case M (denoted by superscript “M”) can be summarized in the lemma as below.

Lemma 2

Under Case M, the firms’ optimal pricing and realized demands are as follows: $w_{2}^{M*} = \frac{{2\left( {1 - f} \right) - c_{1} + 2c_{2} }}{3}$, $p_{b}^{M*} = \frac{2}{9}\left[ {5 + \frac{{2\left( {c_{1} + c_{2} } \right)}}{1 - f}} \right]$, $d_{b}^{M*} = \frac{{8\left[ {2\left( {1 - f} \right) - \left( {c_{1} + c_{2} } \right)} \right]^{2} }}{{81\left( {1 - f} \right)^{2} }}$; the firms’ optimal profits are as follows: $\pi_{e}^{M*} = \frac{{16f\left[ {5\left( {1 - f} \right) + 2\left( {c_{1} + c_{2} } \right)} \right]\left[ {2\left( {1 - f} \right) - \left( {c_{1} + c_{2} } \right)} \right]^{2} }}{{729\left( {1 - f} \right)^{3} }}$, $\pi_{m1}^{M*} = \frac{{16\left[ {2\left( {1 - f} \right) - \left( {c_{1} + c_{2} } \right)} \right]^{3} }}{{729\left( {1 - f} \right)^{2} }}$, $\pi_{m2}^{M*} = \frac{{8\left[ {2\left( {1 - f} \right) - \left( {c_{1} + c_{2} } \right)} \right]^{3} }}{{243\left( {1 - f} \right)^{2} }}$. See all the proofs in the “Appendix”.

Lemma 2 provides the equilibrium outcomes when the base manufacturer adopts a bundling strategy. In this case, firms’ pricing decisions, product demands, and profits are affected by the product’s marginal cost and commission rate. From Lemma 2, we find that the add-on manufacturer decreases its wholesale price as the commission rate increases. Meanwhile, as the commission rate increases, the base manufacturer sets a higher product price, thus decreasing the demand for bundled products. The results show that a higher platform commission rate may discourage consumers’ willingness to purchase a bundle. Moreover, it is intuitive that manufacturers’ profits will decrease as the product’s marginal cost or commission rate increases. However, if the cost is high, the platform’s profit increases in the product’s marginal cost, and the platform might be better off with high-cost bundles under the manufacturer bundling strategy.

4.3 Add-on strategy (Case A)

This subsection considers the case where the base manufacturer adopts an add-on strategy. For analysis, we denote this case by “A.” In this case, the base product and add-on are sold by the base and add-on manufacturers, respectively. Meanwhile, the platform acts as a marketplace for both manufacturers, who in turn pay a commission rate $f$ based on revenue. Recall that the consumer obtains $u_{1} = v_{1} - p_{1} \ge 0$ and $u_{2} = v_{2} - p_{2} < 0$ if buying the base product only, and obtains $u_{1} = v_{1} - p_{1} \ge 0$ and $u_{2} = v_{2} - p_{2} \ge 0$ if buying both products. Accordingly, the sale of the base good is $d_{1} = 1 - p_{1}$ and the add-on is $d_{2} = \left( {1 - p_{1} } \right)\left( {1 - p_{2} } \right)$. Both manufacturers face consumers directly in the platform marketplace. Manufacturer $i$ sets $p_{i}$ to maximize profit:

$$ \pi_{{m_{i} }}^{A} = \left[ {\left( {1 - f} \right)p_{i} - c_{i} } \right]d_{i} $$

(5)

The platform gains profit by charging both manufacturers by $\pi_{e}^{A} = \mathop \sum \limits_{i = 1}^{2} fp_{i} d_{i}$. The firms’ optimal decisions in equilibrium in Case A (denoted by superscript “A”) can be summarized in the lemma as below.

Lemma 3

Under Case A, the firms’ optimal pricing and realized demands are as follows: $p_{1}^{A*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$, $p_{2}^{A*} = \frac{{1 - f + c_{2} }}{{2\left( {1 - f} \right)}}$, $d_{1}^{A*} = \frac{{1 - f - c_{1} }}{{2\left( {1 - f} \right)}}$, $d_{2}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)\left( {1 - f - c_{2} } \right)}}{{4\left( {1 - f} \right)^{2} }}$; the firms’ optimal profits are as follows: $\pi_{e}^{A*} = \frac{{\left[ {3\left( {1 - f} \right)^{2} + 2c_{1} \left( {1 - f} \right) - c_{2}^{2} } \right]f\left( {1 - f - c_{1} } \right)}}{{8\left( {1 - f} \right)^{3} }},$ $\pi_{m1}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)^{2} }}{{4\left( {1 - f} \right)}}$, and $\pi_{m2}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)\left( {1 - f - c_{2} } \right)^{2} }}{{8\left( {1 - f} \right)^{2} }}$.

From the equilibrium outcomes in Lemma 3, we find that when the base product and add-on are sold separately, as the commission rate and the product’s marginal cost increase, all product prices increase. Thus, the base product’s demand decreases as an increase of commission rate. However, for the add-on, its demand will increase as an increase of commission rate only when the commission rate is high. Lemma 3 also shows that manufacturers' profits will decrease as the marginal cost and commission rate increase.

5 Results

In this analysis, we first discuss the base manufacturer’s optimal bundling strategy, that is, BBI or BBP, in Sect. 5.1. Then, we explore the firm’s optimal selling strategy, that is, the bundled or add-on strategy in Sect. 5.2.

5.1 The analysis of the bundling strategy

In this subsection, we first analyze the influences of different bundling strategies on firms’ pricing as well as market demands, and then analyze firms’ profitability performances under the above bundling strategy. Finally, we explore the optimal bundling strategy.

5.1.1 Prices and demands

By comparing the bundle product’s price and demand in cases E and M, we have Proposition 1, which is presented as follows:

Proposition 1

• (i) The bundle product price under the base manufacturer strategy will be lower when the commission rate is low; otherwise, the bundle product price under the platform strategy will be lower. Mathematically, $p_{b}^{M*} \le p_{b}^{E*}$ if $f \le \frac{{2 - \left( {c_{1} + c_{2} } \right)}}{{2 + 3\left( {c_{1} + c_{2} } \right)}}$; otherwise, $p_{b}^{E*} < p_{b}^{M*}$.

• (ii) The product demand under BBBM strategy will be lower when the commission rate is moderate; otherwise, the demand under BBP will be lower. Mathematically, $d_{b}^{M*} < d_{b}^{E*}$ if $\frac{{2 - \left( {c_{1} + c_{2} } \right)}}{{2 + 3\left( {c_{1} + c_{2} } \right)}} < f < \frac{{7\left( {2 - c_{1} - c_{2} } \right)}}{{14 - 3\left( {c_{1} + c_{2} } \right)}}$; otherwise, $d_{b}^{E*} \le d_{b}^{M*}$.

As shown in Proposition 1(i), when the platform charges a low commission rate, the price of bundled products under BBBM strategy will be lower than that under BBP strategy. However, as an increase of commission rate, the price of the product increases under BBBM strategy. There are two reasons for base manufacturers’ pricing decisions. The most obvious difference between the two strategies is the pricing power for the bundle. Under BBP strategy, the base manufacturer and add-on manufacturer first wholesale the base product and add-on to the platform, and then the platform sets the retail price for the bundle. However, under the BBBM, the price of bundles is set by the base manufacturer directly, which can effectively avoid the second markup from the platform. Thus, the manufacturer has the incentive to set product price lower. However, under BBBM strategy, the manufacturer must bear the platform’s commission rate charge. Thus, when the commission rate is set at a low level, the positive effect of alleviating the double marginalization dominates. Thus, the base manufacturer sets a lower price for bundled products. When the commission rate is high, the manufacturer will cover the loss by increasing the product price.

Moreover, Proposition 1(ii) compares the bundle product demands in cases E and M. Specifically, when the commission rate is sufficiently low or high, the product demand in Case M will be higher than that in Case E. Otherwise, the product demand in Case E will be higher. Referring to the result in Proposition 1(i), the base manufacturer in Case M sets a lower product price if the commission rate is low (i.e., $f \le \frac{{2 - \left( {c_{1} + c_{2} } \right)}}{{2 + 3\left( {c_{1} + c_{2} } \right)}}$). Thus, there are more consumers to purchase the bundle owing to the low price. When the commission rate is moderate, the demand in Case M decreases because of a higher product price. Interestingly, our results show that when the commission rate is very high, demand in case M is still higher than that in Case E, even if the manufacturer sets a higher product price. This indicates that when the platform charges a very high commission rate, there is a demand-increasing effect and a price-increasing effect if the base manufacturer adopts a bundling strategy.

5.1.2 Firms’ profits

Then, we develop Propositions 2 and 3 by comparing the profits between the base manufacturer and platform in cases E and M as follows:

Proposition 2

The base manufacturer can benefit from BBI strategy when both the commission rate and the bundle’s cost are low; otherwise, the manufacturer can benefit from BBP strategy. Mathematically, $\pi_{{m_{1} }}^{M*} > \pi_{{m_{1} }}^{E*}$ if $f < \frac{47}{{128}}$ and $c < \frac{{2\left\{ {8\left( {1 - f} \right) - \sqrt[3]{{3\left[ {32\left( {1 - f} \right)} \right]^{2} }}} \right\}}}{{8 - \sqrt[3]{{3\left[ {32\left( {1 - f} \right)} \right]^{2} }}}}\left( { = C_{0} } \right)$; otherwise, $\pi_{{m_{1} }}^{M*} \le \pi_{{m_{1} }}^{E*}$.

As shown in Proposition 2, the results show that the base manufacturer prefers the BBI strategy if the commission rate and marginal cost of the bundle are low. See Fig. 3(1) for an illustration. When the commission rate and the cost are low, the base manufacturer can obtain a higher profit by adopting the BBI strategy owing to the increased product demand. However, when the commission rate or cost is high enough, the base manufacturer is more willing to adopt BBP strategy. Compared with Case E, the manufacturer in Case M needs to shoulder the extra cost of purchasing the add-ons and the commission rate charged by the platform. Therefore, when the commission rate or cost is high, although the BBBM strategy might attract more consumers or force the manufacturer to set a higher price, the extra cost for purchasing the add-ons and the commission rate reduces the base manufacturer’s margin profit. Thus, the manufacturer will be worse off when adopting the bundling strategy by itself. As a result, adopting the BBP strategy is the optimal choice when the commission rate or marginal cost is high.

Fig. 3

The optimal bundling strategy for all participants

Proposition 3

The platform can benefit from the BBBM strategy when (a) the commission rate is low, and the bundle’s cost is moderate or very high, or (b) the commission rate is high, and the bundled product’s marginal cost is very low or very high. Mathematically, $\pi_{e}^{M*} > \pi_{e}^{E*}$ if (a) $f \le \frac{27}{{160}}$, $C_{1} < c < C_{2}$, and $c > C_{3}$, (b) $f > \frac{27}{{160}}$, $c < C_{2}$ and $c > C_{3}$; otherwise, $\pi_{e}^{M*} \le \pi_{e}^{E*}$, where $c = c_{1} + c_{2}$ and $C_{1} < C_{2} < C_{3}$.

Proposition 3 provides the platform’s preference for the bundling strategy. See Fig. 3(2) for an illustration. Specifically, when the platform charges a low commission rate, it prefers BBBM strategy if the product’s marginal cost is moderate. The reason is that in the above region, the price-increasing effect in Case M dominates the demand-increasing effect in Case E. Thus, the platform can profit from the bundled by the manufacturer strategy. Moreover, the platform can earn greater profit from such a strategy if the bundle’s cost is very low while the commission rate is high enough. This is because, under such conditions, the platform can obtain a high commission income from Case M, and the demand in Case M will be higher than in Case E. Consequently, the platform prefers the manufacturer’s bundling strategy. In addition, the results show that when the bundle’s cost is very high, owing to the demand-increasing and price-decreasing effects, the BBBM strategy will be the optimal choice for the platform.

Propositions 2 and 3 show the influences of different bundling strategies on firms’ profitability. It is worth discussing whether the different bundling strategies always influence the base manufacturer and the platform similarly. We obtain Corollary 1 which is shown as below.

Corollary 1

The base manufacturer and platform achieve a win–win situation in Case E in the dark blue region in Fig. 4, a win–win situation in Case M in the yellow region, and a win-lose situation in other parameter regions.

Fig. 4

Firms’ profitability under bundling strategy

Combined with Corollary 1 and Fig. 4, we find that when the commission rate and the bundle’s cost are relatively low (i.e., the win–win in Case M region in Fig. 4), the BBBM strategy will benefit both the base manufacturer and the platform. Meanwhile, when the bundle’s cost is relatively high (i.e., the win in region E in Fig. 4), the BBP strategy will benefit the base manufacturer and platform. Within the above two parameter ranges, the impacts of the bundling strategy on the base manufacturer and platform are the same, which means that the bundling strategy can achieve a “win–win” situation for them. However, in other regions (i.e., the win-lose regions in Fig. 4), the impact of bundling strategies is completely different. Specifically, when the commission rate and the bundle’s cost are very low, the base manufacturer will benefit from Case M, but the platform will benefit from Case E; when the commission rate and the bundle’s cost are moderate, or when the cost is very high, the base manufacturer will benefit from Case E, but the platform will benefit from Case M. All the above regions will result in a win-lose situation for the base manufacturer and the platform. In addition, there is no region in which the base manufacturer and platform suffer from the bundling strategy.

5.1.3 The optimal bundling strategy

We examine the equilibrium bundling strategy, in which a strategic interaction exists between the platform and the base manufacturer. We aim to solve the problems as follows: Is there any bundling strategy in equilibrium? It is worth noting that, to be consistent with the prior studies, we investigated the influence of two key factors only, that is, total marginal cost $c_{1} + c_{2}$ and the platform’s commission rate $f$.

Proposition 4

For any given $c_{1} + c_{2}$, both platform and base manufacturer prefer the base manufacturer to bundle (Case M) for a relatively low commission rate, and both platform and base manufacturer prefer the platform to bundle (Case E) for a relatively high commission rate, and one party prefers the other to bundle otherwise.

Next, we discuss the insights behind this proposition. The profit margin is fixed for any given marginal cost. In addition, a relatively low commission rate leads to a relatively low benefit for the platform. In this case, the platform must increase its sales volume to increase its total profit. According to Proposition 1, under the above conditions, the price of bundled products in Case M is relatively low, resulting in higher demand. Therefore, both the platform and base manufacturers benefit. Similarly, the platform extracts more profits from manufacturers through commission rates for any given marginal cost and a relatively high commission rate. In this case, it is more favorable for the platform to gain more profit through bundling and pricing by itself (i.e., Case E). In addition, the base manufacturer can benefit from this case because commissions are not too high, and the platform is more efficient with bundling. If these two equilibria cannot be achieved, one party expects the other party to implement the bundling strategy. Predictably, if there are no better outside options, both sides will choose a suboptimal strategy. In addition, the stronger party will also dictate the other party to bundle, especially as the rise of the platform’s market power may be dominant; of course, some strong brands will be exceptions, such as Apple and Nike.

5.2 The analysis of optimal strategy: bundling versus add-on

By comparing the prices of the bundle and the demand for the base product under the bundling and add-on strategies, we have Proposition 5, which is described as follows:

Proposition 5

• (i) The product price under the add-on strategy will be lower than that under the bundle by the manufacturer/platform strategy under a low commission rate; otherwise, the product price under the bundle by the manufacturer/platform strategy will be lower. Mathematically, $\left( {p_{1}^{A*} + p_{2}^{A*} } \right) \le p_{b}^{E*}$ if $f \le \frac{{2 - \left( {c_{1} + c_{2} } \right)}}{{2 + 2\left( {c_{1} + c_{2} } \right)}}$; otherwise, $p_{b}^{E*} < \left( {p_{1}^{A*} + p_{2}^{A*} } \right)$; $\left( {p_{1}^{A*} + p_{2}^{A*} } \right) \le p_{b}^{M*}$ if $f \le \frac{{2 - \left( {c_{1} + c_{2} } \right)}}{2}$; otherwise, $p_{b}^{M*} < \left( {p_{1}^{A*} + p_{2}^{A*} } \right)$.

• (ii) The product demand of the base product under the add-on strategy will be lower than that under the bundle by the manufacturer/platform strategy under a high commission rate; otherwise, the product demand of the base product under the bundle by the manufacturer/platform strategy will be lower. Mathematically, $d_{1}^{A*} < d_{b}^{E*}$ if $f > 1 - \frac{{9c_{1} }}{{\left( {5 - c_{1} - c_{2} } \right)\left( {1 + c_{1} + c_{2} } \right)}}$; otherwise, $d_{b}^{E*} \le d_{1}^{A*}$; $d_{1}^{A*} < d_{b}^{M*}$ if $f > \frac{{\left( {34 - 17c_{1} + 64c_{2} - 9\sqrt {17c_{1}^{2} + 64c_{2}^{2} } } \right)}}{34}$; otherwise, $d_{b}^{M*} \le d_{1}^{A*}$.

The result in Proposition 5(i) indicates that when the platform sets a low commission rate, the sum price of both products under the add-on model is lower than that under the bundling model; otherwise, the price under the bundling model is lower. The results show that selling the base product and add-on separately might result in a lower product price with a low commission rate. For instance, in the app stores of operating systems like Apple’s iOS or Google’s Android, developers can sell their apps (add-ons) separately from the base software, allowing users to customize their software experience without the burden of a higher bundled price (Kawaguchi et al., 2023). This can attract more consumers to purchase products, thus increasing in demand, as shown in Proposition 5(ii). Moreover, when the platform’s commission rate is high, manufacturers are incentivized to increase product prices under the add-on strategy. Thus, the bundle’s price in the add-on strategy is higher than that in the bundling strategy. Consequently, consumers are less willing to purchase a bundle under the add-on strategy. For example, car manufacturers often provide basic models with the option to add extra features (add-ons) such as GPS navigation systems, leather seats, or upgraded sound systems (Zhang et al., 2023).

6 Sensitivity analysis

It is difficult to compare the outcomes of the add-on model because of the mathematical complexity. Thus, in this section, we employ some numerical experiments to analyze the impacts of relevant parameters on the supply chain and its member’s profitability, and further explore the optimal selling strategies.

6.1 The effect of the marginal cost

In this subsection, we first consider the effect of products’ marginal costs under different selling strategies. We fix the commission rate at $f = 0.2$, and the marginal cost of the base product $c_{1} = xc_{2}$, where $x = 0.2$, $x = 0.5$, $x = 0.8$, respectively. Further, we vary the marginal cost of add-ons $c_{2}$ from 0 to 1 to ensure that all equilibrium outcomes are positive. Thus, we obtain Figs. 5, 6 and 7.

Fig. 5

The effect of $c_{2}$ on the base manufacturer’s profit

Fig. 6

The effect of $c_{2}$ on the platform’s profit

Fig. 7

The effect of $c_{2}$ on the supply chain profitability

Figure 5 shows that when the base product’s cost is relatively low, the base manufacturer prefers the add-on strategy, that is, selling the base good and add-on separately. When the base product’s marginal cost is very high, the manufacturer prefers the add-on strategy only if the add-on’s marginal cost is low; otherwise, the manufacturer is more likely to choose the bundling strategy. This indicates that for a high-cost base product and high-cost add-ons, the bundling strategy would be the optimal choice for the base manufacturer. Figure 6 shows the platform’s profits under different strategies. We find that the platform always becomes worse off under the BBI strategy. The result indicates that when the platform sets a relatively high commission rate, i.e., $f = 0.2$, the platform should refuse to provide bundle-selling service for the base manufacturer. Furthermore, for the platform, the BBBM strategy is the optimal choice when the marginal cost of add-ons is relatively low, otherwise, the add-on strategy is optimal.

Figure 7 shows the optimal product strategy from the supply chain perspective. It signifies that when the base product’s marginal cost is low, the supply chain can benefit from the bundle using the base manufacturer strategy. When the cost is high, the supply chain should adopt the add-on strategy and switch to the BBP strategy, which mainly depends on the add-ons’ marginal cost.

6.2 The effect of the commission rate

Next, we further analyses the effect of the commission rate on firms’ profitability. We set the marginal cost of add-ons $c_{2} = 0.2$, and the marginal cost of the base product $c_{1} = xc_{2}$, where $x = 0.2$, $x = 0.5$, $x = 0.8$, respectively. Further, we vary the commission rate from 0 to 0.7 to ensure that all equilibrium outcomes are positive. Thus, we obtain Figs. 8, 9 and 10.

Fig. 8

The impact of $f$ on the base manufacturer’s profit

Fig. 9

The effect of $f$ on the platform’s profit

Fig. 10

The effect of $f$ on the supply chain’s total profit

As shown in Fig. 8, the base manufacturer’s profit in cases M and A decreases as the commission rate increases. As the base product’s marginal cost increases, the base manufacturer prefers the add-on strategy under a low commission rate; otherwise, it is more likely to choose the bundle using the platform strategy. The reason is that, when the commission rate is very high, the negative effect of a high commission rate in Case A dominates the positive effect of the double marginalization mitigation. Thus, the base manufacturer will forgo the add-on strategy. Figure 9 shows the effect of the commission rate and indicates that, when the commission rate is very low, the platform would be better off under the BBI strategy. However, as the commission rate increases, the platform can benefit from the add-on strategy, which is consistent with Fig. 6.

Figure 10 illustrates the effect of the commission rate on the supply chain profitability. It indicates that the total profit of the supply chain decreases as the commission rate increases. Moreover, when the base product’s marginal cost is low, the add-on strategy is always the optimal strategy. As the cost increases, bundling by the base manufacturer might be the optimal choice under a low commission rate.

7 Extensions

We consider independent valuations of the base and add-on products in the main model. One may question that customers may value the base product more while value the add-on more (or less), corresponding to positive (or negative) correlations. In this section, we analyze two extensions of correlated valuations. All other modeling elements remained the same as those in the main model. We test these two extensions respectively, since the derivation methods under positive and negative correlation scenarios are different.

7.1 Positive correlation between base product valuation and add-on valuation

We first consider the positive correlation between base product valuation and add-on valuation. In reality, for example, it stands to reason that consumers who wants to buy an expensive shampoo (the base product) might also be willing to pay more for conditioner (the add-on). Using the settings of McCardle et al. (2007) and Yin et al. (2021), we assumed that the value $v_{1}$ of the base product is distributed between 0 and 1 uniformly, whereas the value $v_{2}$ of the add-on is distributed between $l$ and $h$ uniformly, where $0 \le l < h \le 1$. Accordingly, product costs satisfy $c_{1} \in \left[ {0,1} \right]$ and $c_{2} \in \left[ {l,h} \right]$. Then, $v_{2} = l + \left( {h - l} \right)v_{1}$ is used to capture the perfectly positive correlation between both products’ valuations.

Therefore, for the bundled product, its market demand becomes $d_{b} = 1 - \frac{{p_{b} - l}}{1 + h - l}$, and the price of the bundled product $p_{b}$ is between $l$ and ($1 + h$). Under the add-on model, given $p_{1}$, the base product’s market demand is $d_{1} = 1 - p_{1}$, where the marginal consumer $\tilde{v}_{1} = p_{1}$. Hence, (i) when the marginal consumer buys the add-on, we have $v_{2} - p_{2} = l + \left( {h - l} \right)p_{1} - p_{2} \ge 0$, which leads to $p_{1} \ge \frac{{p_{2} - l}}{h - l}$. In this scenario, all consumers who consume the base product will own the add-on as well, so we have $d_{2} = d_{1}$. (ii) When the marginal consumer does not buy the add-on, there exists $p_{1} < \frac{{p_{2} - l}}{h - l}$. In this scenario, consumers will buy the add-on if they obtain positive utility, that is, $v_{2} - p_{2} = l + \left( {h - l} \right)v_{1} - p_{2} \ge 0,$ which leads to the marginal consumer $\tilde{v}_{1} \ge \frac{{p_{2} - l}}{h - l}$ and $d_{2} = 1 - \frac{{p_{2} - l}}{h - l}$. Combining scenarios (i) and (ii), we obtain $d_{2} = 1 - \max \left\{ {\frac{{p_{2} - l}}{h - l},p_{1} } \right\}$.

To examine the impact of the positive correlation between base product valuation and add-on valuation, we compared firms’ profits under the bundling strategy. The results are summarized in Proposition 6. All analysis and equilibrium results are presented in the “Appendix”.

Proposition 6

When there exist perfectly positively correlated product valuations, such that:

• (1) For any given $\frac{{c_{1} + c_{2} }}{1 + h}$, both platform and base manufacturer prefer the platform to bundle (Case E) for a relatively low or high commission rate; mathematically, $\pi_{e}^{M*} \le \pi_{e}^{E*}$ and $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$ if $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} < \frac{{c_{1} + c_{2} }}{1 + h} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$ or $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{{c_{1} + c_{2} }}{1 + h} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$;

• (2) Both platform and base manufacturer never simultaneously prefer the base manufacturer to bundle (Case M).

• (3) For any given $\frac{{c_{1} + c_{2} }}{1 + h}$, if the commission rate reaches a relatively high level, a win–win-win situation achieves for the platform and both manufacturers in Case E compared to Case M; mathematically, $\pi_{e}^{M*} \le \pi_{e}^{E*}$ and $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$ and $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$ if $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{{c_{1} + c_{2} }}{1 + h} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}$.

Proposition 6 echoes a part of the conclusion in Proposition 4. That is, for any given sum of marginal costs, both the platform and base manufacturers tend to favor platform bundling (i.e., Case E) when the commission rate reaches a relatively high level. The difference is that, in this case, there is some variation in marginal cost, which needs to be divided by the sum of the maximum values of products (i.e., $1 + h$). This is because when the value of the two products is positively correlated, cost-effectiveness can be reconciled by the magnitude of the product values. In addition, Proposition 6 provides new insights. First, for any given marginal cost, both platform and base manufacturers tend to prefer platform bundling when the commission rate is relatively low. Even more extreme, as shown in Proposition 6(2), both platform and base manufacturers never simultaneously prefer the base manufacturer to bundle. This is because when there exist perfectly positively correlated product valuations, the demand pooling effect of bundled products decreases. Thus, platform bundling can double the marginal effect compared with base manufacturer bundling, which benefits both the platform itself and the base manufacturer. Moreover, Proposition 6(3) shows that a win–win-win situation is occurred in Case E when the commission rate reaches a relatively high level. This is because a high commission rate ensures the platform’s earnings, the platform sales of the bundled product are conducive to expanding demand, and the correlation between the base product and add-on benefits both manufacturers under the given conditions.

Proposition 7

When there exist perfectly positively correlated product valuations, the base manufacturer prefers add-on strategy to bundling strategy if the pair of $\left( {c_{1} ,c_{2} } \right)$ satisfies $c_{2} \ge Max\left\{ {1 + h + \left( {\frac{3}{2}\sqrt {\frac{{2\left( {1 + h - l} \right)}}{1 - f}} - 1} \right)c_{1} - \frac{3}{2}\left( {1 - f} \right)\sqrt {\frac{{2\left( {1 + h - l} \right)}}{1 - f}} ,\left( {1 - f} \right)\left( {1 + h} \right) + \left( {2\sqrt {1 + h - l} - 1} \right)c_{1} - 2\left( {1 - f} \right)\sqrt {1 + h - l} } \right\}$.

Proposition 7 shows that the base manufacturer prefers the add-on strategy to the bundle strategy when the base product’s cost is relatively low and the add-on’s cost is relatively high. Since the products’ valuations are positively correlated, the demand pooling effect of bundled products decreases. However, the base manufacturer’s incentive to adopt the bundling strategy remains. The relatively low marginal cost of the base product implies a relatively high-profit margin for the base manufacturer. At this time, the relatively high marginal cost of the add-on means a relatively low-profit margin for the add-on manufacturer. In this circumstance, the adoption of an add-on strategy is conducive to stimulating add-on manufacturers to expand the market through aggressive pricing, and the effect of price competition is lower than the enhancement of the market effect and the weakening of the double marginal effect.

7.2 Negative correlation between base product valuation and add-on valuation

In this subsection, we consider that consumers who value the base product more value the add-on less, that is, there exist negatively correlated product valuations. In reality, for example, a consumer who has already purchased a luxury brand car (i.e., the base product) may not be willing to spend time buying decorative accessories (i.e., the add-on). Consistent with the model in Sect. 7.1, we assume that the value $v_{1}$ is distributed between 0 and 1 uniformly, whereas the value $v_{2}$ is distributed between $l$ and $h$ uniformly, where $0 \le l < h \le 1$. Accordingly, product costs satisfy $c_{1} \in \left[ {0,1} \right]$ and $c_{2} \in \left[ {l,h} \right]$. Then, $v_{2} = h - \left( {h - l} \right)v_{1}$ is used to capture the perfectly negative correlation between both products’ valuations.

Therefore, the market demand for the bundled product becomes $d_{b} = 1 - \frac{{p_{b} - h}}{1 + l - h}$, and the price of the bundled product is between $h$ and ($1 + l$). Under the add-on model, given $p_{1}$, the base product’s market demand is $d_{1} = 1 - p_{1}$, where the marginal consumer is $\tilde{v}_{1} = p_{1}$. Hence, (i) when the marginal consumer buys the add-on, we have $v_{2} - p_{2} = h - \left( {h - l} \right)p_{1} - p_{2} \ge 0$, which leads to $p_{1} \le \frac{{h - p_{2} }}{h - l}$. In this scenario, all consumers who own the base product will be not willing to pay for the add-on, so the add-on’s market demand is $d_{2} = 0$. (ii) When the marginal consumer does not buy an add-on, $p_{1} > \frac{{h - p_{2} }}{h - l}$ holds. In this scenario, consumers who buy the base product will purchase the add-on if $v_{2} - p_{2} = h - \left( {h - l} \right)v_{1} - p_{2} \ge 0$, where the marginal consumer is $\tilde{v}_{1} \le \frac{{h - p_{2} }}{h - l}$. Thus, add-on market demand is $d_{2} = \frac{{h - p_{2} }}{h - l} - p_{1}$. Combining scenarios (i) and (ii), we have $d_{2} = \max \left\{ {\frac{{h - p_{2} }}{h - l} - p_{1} ,0} \right\}$.

To examine the impact of the negatively correlated valuations, we compare firms’ profits under the bundling strategy. The results are summarized in Proposition 8. All  analysis and equilibrium results are presented in the “Appendix”.

Proposition 8

When there exist perfectly negatively correlated product valuations, such that:

• (1) For any given $\frac{{c_{1} + c_{2} }}{1 + l}$, both platform and base manufacturer prefer the platform to bundle (Case E) for a relatively low or high commission rate; mathematically, $\pi_{e}^{M*} \le \pi_{e}^{E*}$ and $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$ if $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} < \frac{{c_{1} + c_{2} }}{1 + l} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$ or $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{{c_{1} + c_{2} }}{1 + l} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$;

• (2) Both platform and base manufacturer never simultaneously prefer the base manufacturer to bundle (Case M).

• (3) For any given $\frac{{c_{1} + c_{2} }}{1 + l}$, if the commission rate reaches a relatively high level, a win–win-win situation achieves in Case E compared to Case M; mathematically, $\pi_{e}^{M*} \le \pi_{e}^{E*}$ and $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$ and $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$ if $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{{c_{1} + c_{2} }}{1 + l} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}$.

Proposition 8 is similar to Proposition 6. The difference is that, in this case, the ratio of the sum of the marginal costs to ($1 + l$) plays an important role. It can be seen that the lower threshold of the add-on value plays a key role in this case. In the case of a perfectly positive correlation, the upper threshold of the add-on value plays a key role. In addition, for the case of negatively correlation, the bundling strategy’s demand pooling effect is enhanced, which is exactly the opposite of the situation in which the valuation of both products is perfectly positively correlated. Under the combined influence of the two opposite effects, Proposition 8 in this case is so close to Proposition 6 that no further explanation is required. Similarly, the conclusion obtained from Proposition 7 is valid in this case, which also shows that our conclusion is robust.

8 Managerial insights and practical implications

Regarding the bundling strategy, our findings show that, when the commission rate is relatively low, both the platform and base manufacturer can benefit from the BBBM strategy. This indicates that when the base manufacturer provides the bundle by itself, the platform should lower the commission rate so that all participants can earn a greater profit. However, one party expects the other party to implement a bundling strategy at a relatively high commission rate. In the real world, the stronger party will also dictate the other party to bundle, especially with the rise of the market power of the platform, which may be dominant. Of course, some strong brands will be exceptions, such as Apple and Nike.

The second finding is that, if the platform sets a low commission rate, the sum of the products’ prices is lower than that of the bundle; otherwise, the bundle’s price would be lower. These results provide pricing guidance for consumers when they tend to purchase base products and add-ons. In addition, the above finding indicates that the platform could play an important role in the firm’s pricing decisions and even the consumer’s purchase decisions by adjusting the commission rate. Taking a ride-hailing service like Uber or Lyft for example, the drivers provide the base service (rides) and can offer additional services or amenities (add-ons), such as premium car types or in-car entertainment options, for an extra cost. If the platform sets a low commission rate, drivers might price their additional services competitively. Customers interested in specific add-ons may find it cheaper to select and pay for these services individually rather than opting for a pre-packaged bundle, especially if they don't need all the bundled services.

In addition, by analyzing the impacts of relative factors (i.e., the commission rate and product cost) on the firms’ and supply chain’s profits under different selling strategies, we find that not only the platform’s commission rate but also the product cost can have a significant influence on the firms’ decisions. For example, as the product’s marginal cost (or commission rate) increases, the base manufacturer’s profit decreases, whereas the platform’s profit increases in the commission rate. These observations show that the base manufacturer can increase profits by reducing the product’s cost, whereas the platform can benefit from a higher commission rate. In the e-commerce industry, platforms like Amazon charge varying commission rates to third-party sellers. When sellers analyze the impact of these commission rates and their product costs, they may find that adjusting their product costs through bunding or add-on to affect their profitability.

9 Conclusions and outlook

In business practices, bundling or add-on strategies are widely used for selling base products and add-on products, such as mobile phones and earphones, cars and their post-sale services, and hotel accommodations and additional services. Under the bundling strategy, the add-on product can be sold with the base product, while the firm must give up the additional profits created by the add-ons. Under the add-on strategy, the firm could impose price discrimination on consumers, although some consumers may not purchase add-ons. Thus, firms should weigh the advantages and disadvantages of the aforementioned strategies when selling these products. Moreover, e-commerce growth has forced more platforms to engage in bundled sales, which may affect the manufacturer’s distribution strategy. Motivated by these results, we propose a game-theoretical model to explore bundling or add-on strategy’s implementation in the context of a platform supply chain and obtain some results.

The results of this research and managerial insights are as follows:

• The base manufacturer can benefit from the BBI strategy only when both the commission rate and the bundle’s cost are at a low level, which results from increasing product demand. Meanwhile, the platform is better off under such a strategy when the commission rate is low while the bundle’s cost is moderate or very high, or the commission rate is high and the bundle’s cost is very low or very high. These results also indicate that under certain conditions, the base manufacturer and platform can agree on the bundling strategy and achieve a win–win outcome.

• Our study analyzes the impact of the add-on and bundling strategies on firms’ pricing decisions. The results show that, when the commission rate stays at a low level, the bundled product’s price under the BBP strategy is higher than that under the BBBM strategy, thus leading to higher product demand. However, both the price and demand for bundled product under the BBBM strategy will be higher than those under the BBP strategy if the commission rate stays sufficiently high level. Moreover, we find that, compared with the bundling strategy, the bundle’s price under the add-on strategy will be lower if the platform charges a low commission rate, thus attracting more consumers to purchase the bundles.

• We explore the impacts of the relevant factors, that is, the commission rate and marginal cost of the base product and add-on, on the firms’ and supply chain’s profits under the above strategies. The observations show that as the product’s marginal cost (or commission rate) increases, the base manufacturer’s profit decreases while the platform’s profit increases as the commission rate increases. Moreover, by maximizing the supply chain’s total profit, we find that the add-on strategy is always optimal when the base product’s marginal cost is low. When the cost increases, bundling by the base manufacturer might be the optimal choice if the commission rate is low.

Although we obtained the above findings, our study has some limitations, which can be studied in future research. First, we assume that the base manufacturer is more powerful in the platform supply chain, whereas some powerful platforms exist in the real world. Thus, exploring a manufacturer’s selling strategy in a platform-led supply chain is also interesting. Second, our study focuses on a supply chain with a monopoly platform. Investigating a competitive environment with multiple platforms will be interesting. In addition, our study assumes that the platform charges the same commission rate to all manufacturers. Therefore, our study can extend to the case where the platform sets different commission rates for manufacturers. Finally, with the emergence of various emerging technologies, such as Internet-ofThings (Goli et al., 2023b) and blockchain (Lotfi et al., 2022b, 2023), and new algorithms (Goli et al., 2023a; Lotfi et al., 2022a), the impact of these new technologies can be considered in models, and more complex models can be solved with these more advanced algorithms.

Appendices

Appendix

Proof of Lemma 1

In Case E, we can formulate the platform’s and manufacturers’ profit functions and solve the game by backward induction. In the second stage of the game, given a wholesale price $w_{i}$, the platform’s problem is as follows:

$$ \mathop {\max }\limits_{{p_{b} }} \pi_{e}^{E} = \left( {p_{b} - w_{1} - w_{2} } \right)d_{b} $$

(6)

where $d_{b}$ is given in Sect. 3.2. Specifically, we consider the following two scenarios: (1) When $p_{b} \le 1$, the platform’s profit is $\pi_{e}^{E} = \left( {p_{b} - w_{1} - w_{2} } \right)\frac{1}{2}\left( {2 - p_{b}^{2} } \right)$. By setting $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} = 0$, we have $p_{b} = p_{b1} \equiv \frac{1}{3}\left( {w_{1} + w_{2} - \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)$ or $p_{b} = p_{b2} \equiv \frac{1}{3}\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)$. Note that $p_{b1} < 0$ and $p_{b2} > 0$, and thus $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} > 0$ if $p_{b} \le {\text{min}}\left\{ {p_{b2} ,1} \right\}$, while $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} < 0$ otherwise. Since $p_{b2} \le 1$, and thus $w_{1} + w_{2} \le \frac{1}{2}$. Thus, we can obtain the platform’s local optimal solution, which is as follows: $p_{b}^{*} \left( {w_{1} , w_{2} } \right) = \frac{1}{3}\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)$ if $w_{1} + w_{2} \le \frac{1}{2}$, and $p_{b}^{*} \left( {w_{1} , w_{2} } \right) = 1$ otherwise. (2) When $1 < p_{b} \le 2$, the platform’s profit is $\pi_{e}^{E} = \left( {p_{b} - w_{1} - w_{2} } \right)\frac{1}{2}\left( {2 - p_{b} } \right)^{2}$. By setting $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} = 0$, we have $p_{b} = p_{b1}{\prime} \equiv \frac{{2\left( {1 + w_{1} + w_{2} } \right)}}{3}$ or $p_{b} = p_{b2}{\prime} \equiv 2$. Note that $1 < p_{b1}{\prime} < p_{b2}{\prime}$ if $w_{1} + w_{2} > \frac{1}{2}$, and $p_{b1}{\prime} \le 1 < p_{b2}{\prime}$ if $w_{1} + w_{2} \le \frac{1}{2}$, and thus $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} < 0$ if $p_{b} > \max \left\{ {p_{b1}{\prime} ,1} \right\}$, while $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} > 0$ otherwise. Since $p_{b2}{\prime} \le 1$, and thus $w_{1} + w_{2} \le \frac{1}{2}$. Thus, we can obtain the platform’s local optimal solution, which is as follows: $p_{b}^{*} \left( {w_{1} , w_{2} } \right) = \frac{{2\left( {1 + w_{1} + w_{2} } \right)}}{3}$ if $w_{1} + w_{2} > \frac{1}{2}$, and $p_{b}^{*} \left( {w_{1} , w_{2} } \right) = 1$ otherwise.

By comparing the above two scenarios, we can obtain the platform’s global optimal solution, which is as follows: $p_{b}^{*} \left( {w_{1} , w_{2} } \right) = \frac{1}{3}\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)$ if $w_{1} + w_{2} \le \frac{1}{2}$, and $p_{b}^{*} \left( {w_{1} , w_{2} } \right) = \frac{{2\left( {1 + w_{1} + w_{2} } \right)}}{3}$ otherwise. By substituting the optimal price of bundle price into the demand function in case E, we have $d_{b}^{*} \left( {w_{1} , w_{2} } \right) = \frac{1}{9}\left( {6 - \left( {w_{1} + w_{2} } \right)\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)} \right)$ if $w_{1} + w_{2} \le \frac{1}{2}$ and $d_{b}^{*} \left( {w_{1} , w_{2} } \right) = \frac{2}{9}\left( {w_{1} + w_{2} - 2} \right)^{2}$ otherwise.

In the first stage of the game, the manufacturers set $w_{i}$ to maximize its profit simultaneously:

$$ \mathop {\max }\limits_{{w_{i} }} \pi_{{m_{i} }}^{E} = \left( {w_{i} - c_{i} } \right)d_{b} ,{ } i = 1,2 $$

(7)

Inserting $d_{b}^{*} \left( {w_{1} , w_{2} } \right)$, we have

$$ \pi_{mi}^{E} = \left\{ {\begin{array}{*{20}l} {\frac{1}{9}\left( {w_{i} - c_{i} } \right)\left( {6 - \left( {w_{1} + w_{2} } \right)\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)} \right),} \hfill & { w_{1} + w_{2} \le \frac{1}{2}} \hfill \\ {\frac{2}{9}\left( {w_{i} - c_{i} } \right)\left( {w_{1} + w_{2} - 2} \right)^{2} ,} \hfill & { otherwise} \hfill \\ \end{array} } \right. $$

(8)

Next, we consider the following two scenarios: (1) When $w_{1} + w_{2} \le \frac{1}{2}$, the manufacturer $i$’s profit is $\pi_{mi}^{E} = \frac{1}{9}\left( {w_{i} - c_{i} } \right)\left( {6 - \left( {w_{1} + w_{2} } \right)\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)} \right)$. We have $\begin{aligned} \frac{{d\pi_{mi}^{E} }}{{dw_{i} }} & = \frac{1}{9}\left[ {6 - \left( {w_{1} + w_{2} } \right)\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right) - \frac{{\left( {w_{i} - c_{i} } \right)\left( {w_{1} + w_{2} + \sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{\sqrt {6 + \left( {w_{1} + w_{2} } \right)^{2} } }}} \right] \\ & > \frac{1}{9}\left( {6 - 1.5 - 3.6\left( {w_{i} - c_{i} } \right)} \right) > 0 \\ \end{aligned}$, which implies that manufacturer $m_{i}$’s profit increases in its wholesale price $w_{i}$, so we need to consider the scenario with $w_{1} + w_{2} > \frac{1}{2}$ solely. (2) When $w_{1} + w_{2} > \frac{1}{2}$, the manufacturer $i$’s profit is $\pi_{mi}^{E} = \frac{2}{9}\left( {w_{i} - c_{i} } \right)\left( {w_{1} + w_{2} - 2} \right)^{2}$. Since the second-order condition $\frac{{d^{2} \pi_{mi}^{E} }}{{dw_{i}^{2} }} = - \frac{4}{9}\left( {4 + c_{i} - w_{i} - 2w_{i} - 2w_{3 - i} } \right) < 0$ is satisfied, we can obtain that the manufacturer $i$’s profit function is concave in $w_{i}$. By jointly solving $\frac{{d\pi_{m1}^{E} }}{{dw_{1} }} = \frac{{d\pi_{m2}^{E} }}{{dw_{2} }} = 0$, we have $w_{1}^{*} = \frac{1}{4}\left( {2 + 3c_{1} - c_{2} } \right)$ and $w_{2}^{*} = \frac{1}{4}\left( {2 - c_{1} + 3c_{2} } \right)$. Inserting the expressions of $w_{1}^{*}$ and $w_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case E as summarized in Lemma 1. □

Proof of Lemma 2

In Case M, we can formulate the platform’s and manufacturers’ profit functions and solve the game by backward induction. In the second stage of the game, given a wholesale price $w_{2}$ by the add-on manufacturer, the base good manufacturer’s problem is as follows:

$$ \mathop {\max }\limits_{{p_{b} }} \pi_{m1}^{M} = \left( {\left( {1 - f} \right)p_{b} - w_{2} - c_{1} } \right)d_{b} $$

(9)

where $d_{b}$ is given in Sect. 3.2. Specifically, we consider the following two scenarios: (1) When $p_{b} \le 1$, the base manufacturer’s profit is $\pi_{m1}^{M} = \left( {\left( {1 - f} \right)p_{b} - w_{2} - c_{1} } \right)\frac{1}{2}\left( {2 - p_{b}^{2} } \right)$. By setting $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} = 0$, we have $p_{b} = p_{b1}^{\left( 3 \right)} \equiv \frac{{c_{1} + w_{2} - \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }}{{3\left( {1 - f} \right)}}$ or $p_{b} = p_{b2}^{\left( 3 \right)} \equiv \frac{{c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }}{{3\left( {1 - f} \right)}}$. Note that $p_{b1}^{\left( 3 \right)} < 0$ and $p_{b2}^{\left( 3 \right)} > 0$, and thus $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} > 0$ if $p_{b} \le {\text{min}}\left\{ {p_{b2}^{\left( 3 \right)} ,1} \right\}$, while $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} < 0$ otherwise. Since $p_{b2}^{\left( 3 \right)} \le 1$, and thus $w_{2} \le \frac{1 - f}{2} - c_{1}$. Thus, we can obtain the base manufacturer’s local optimal solution, which is as follows: $p_{b}^{*} \left( { w_{2} } \right) = \frac{{c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }}{{3\left( {1 - f} \right)}}$ if $w_{2} \le \frac{1 - f}{2} - c_{1}$, and $p_{b}^{*} \left( { w_{2} } \right) = 1$ otherwise. (2) When $1 < p_{b} \le 2$, the base manufacturer’s profit is $\pi_{m1}^{M} = \left( {\left( {1 - f} \right)p_{b} - w_{2} - c_{1} } \right)\frac{1}{2}\left( {2 - p_{b} } \right)^{2}$. By setting $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} = 0$, we have $p_{b} = p_{b1}^{\left( 4 \right)} \equiv \frac{{2\left( {1 - f + c_{1} + w_{2} } \right)}}{{3\left( {1 - f} \right)}}$ or $p_{b} = p_{b2}^{\left( 4 \right)} \equiv 2$. Note that $1 < p_{b1}^{\left( 4 \right)} < p_{b2}^{\left( 4 \right)}$ if $w_{2} > \frac{1 - f}{2} - c_{1}$, and $p_{b1}^{\left( 4 \right)} \le 1 < p_{b2}^{\left( 4 \right)}$ if $w_{2} \le \frac{1 - f}{2} - c_{1}$. Thus, $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} < 0$ if $p_{b} > \max \left\{ {p_{b1}^{\left( 4 \right)} ,1} \right\}$ while $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} > 0$ otherwise. Since $w_{2} \le \frac{1 - f}{2} - c_{1}$ given $p_{b2}^{\left( 4 \right)} \le 1$, hence we can obtain the base manufacturer’s local optimal solution, which is as follows: $p_{b}^{*} \left( { w_{2} } \right) = \frac{{2\left( {1 - f + c_{1} + w_{2} } \right)}}{{3\left( {1 - f} \right)}}$ if $w_{2} > \frac{1 - f}{2} - c_{1}$, and $p_{b}^{*} \left( { w_{2} } \right) = 1$ otherwise.

Then, by comparing the two scenarios, we can obtain the base manufacturer’s global optimal solution, which is as follows: $p_{b}^{*} \left( { w_{2} } \right) = \frac{{c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }}{{3\left( {1 - f} \right)}}$ if $w_{2} \le \frac{1 - f}{2} - c_{1}$, and $p_{b}^{*} \left( { w_{2} } \right) = \frac{{2\left( {1 - f + c_{1} + w_{2} } \right)}}{{3\left( {1 - f} \right)}}$ otherwise. By substituting the optimal price of bundle price into the demand function in case M, we have $d_{b}^{*} \left( { w_{2} } \right) = 1 - \frac{{\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{18\left( {1 - f} \right)^{2} }}$ if $w_{2} \le \frac{1 - f}{2} - c_{1}$ and $d_{b}^{*} \left( { w_{2} } \right) = \frac{{2\left( {2 - 2f - c_{1} - w_{2} } \right)^{2} }}{{9\left( {1 - f} \right)^{2} }}$ otherwise.

In the first stage of the game, the add-on manufacturer set $w_{2}$ to maximize its profit as follows:

$$ \mathop {\max }\limits_{2} \pi_{m2}^{M} = \left( {w_{2} - c_{2} } \right)d_{b} $$

(10)

Inserting $d_{b}^{*} \left( {w_{2} } \right)$, we have

$$ \pi_{m2}^{M} = \left\{ {\begin{array}{*{20}l} {\left( {w_{2} - c_{2} } \right)\left( {1 - \frac{{\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{18\left( {1 - f} \right)^{2} }}} \right), } \hfill & { w_{2} \le \frac{1 - f}{2} - c_{1} } \hfill \\ {\left( {w_{2} - c_{2} } \right)\frac{{2\left( {2 - 2f - c_{1} - w_{2} } \right)^{2} }}{{9\left( {1 - f} \right)^{2} }},} \hfill & {otherwise} \hfill \\ \end{array} } \right. $$

(11)

Next, we consider the following two scenarios: (1) When $w_{2} \le \frac{1 - f}{2} - c_{1}$, the add-on manufacturer’s profit is $\pi_{m2}^{M} = \left( {w_{2} - c_{2} } \right)\left( {1 - \frac{{\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{18\left( {1 - f} \right)^{2} }}} \right)$, we have $\begin{aligned} \frac{{d\pi_{m2}^{M} }}{{dw_{2} }} & = 1 - \frac{{\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{18\left( {1 - f} \right)^{2} }} - \frac{{\left( {w_{2} - c_{2} } \right)\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{9\left( {1 - f} \right)^{2} \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }} \\ & \ge 1{-}\frac{{\left( {c_{1}{+} w_{2}{+}\sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{18\left( {1 - f} \right)^{2} }} - \frac{{\left( {\frac{1 - f}{2} - c_{1} - c_{2} } \right)\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{9\left( {1 - f} \right)^{2} \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }} \\ & > 1 - \frac{{\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{18\left( {1 - f} \right)^{2} }}\\ &\quad - \frac{{\left( {\frac{1 - f}{2} - c_{1} - w_{2} } \right)\left( {c_{1} + w_{2} + \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } } \right)^{2} }}{{9\left( {1 - f} \right)^{2} \sqrt {6\left( {1 - f} \right)^{2} + \left( {c_{1} + w_{2} } \right)^{2} } }}> 0 \\ \end{aligned}$ on the parameter regions of $f \in \left[ {0,1} \right]$ and $c_{1} + w_{2} \in \left[ {0,\frac{1 - f}{2}} \right]$, which implies that manufacturer $m_{2}$’s profit increases in its wholesale price $w_{2}$, so we only need to consider the scenario with $w_{2} > \frac{1 - f}{2} - c_{1}$. (2) When $w_{2} > \frac{1 - f}{2} - c_{1}$, the add-on manufacturer’s profit is $\pi_{m2}^{M} = \left( {w_{2} - c_{2} } \right)\frac{{2\left( {2 - 2f - c_{1} - w_{2} } \right)^{2} }}{{9\left( {1 - f} \right)^{2} }}$. Since the second-order condition $\frac{{d^{2} \pi_{m2}^{M} }}{{dw_{2}^{2} }} = - \frac{{4\left( {4 - 2c_{1} + c_{2} - 4f - 3w_{2} } \right)}}{{9\left( {1 - f} \right)^{2} }} < 0$ is satisfied, we can obtain the add-on manufacturer’s profit function is concave in $w_{2}$. By solving $\frac{{d\pi_{m2}^{M} }}{{dw_{2} }} = 0$, we have $w_{2}^{*} = \frac{{2\left( {1 - f} \right) - c_{1} + 2c_{2} }}{3}$. Inserting the expression of $w_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case M as summarized in Lemma 2. □

Proof of Proposition 1

We first compare the price of bundled products in cases E and M, we have $p_{b}^{E*} \ge p_{b}^{M*}$ if $f \le \frac{{2 - \left( {c_{1} + c_{2} } \right)}}{{2 + 3\left( {c_{1} + c_{2} } \right)}}$; otherwise, $p_{b}^{E*} < p_{b}^{M*}$. We now compare the demand of bundled products in cases E and M, we have $d_{b}^{E*} - d_{b}^{M*} = - \frac{{\left( {\left( {2 + 3\left( {c_{1} + c_{2} } \right)} \right)f - 2 + c_{1} + c_{2} } \right)\left( {\left( {14 - 3\left( {c_{1} + c_{2} } \right)} \right)f - 7\left( {2 - \left( {c_{1} + c_{2} } \right)} \right)} \right)}}{{162\left( {1 - f} \right)^{2} }} > 0$ if $\frac{{2 - c_{1} - c_{2} }}{{2 + 3\left( {c_{1} + c_{2} } \right)}} < f < \frac{{7\left( {2 - c_{1} - c_{2} } \right)}}{{14 - 3\left( {c_{1} + c_{2} } \right)}}$; otherwise, $d_{b}^{E*} - d_{b}^{M*} \le 0$. Hence, we have the result in Proposition 1. □

Proof of Proposition 2

In this proposition, we focus on the impact of different bundling strategies on the base manufacturer’s profit. By comparing the base manufacturer’s profit in cases E and M, we have $\pi_{m1}^{M*} > \pi_{{m_{1} }}^{E*}$ if $0 < f < \frac{47}{{128}}$ and $0 < c_{1} + c_{2} < \frac{{2\left( {8\left( {1 - f} \right) - \sqrt[3]{{3\left( {32\left( {1 - f} \right)} \right)^{2} }}} \right)}}{{8 - \sqrt[3]{{3\left( {32\left( {1 - f} \right)} \right)^{2} }}}}$. Hence, we have the result in Proposition 2. □

Proof of Proposition 3

In this proposition, we focus on the impact of different bundling strategies on the platform’s profit. For analysis, we denote $h\left( c \right) = - \left[ {12c\left( { - 1 + f} \right)^{2} \left( { - 27 + 91f} \right) + 8\left( { - 1 + f} \right)^{3} \left( { - 27 + 160f} \right) + c^{3} \left( { - 27 - 47f - 81f^{2} + 27f^{3} } \right) - 6c^{2} \left( { - 27 + 49f - 49f^{2} + 27f^{3} } \right)} \right]$, and $c = c_{1} + c_{2}$. By comparing the platform’s profits in cases E and M, we have $\pi_{e}^{M*} - \pi_{e}^{E*} = \frac{h\left( c \right)}{{2916\left( {1 - f} \right)^{3} }}$, and denote $g\left( c \right) = h^{\prime}\left( c \right) = - 12\left( { - 1 + f} \right)^{2} \left( { - 27 + 91f} \right) - 3c^{2} \left( { - 27 - 47f - 81f^{2} + 27f^{3} } \right) + 12c\left( { - 27 + 49f - 49f^{2} + 27f^{3} } \right)$. We have $g\left. {\left( c \right)} \right|_{c = 0} \ge 0$ if $f \le \frac{1}{3}$, $g\left. {\left( c \right)} \right|_{c = 0} < 0$ if $f > \frac{1}{3}$, and $g\left. {\left( c \right)} \right|_{c = 2} > 0$. Thus, we can obtain that (1) when $f \le \frac{1}{3}$, there exist two thresholds $C_{1}$ and $C_{2}$, so that $g\left( c \right) < 0$ if $C_{1} < c < C_{2}$, otherwise, $g\left( c \right) \ge 0$; when $f > \frac{1}{3}$, there exists a threshold $C_{2}$, so that $g\left( c \right) < 0$ if $c < C_{2}$, otherwise, $g\left( c \right) \ge 0$. Thus, we can obtain that: (1) when $f \le \frac{1}{3}$, with $c$ increases, $h\left( c \right)$ will increase at first, and then will decrease, and finally will increase; (2) when $f > \frac{1}{3}$, with $c$ increases, $h\left( c \right)$ will decrease at first and then will increase.

Moreover, we have $h\left. {\left( c \right)} \right|_{c = 0} > 0$ if $f > \frac{27}{{160}}$, $h\left. {\left( c \right)} \right|_{c = 0} \le 0$ if $f \le \frac{27}{{160}}$, $h\left. {\left( c \right)} \right|_{{c = C_{1} }} > 0$, $h\left. {\left( c \right)} \right|_{{c = C_{2} }} < 0$, and $h\left. {\left( c \right)} \right|_{c = 2} > 0$. Thus, we can conclude that: (1) when $f \le \frac{27}{{160}}$, there exists three thresholds $C_{1}$, $C_{2}$, and $C_{3}$, so that $h\left( c \right) > 0$ if $C_{1} < c < C_{2}$ and $c > C_{3}$; otherwise, $h\left( c \right) \le 0$. (2) when $f > \frac{27}{{160}}$, there exists two thresholds $C_{2}$ and $C_{3}$, so that $h\left( c \right) > 0$ if $c < C_{2}$ and $c > C_{3}$; otherwise, $h\left( c \right) \le 0$. Where $C_{1}$, $C_{2}$, $C_{3}$ are solutions of $\pi_{e}^{M*} - \pi_{e}^{E*} = 0$. Hence, we have the result in Proposition 3. □

Proof of Lemma 3

In Case A, given a commission rate $f$, the manufacturers’ problems are as follows:

$$ \mathop {\max }\limits_{{p_{1} }} \pi_{m1}^{A} = \left( {\left( {1 - f} \right)p_{1} - c_{1} } \right)\left( {1 - p_{1} } \right) $$

(12)

$$ \mathop {\max }\limits_{{p_{2} }} \pi_{m2}^{A} = \left( {\left( {1 - f} \right)p_{2} - c_{2} } \right)\left( {1 - p_{1} } \right)\left( {1 - p_{2} } \right) $$

(13)

By jointly solving $\frac{{d\pi_{m1}^{A} }}{{dp_{1} }} = \frac{{d\pi_{m2}^{A} }}{{dp_{2} }} = 0$, we have $p_{1}^{*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}, p_{2}^{*} = \frac{{1 - f + c_{2} }}{{2\left( {1 - f} \right)}}$. Inserting the expression of $p_{1}^{*}$ and $p_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case A as summarized in Lemma 3. □

Proof of Proposition 4

Since $C_{i}$ is not analytically tractable, we resort to numerical analysis to derive the equilibrium configuration presented in Fig. 4. □

Proof of Proposition 5

Proposition 5 can be derived by using a similar logic in Proposition 1, and thus we omit it. □

Proof of Proposition 6

In Case E, we can formulate the platform’s and manufacturers’ profit functions and solve the game by backward induction. In the second stage of the game, given a wholesale price $w_{i}$, the platform’s problem is as follows:

$$ \mathop {\max }\limits_{{p_{b} }} \pi_{e}^{E} = \left( {p_{b} - w_{1} - w_{2} } \right)d_{b} $$

(14)

where $d_{b} = 1 - \frac{{p_{b} - l}}{1 + h - l}$. By setting $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} = 0$, we have $p_{b} \left( {w_{1} , w_{2} } \right) = \frac{1}{2}\left( {1 + h + w_{1} + w_{2} } \right)$. In the first stage of the game, the manufacturers set $w_{i}$ to maximize its profit simultaneously:

$$ \mathop {\max }\limits_{{w_{i} }} \pi_{{m_{i} }}^{E} = \left( {w_{i} - c_{i} } \right)d_{b} ,{ } i = 1,2 $$

(15)

Inserting $p_{b} \left( {w_{1} , w_{2} } \right)$, since the second-order condition $\frac{{d^{2} \pi_{mi}^{E} }}{{dw_{i}^{2} }} < 0$ is satisfied, we can obtain that the manufacturer $i$’s profit function is concave in $w_{i}$. By jointly solving $\frac{{d\pi_{m1}^{E} }}{{dw_{1} }} = \frac{{d\pi_{m2}^{E} }}{{dw_{2} }} = 0$, we have $w_{1}^{*} = \frac{1}{3}\left( {1 + h + 2c_{1} - c_{2} } \right)$ and $w_{2}^{*} = \frac{1}{3}\left( {1 + h - c_{1} + 2c_{2} } \right)$. Inserting the expressions of $w_{1}^{*}$ and $w_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case E as summarized in Lemma 4.

Lemma 4

Under case E, the firms’ optimal pricing decisions and product demands are as follows: $w_{i}^{E*} = \frac{{\left( {1 + h + 2c_{i} - c_{3 - i} } \right)}}{3}$, $p_{b}^{E*} = \frac{{5\left( {1 + h} \right) + c_{1} + c_{2} }}{6}$, $d_{b}^{E*} = \frac{{1 + h - c_{1} - c_{2} }}{{6\left( {1 + h - l} \right)}}$; the firms’ optimal profits are as follows: $\pi_{e}^{E*} = \frac{{\left( {1 + h - c_{1} - c_{2} } \right)^{2} }}{{36\left( {1 + h - l} \right)}}$ and $\pi_{{m_{i} }}^{E*} = \frac{{\left( {1 + h - c_{1} - c_{2} } \right)^{2} }}{{18\left( {1 + h - l} \right)}}$.

In Case M, we can formulate the platform’s and manufacturers’ profit functions and solve the game by backward induction. In the second stage of the game, given a wholesale price $w_{2}$ by the add-on manufacturer, the base good manufacturer’s problem is as follows:

$$ \mathop {\max }\limits_{{p_{b} }} \pi_{m1}^{M} = \left( {\left( {1 - f} \right)p_{b} - w_{2} - c_{1} } \right)d_{b} $$

(16)

where $d_{b} = 1 - \frac{{p_{b} - l}}{1 + h - l}$. By setting $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} = 0$, we have $p_{b} \left( { w_{2} } \right) = \frac{{\left( {1 - f} \right)\left( {1 + h} \right) + c_{1} + w_{2} }}{{2\left( {1 - f} \right)}}$. In the first stage of the game, the add-on manufacturer set $w_{2}$ to maximize its profit as follows:

$$ \mathop {\max }\limits_{2} \pi_{m2}^{M} = \left( {w_{2} - c_{2} } \right)d_{b} $$

(17)

Inserting $d_{b}^{*} \left( {w_{2} } \right)$, since the second-order condition $\frac{{d^{2} \pi_{m2}^{M} }}{{dw_{2}^{2} }} = - \frac{1}{{\left( {1 - f} \right)\left( {1 + h - l} \right)}} < 0$ is satisfied, we can obtain the add-on manufacturer’s profit function is concave in $w_{2}$. By solving $\frac{{d\pi_{m2}^{M} }}{{dw_{2} }} = 0$, we have $w_{2}^{*} = \frac{{\left( {1 - f} \right)\left( {1 + h} \right) - c_{1} + c_{2} }}{2}$. Inserting the expression of $w_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case M as summarized in Lemma 5.

Lemma 5

Under case M, the firms’ optimal pricing decisions and product demands are as follows: $w_{2}^{M*} = \frac{{\left( {1 - f} \right)\left( {1 + h} \right) - c_{1} + c_{2} }}{2}$, $p_{b}^{M*} = \frac{{3\left( {1 - f} \right)\left( {1 + h} \right) + c_{1} + c_{2} }}{{4\left( {1 - f} \right)}}$, $d_{b}^{M*} = \frac{{\left( {1 - f} \right)\left( {1 + h} \right) - \left( {c_{1} + c_{2} } \right)}}{{4\left( {1 - f} \right)\left( {1 + h - l} \right)}}$; the firms’ optimal profits are as follows: $\pi_{e}^{M*} = \frac{{f\left( {\left( {1 - f} \right)\left( {1 + h} \right) - \left( {c_{1} + c_{2} } \right)} \right)\left( {3\left( {1 - f} \right)\left( {1 + h} \right) + \left( {c_{1} + c_{2} } \right)} \right)}}{{16\left( {1 - f} \right)^{2} \left( {1 + h - l} \right)}}$, $\pi_{m1}^{M*} = \frac{{\left( {\left( {1 - f} \right)\left( {1 + h} \right) - \left( {c_{1} + c_{2} } \right)} \right)^{2} }}{{16\left( {1 - f} \right)\left( {1 + h - l} \right)}}$, $\pi_{m2}^{M*} = \frac{{\left( {\left( {1 - f} \right)\left( {1 + h} \right) - \left( {c_{1} + c_{2} } \right)} \right)^{2} }}{{8\left( {1 - f} \right)\left( {1 + h - l} \right)}}$.

In Case A, given a commission rate $f$, the manufacturers’ problems are as follows:

$$ \mathop {\max }\limits_{{p_{1} }} \pi_{m1}^{A} = \left( {\left( {1 - f} \right)p_{1} - c_{1} } \right)\left( {1 - p_{1} } \right) $$

(18)

$$ \mathop {\max }\limits_{{p_{2} }} \pi_{m2}^{A} = \left( {\left( {1 - f} \right)p_{2} - c_{2} } \right)\left( {1 - \max \left\{ {\frac{{p_{2} - l}}{h - l},p_{1} } \right\}} \right) $$

(19)

By jointly solving $\frac{{d\pi_{m1}^{A} }}{{dp_{1} }} = 0$, we have $p_{1}^{*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$. We consider the following two scenarios: (1) When $p_{2} \le l + \left( {h - l} \right)p_{1} \to p_{1} \ge \frac{{p_{2} - l}}{h - l}$, we have $\pi_{m2}^{A} = \left( {\left( {1 - f} \right)p_{2} - c_{2} } \right)\left( {1 - p_{1} } \right)$, so $p_{2}^{*\left( 1 \right)} = l + \left( {h - l} \right){ }p_{1}^{*} = \frac{{\left( {1 - f} \right)\left( {h + l} \right) + \left( {h - l} \right)c_{1} }}{{2\left( {1 - f} \right)}}$, inserting $p_{2}^{*\left( 1 \right)}$ we have $\pi_{m2}^{A*\left( 1 \right)} = \frac{{\left( {1 - f - c_{1} } \right)\left( {\left( {1 - f + c_{1} } \right)h + \left( {1 - f - c_{1} } \right)l - 2c_{2} } \right)}}{{4\left( {1 - f} \right)}}$. (2) When $p_{2} \ge l + \left( {h - l} \right)p_{1}$, we have $\pi_{m2}^{A} = \left( {\left( {1 - f} \right)p_{2} - c_{2} } \right)\left( {1 - \frac{{p_{2} - l}}{h - l}} \right)$. By solving $\frac{{d\pi_{m2}^{A} }}{{dp_{2} }} = 0$, we have $p_{2}^{*\left( 2 \right)} = \frac{1}{2}\left( {\frac{{c_{2} }}{1 - f} + h} \right)$, inserting $p_{2}^{*\left( 2 \right)}$ we have $\pi_{m2}^{A*\left( 2 \right)} = \frac{{\left( {\left( {1 - f} \right)h - c_{2} } \right)^{2} }}{{4\left( {1 - f} \right)\left( {h - l} \right)}}$. Since $\pi_{m2}^{A*\left( 2 \right)} - \pi_{m2}^{A*\left( 1 \right)} = \frac{{\left( {c_{2} - hc_{1} - \left( {1 - f - c_{1} } \right)l} \right)^{2} }}{{4\left( {1 - f} \right)\left( {h - l} \right)}} \ge 0$, hence the scenario (2) dominates scenario (1), so we have $p_{1}^{*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$ and $p_{2}^{*} = \frac{1}{2}\left( {\frac{{c_{2} }}{1 - f} + h} \right)$ in equilibrium. Inserting the expression of $p_{1}^{*}$ and $p_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case A as summarized in Lemma 6.

Lemma 6

Under case A, the firms’ optimal pricing decisions and product demands are as follows: $p_{1}^{A*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$, $p_{2}^{A*} = \frac{1}{2}\left( {\frac{{c_{2} }}{1 - f} + h} \right)$, $d_{1}^{A*} = \frac{{1 - f - c_{1} }}{{2\left( {1 - f} \right)}}$, $d_{2}^{A*} = \frac{{\left( {1 - f} \right)h - c_{2} }}{{2\left( {1 - f} \right)\left( {h - l} \right)}}$; the firms’ optimal profits are as follows: $\pi_{e}^{A*} = \frac{{f\left( {\left( {1 - f} \right)^{2} h\left( {1 + h} \right) - \left( {1 - f + c_{1} } \right)\left( {1 - f - c_{1} } \right)l - c_{2}^{2} - c_{1}^{2} h} \right)}}{{4\left( {1 - f} \right)^{2} \left( {h - l} \right)}},$ $\pi_{m1}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)^{2} }}{{4\left( {1 - f} \right)}}$, and $\pi_{m2}^{A*} = \frac{{\left( {\left( {1 - f} \right)h - c_{2} } \right)^{2} }}{{4\left( {1 - f} \right)\left( {h - l} \right)}}$.

Let $c = c_{1} + c_{2}$, we have $\pi_{e}^{M*} - \pi_{e}^{E*} = \frac{1}{{144\left( {1 + h - l} \right)}}\left( { - \left( {4 + \frac{9f}{{\left( {1 - f} \right)^{2} }}} \right)c^{2} + \frac{{2\left( {4 - 13f} \right)\left( {1 + h} \right)}}{1 - f}c - \left( {4 - 27f} \right)\left( {1 + h} \right)^{2} } \right) = - \frac{{\left( {1 + h} \right)^{2} }}{{144\left( {1 + h - l} \right)}}\left( {\frac{c}{1 + h} - \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 - 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}} \right)\left( {\frac{c}{1 + h} - \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}} \right)$, one can obtain that $\pi_{e}^{M*} \ge \pi_{e}^{E*}$ if $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} \le \frac{c}{1 + h} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 - 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$, and $\pi_{e}^{M*} \le \pi_{e}^{E*}$ otherwise.

Similarily, we have $\pi_{m1}^{M*} - \pi_{m1}^{E*} = \frac{1}{{144 \left( {1 + h - l} \right)}}\left( { \left( {\frac{9}{1 - f} - 8} \right)c^{2} - 2\left( {1 + h} \right)c + \left( {1 - 9f} \right)\left( {1 + h} \right)^{2} } \right) = \frac{{\left( {1 + h} \right)^{2} }}{{144 \left( {1 + h - l} \right)}}\left( {\frac{c}{1 + h} - \frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}} \right)\left( {\frac{c}{1 + h} - \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}} \right)$, one can obtain that $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$ if $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} \le \frac{c}{1 + h} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, and $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$ otherwise.

Similarily, we have $\pi_{m2}^{M*} - \pi_{m2}^{E*} = \frac{1}{{72\left( {1 + h - l} \right)}}\left( { \left( {\frac{9}{1 - f} - 4} \right)c^{2} - 10\left( {1 + h} \right)c + \left( {5 - 9f} \right)\left( {1 + h} \right)^{2} } \right) = \frac{{\left( {1 + h} \right)^{2} }}{{72\left( {1 + h - l} \right)}}\left( {\frac{c}{1 + h} - \frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f}} \right)\left( {\frac{c}{1 + h} - \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}} \right)$, one can obtain that $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$ if $\frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f} \le \frac{c}{1 + h} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$ otherwise.

Given $0 \le f \le 1$ and $0 \le \frac{c}{1 + h} \le 1$, we have $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} \le \frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, on can easily obtain that: (1). when $0 \le \frac{c}{1 + h} \le \frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (2). when $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} < \frac{c}{1 + h} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (3). when $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{c}{1 + h} \le \frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f}$, such that $\pi_{e}^{M*} \ge \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (4). when $\frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f} < \frac{c}{1 + h} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$, such that $\pi_{e}^{M*} \ge \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$; (5). when $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{c}{1 + h} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$; (6). when $\frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f} < \frac{c}{1 + h} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (7). when $\frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} < \frac{c}{1 + h} \le 1$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$. Hence, we have the result in Proposition 6. □

Proof of Proposition 7

Since $\pi_{m1}^{E*} - \pi_{m1}^{A*} = \frac{{\left( {1 + h - c_{1} - c_{2} } \right)^{2} }}{{18\left( {1 + h - l} \right)}} - \frac{{\left( {1 - f - c_{1} } \right)^{2} }}{{4\left( {1 - f} \right)}}$, one can obtain that $\pi_{m1}^{E*} \le \pi_{m1}^{A*}$ if $c_{2} \ge 1 + h + \left( {\frac{3}{2}\sqrt {\frac{{2\left( {1 + h - l} \right)}}{1 - f}} - 1} \right)c_{1} - \frac{3}{2}\left( {1 - f} \right)\sqrt {\frac{{2\left( {1 + h - l} \right)}}{1 - f}}$, and $\pi_{m1}^{E*} > \pi_{m1}^{A*}$ otherwise. Since $\pi_{m1}^{M*} - \pi_{m1}^{A*} = \frac{1}{{16\left( {1 - f} \right)}}\left( {\frac{{\left( {\left( {1 - f} \right)\left( {1 + h} \right) - c_{1} - c_{2} } \right)^{2} }}{1 + h - l} - 4\left( {1 - f - c_{1} } \right)^{2} } \right)$, one can obtain that $\pi_{m1}^{M*} \le \pi_{m1}^{A*}$ if $c_{2} \ge \left( {1 - f} \right)\left( {1 + h} \right) + \left( {2\sqrt {1 + h - l} - 1} \right)c_{1} - 2\left( {1 - f} \right)\sqrt {1 + h - l}$, and $\pi_{m1}^{M*} > \pi_{m1}^{A*}$ otherwise. So, we have $\pi_{m1}^{E*} \le \pi_{m1}^{A*}$ and $\pi_{m1}^{M*} \le \pi_{m1}^{A*}$ if $c_{2} \ge Max\left\{ {1 + h + \left( {\frac{3}{2}\sqrt {\frac{{2\left( {1 + h - l} \right)}}{1 - f}} - 1} \right)c_{1} - \frac{3}{2}\left( {1 - f} \right)\sqrt {\frac{{2\left( {1 + h - l} \right)}}{1 - f}} ,\left( {1 - f} \right)\left( {1 + h} \right) + \left( {2\sqrt {1 + h - l} - 1} \right)c_{1} - 2\left( {1 - f} \right)\sqrt {1 + h - l} } \right\}$. Hence, we have the result in Proposition 7. □

Proof of Proposition 8

In Case E, we can formulate the platform’s and manufacturers’ profit functions and solve the game by backward induction. In the second stage of the game, given a wholesale price $w_{i}$, the platform’s problem is as follows:

$$ \mathop {\max }\limits_{{p_{b} }} \pi_{e}^{E} = \left( {p_{b} - w_{1} - w_{2} } \right)d_{b} $$

(20)

where $d_{b} = 1 - \frac{{p_{b} - h}}{1 + l - h}$. By setting $\frac{{d\pi_{e}^{E} }}{{dp_{b} }} = 0$, we have $p_{b} \left( {w_{1} , w_{2} } \right) = \frac{1}{2}\left( {1 + l + w_{1} + w_{2} } \right)$. In the first stage of the game, the manufacturers set $w_{i}$ to maximize its profit simultaneously:

$$ \mathop {\max }\limits_{{w_{i} }} \pi_{{m_{i} }}^{E} = \left( {w_{i} - c_{i} } \right)d_{b} ,{ } i = 1,2 $$

(21)

Inserting $p_{b} \left( {w_{1} , w_{2} } \right)$, since the second-order condition $\frac{{d^{2} \pi_{mi}^{E} }}{{dw_{i}^{2} }} < 0$ is satisfied, we can obtain that the manufacturer $i$’s profit function is concave in $w_{i}$. By jointly solving $\frac{{d\pi_{m1}^{E} }}{{dw_{1} }} = \frac{{d\pi_{m2}^{E} }}{{dw_{2} }} = 0$, we have $w_{1}^{*} = \frac{1}{3}\left( {1 + l + 2c_{1} - c_{2} } \right)$ and $w_{2}^{*} = \frac{1}{3}\left( {1 + l - c_{1} + 2c_{2} } \right)$. Inserting the expressions of $w_{1}^{*}$ and $w_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case E as summarized in Lemma 7.

Lemma 7

Under case E, the firms’ optimal pricing decisions and product demands are as follows: $w_{i}^{E*} = \frac{{\left( {1 + l + 2c_{i} - c_{3 - i} } \right)}}{3}$, $p_{b}^{E*} = \frac{{5\left( {1 + l} \right) + c_{1} + c_{2} }}{6}$, $d_{b}^{E*} = \frac{{1 + l - c_{1} - c_{2} }}{{6\left( {1 - h + l} \right)}}$; the firms’ optimal profits are as follows: $\pi_{e}^{E*} = \frac{{\left( {1 + l - c_{1} - c_{2} } \right)^{2} }}{{36\left( {1 - h + l} \right)}}$ and $\pi_{{m_{i} }}^{E*} = \frac{{\left( {1 + l - c_{1} - c_{2} } \right)^{2} }}{{18\left( {1 - h + l} \right)}}$.

In Case M, we can formulate the platform’s and manufacturers’ profit functions and solve the game by backward induction. In the second stage of the game, given a wholesale price $w_{2}$ by the add-on manufacturer, the base good manufacturer’s problem is as follows:

$$ \mathop {\max }\limits_{{p_{b} }} \pi_{m1}^{M} = \left( {\left( {1 - f} \right)p_{b} - w_{2} - c_{1} } \right)d_{b} $$

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where $d_{b} = 1 - \frac{{p_{b} - h}}{1 + l - h}$. By setting $\frac{{d\pi_{m1}^{M} }}{{dp_{b} }} = 0$, we have $p_{b} \left( { w_{2} } \right) = \frac{{\left( {1 - f} \right)\left( {1 + l} \right) + c_{1} + w_{2} }}{{2\left( {1 - f} \right)}}$. In the first stage of the game, the add-on manufacturer set $w_{2}$ to maximize its profit as follows:

$$ \mathop {\max }\limits_{2} \pi_{m2}^{M} = \left( {w_{2} - c_{2} } \right)d_{b} $$

(23)

Inserting $d_{b}^{*} \left( {w_{2} } \right)$, since the second-order condition $\frac{{d^{2} \pi_{m2}^{M} }}{{dw_{2}^{2} }} = - \frac{1}{{\left( {1 - f} \right)\left( {1 - h + l} \right)}} < 0$ is satisfied, we can obtain the add-on manufacturer’s profit function is concave in $w_{2}$. By solving $\frac{{d\pi_{m2}^{M} }}{{dw_{2} }} = 0$, we have $w_{2}^{*} = \frac{{\left( {1 - f} \right)\left( {1 + l} \right) - c_{1} + c_{2} }}{2}$. Inserting the expression of $w_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case M as summarized in Lemma 8.

Lemma 8

Under case M, the firms’ optimal pricing decisions and product demands are as follows: $w_{2}^{M*} = \frac{{\left( {1 - f} \right)\left( {1 + l} \right) - c_{1} + c_{2} }}{2}$, $p_{b}^{M*} = \frac{{3\left( {1 - f} \right)\left( {1 + l} \right) + c_{1} + c_{2} }}{{4\left( {1 - f} \right)}}$, $d_{b}^{M*} = \frac{{\left( {1 - f} \right)\left( {1 + l} \right) - \left( {c_{1} + c_{2} } \right)}}{{4\left( {1 - f} \right)\left( {1 - h + l} \right)}}$; the firms’ optimal profits are as follows: $\pi_{e}^{M*} = \frac{{f\left( {4\left( {1 - f} \right)^{2} \left( {1 + l} \right)^{2} - \left( {c_{1} + c_{2} + \left( {1 - f} \right)\left( {1 + l} \right)} \right)^{2} } \right)}}{{16\left( {1 - f} \right)^{2} \left( {1 - h + l} \right)}}$, $\pi_{m1}^{M*} = \frac{{\left( {\left( {1 - f} \right)\left( {1 + l} \right) - \left( {c_{1} + c_{2} } \right)} \right)^{2} }}{{16\left( {1 - f} \right)\left( {1 - h + l} \right)}}$, $\pi_{m2}^{M*} = \frac{{\left( {\left( {1 - f} \right)\left( {1 + l} \right) - \left( {c_{1} + c_{2} } \right)} \right)^{2} }}{{8\left( {1 - f} \right)\left( {1 - h + l} \right)}}$.

In Case A, given a commission rate $f$, the manufacturers’ problems are as follows:

$$ \mathop {\max }\limits_{{p_{1} }} \pi_{m1}^{A} = \left( {\left( {1 - f} \right)p_{1} - c_{1} } \right)\left( {1 - p_{1} } \right) $$

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$$ \mathop {\max }\limits_{{p_{2} }} \pi_{m2}^{A} = \left( {\left( {1 - f} \right)p_{2} - c_{2} } \right)\max \left\{ {\frac{{h - p_{2} }}{h - l} - p_{1} ,0} \right\} $$

(25)

By jointly solving $\frac{{d\pi_{m1}^{A} }}{{dp_{1} }} = 0$, we have $p_{1}^{*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$. We consider the following two scenarios: (1) When $p_{2} \le h - \left( {h - l} \right)p_{1} \to p_{1} \le \frac{{h - p_{2} }}{h - l}$, we have $\pi_{m2}^{A*\left( 1 \right)} = 0$. (2) When $p_{2} > h - \left( {h - l} \right)p_{1}$, we have $\pi_{m2}^{A} = \left( {\left( {1 - f} \right)p_{2} - c_{2} } \right)\left( {\frac{{h - p_{2} }}{h - l} - p_{1} } \right)$. By solving $\frac{{d\pi_{m2}^{A} }}{{dp_{2} }} = 0$, we have $p_{2}^{*} = \frac{1}{2}\left( {\frac{{c_{2} }}{1 - f} + h - \left( {h - l} \right)p_{1}^{*} } \right) = \frac{{\left( {1 - f - c_{1} } \right)h + \left( {1 - f + c_{1} } \right)l + 2c_{2} }}{{4\left( {1 - f} \right)}}$, inserting $p_{1}^{*}$ and $p_{2}^{*}$ we have $\pi_{m2}^{A*\left( 2 \right)} = \frac{{\left( {\left( {1 - f - c_{1} } \right)h + \left( {1 - f + c_{1} } \right)l - 2c_{2} } \right)^{2} }}{{16\left( {1 - f} \right)\left( {h - l} \right)}} > 0$, hence $\pi_{m2}^{A*\left( 2 \right)} > \pi_{m2}^{A*\left( 1 \right)}$; therefore, the scenario (2) dominates scenario (1), so we have $p_{1}^{*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$ and $p_{2}^{*} = \frac{{\left( {1 - f - c_{1} } \right)h + \left( {1 - f + c_{1} } \right)l + 2c_{2} }}{{4\left( {1 - f} \right)}}$ in equilibrium. Inserting the expression of $p_{1}^{*}$ and $p_{2}^{*}$, we then get the firm’s optimal decisions, product demands, and firms’ profits under Case A as summarized in Lemma 9.

Lemma 9

Under case A, the firms’ optimal pricing decisions and product demands are as follows: $p_{1}^{A*} = \frac{{1 - f + c_{1} }}{{2\left( {1 - f} \right)}}$, $p_{2}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)h + \left( {1 - f + c_{1} } \right)l + 2c_{2} }}{{4\left( {1 - f} \right)}}$, $d_{1}^{A*} = \frac{{1 - f - c_{1} }}{{2\left( {1 - f} \right)}}$, $d_{2}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)h + \left( {1 - f + c_{1} } \right)l - 2c_{2} }}{{4\left( {1 - f} \right)\left( {h - l} \right)}}$; the firms’ optimal profits are as follows: $\pi_{e}^{A*} = \frac{1}{{16\left( {1 - f} \right)^{2} \left( {h - l} \right)}}f\left( {\left( {1 - f - c_{1} } \right)h\left( {c_{1} \left( {4 - h} \right) + \left( {1 - f} \right)\left( {4 + h} \right)} \right) - 2\left( {1 - f + c_{1} } \right)\left( {1 - f - c_{1} } \right)\left( {2 - h} \right)l + \left( {1 - f + c_{1} } \right)^{2} l^{2} - 4c_{2}^{2} } \right),$ $\pi_{m1}^{A*} = \frac{{\left( {1 - f - c_{1} } \right)^{2} }}{{4\left( {1 - f} \right)}}$, and $\pi_{m2}^{A*} = \frac{{\left( {\left( {1 - f - c_{1} } \right)h + \left( {1 - f + c_{1} } \right)l - 2c_{2} } \right)^{2} }}{{16\left( {1 - f} \right)\left( {h - l} \right)}}$.

Let $c = c_{1} + c_{2}$, we have $\pi_{e}^{M*} - \pi_{e}^{E*} = - \frac{1}{{36\left( {1 - h + l} \right)}}\left( {\left( {1 + \frac{9f}{{4\left( {1 - f} \right)^{2} }}} \right)c^{2} - \frac{{\left( {4 - 13f} \right)\left( {1 + l} \right)}}{{2\left( {1 - f} \right)}}c + \frac{1}{4}\left( {4 - 27f} \right)\left( {1 + l} \right)^{2} } \right) = - \frac{{\left( {1 + l} \right)^{2} }}{{36\left( {1 - h + l} \right)}}\left( {\frac{c}{1 + l} - \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 - 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}} \right)\left( {\frac{c}{1 + l} - \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}} \right)$, one can obtain that $\pi_{e}^{M*} \ge \pi_{e}^{E*}$ if $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} \le \frac{c}{1 + l} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 - 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$, and $\pi_{e}^{M*} \le \pi_{e}^{E*}$ otherwise.

Similarly, we have $\pi_{m1}^{M*} - \pi_{m1}^{E*} = \frac{1}{{144 \left( {1 - h + l} \right)}}\left( { \left( {\frac{9}{1 - f} - 8} \right)c^{2} - 2\left( {1 + l} \right)c + \left( {1 - 9f} \right)\left( {1 + l} \right)^{2} } \right) = \frac{{\left( {1 + l} \right)^{2} }}{{144 \left( {1 - h + l} \right)}}\left( {\frac{c}{1 + l} - \frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}} \right)\left( {\frac{c}{1 + l} - \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}} \right)$, one can obtain that $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$ if $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} \le \frac{c}{1 + l} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, and $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$ otherwise.

Similarly, we have $\pi_{m2}^{M*} - \pi_{m2}^{E*} = \frac{1}{{72\left( {1 - h + l} \right)}}\left( { \left( {\frac{9}{1 - f} - 4} \right)c^{2} - 10\left( {1 + l} \right)c + \left( {5 - 9f} \right)\left( {1 + l} \right)^{2} } \right) = \frac{{\left( {1 + l} \right)^{2} }}{{72\left( {1 - h + l} \right)}}\left( {\frac{c}{1 + l} - \frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f}} \right)\left( {\frac{c}{1 + l} - \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}} \right)$, one can obtain that $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$ if $\frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f} \le \frac{c}{1 + l} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$ otherwise.

Given $0 \le f \le 1$ and $0 \le \frac{c}{1 + l} \le 1$, we have $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} \le \frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, on can easily obtain that: (1). when $0 \le \frac{c}{1 + l} \le \frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (2). when $\frac{{1 - f - 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} < \frac{c}{1 + l} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (3). when $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{c}{1 + l} \le \frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f}$, such that $\pi_{e}^{M*} \ge \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (4). when $\frac{{5 - \left( {5 + 6\sqrt {1 - f} } \right)f}}{5 + 4f} < \frac{c}{1 + l} \le \frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }}$, such that $\pi_{e}^{M*} \ge \pi_{e}^{E*}$, $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$; (5). when $\frac{{\left( {1 - f} \right)\left( {4 - f\left( {13 + 6\sqrt {5 + 3f} } \right)} \right)}}{{4 + f + 4f^{2} }} < \frac{c}{1 + l} \le \frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \le \pi_{m2}^{E*}$; (6). when $\frac{{5 - \left( {5 - 6\sqrt {1 - f} } \right)f}}{5 + 4f} < \frac{c}{1 + l} \le \frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f}$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \le \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$; (7). when $\frac{{1 - f + 6f\sqrt {2\left( {1 - f} \right)} }}{1 + 8f} < \frac{c}{1 + l} \le 1$, such that $\pi_{e}^{M*} \le \pi_{e}^{E*}$, $\pi_{m1}^{M*} \ge \pi_{m1}^{E*}$, and $\pi_{m2}^{M*} \ge \pi_{m2}^{E*}$. Hence, we have the result in Proposition 8. □