Supply Chain Management

Vol.:(0123456789)

Electronic Commerce Research (2021) 21:393–422 https://doi.org/10.1007/s10660-019-09370-7

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A choice of selling format in the online marketplace with cross‑sales supply chain: Platform selling or traditional reselling?

Xiaojing Li1  · Xingzheng Ai2

Published online: 14 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019

Abstract In the online retail market, how to work with upstream suppliers is a key issue for downstream online retailers (e-retailers). Online retailers can choose between func- tioning either as the “two-sided platforms” (e.g., Taobao.com or eBay.com) allow- ing suppliers to sell directly to customers by paying a revenue-sharing fee, or as the “resellers” (e.g., Wal-mart.com or JingDong.com) that purchase products from suppliers, and then resell them to customers. Given the rapid growth of e-commerce over past few years, this choice, which is the focus of this article, has become an important practice-based decision. We develop a game-theoretic model for a cross- sales supply chain in which two suppliers deal with two common online retailers. As Stackelberg leaders, online retailers can operate either as a two-sided platform (serving both suppliers and customers) or as a reseller (ordering from suppliers and selling competing products on its own platform). Each supplier adopts either an exclusive-sales strategy, selling products through an exclusive e-retailer, or a cross- sales strategy, selling products through two e-retailers. We analyze the optimal deci- sions for both e-retailers and suppliers in competing supply chains and describe the system equilibrium for the online marketplace.

Keywords E-business · Selling format choice · Supply chain competition · Online marketplace · Game theory

* Xiaojing Li [email protected]

1 School of Business, Sichuan Normal University, Chengdu, China 2 School of Economics and Management, University of Electronic Science and Technology

of China, Chengdu, China

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1 Introduction

In the past few decades, online shopping has become an important form of con- sumption for people. According to a report by Statista.com, global online retail sales reached $2.3 trillion in 2017, and e-retail revenue is expected to increase to $4.88 trillion in 2021.1 In UK, the Capgemini IMRG reports that online retail sales growth reached 16.8% in the first half of 2018, which is the strongest growth rate in the first half of the past 8 years.2 According to the China Department of Commerce, online retail sales reached ¥4.08 trillion in the first half of 2018 and increased by 30.1% year-on-year.3 These growths indicate that online retailing plays an significant role in suppliers and retailers.

There are two main online selling formats for online retail in the existing market [2, 17], one of which is that online retailers serve as a bilateral platform to match consumers and suppliers, and the other of which is online retailers operating in a manner resembling traditional “resellers”. Under the former selling format, the sup- pliers can set retail prices and sell directly to consumers at the expense of paying retailers, such as a revenue-sharing requirement. For example, Huawei and Xiaomi, as two largest mobile phone suppliers, give share of every sale to the platform owner Taobao.com. Under the latter selling format, online retailers orders prod- ucts from suppliers at a wholesale price (set by the supplier) and sells products at a retail price (set by the e-retailer itself), Such as the transaction between JingDong. com and Haier, the largest online retailer and supplier for home appliance in China, respectively.

Motivated by online retail practice, which selling format is most beneficial has attracted extensive attention from scholars. Some studies have analyzed the selling format choice of e-retailers, but they focus on three structures. First is a monopoly supply chain system in which there are one supplier and one e-retailer [15, 30]; sec- ond is a downstream competitive supply chain system in which there are one supplier and multiple e-retailers [2, 26]. In fact, both upstream suppliers and downstream e-retailers have members competing with each other; the third is a upstream com- petitive supply chain system in which there are multiple suppliers and one e-retailer [4, 13, 24]. Taking household appliances as an example, a lot of brands for washing machine, such as SIEMENS, Panasonic, LG, etc., and multiple e-retailers for selling washing machine, such as JingDong.com, Taobao.com, Amazon.com, etc.

Cross-sales, formed by multiple manufacturers and multiple common retail- ers, is a common phenomenon in supply chain under the Internet economy [20, 27]. For instance, Dell and HP sell their computers through the same e-retail- ers, Taobao and Amazon. As suppliers, selling products through multiple retail- ers can not only reduce inventory costs, but also enhance business insight into market information. As e-retailers, attracting multiple suppliers can establish a large-scale centralized control system to realize the simultaneous management

1 Available at http://www.stati sta.com/stati stics /37904 6/world wide-retai l-e-comme rce-sales . 2 Available at http://www.199it .com/archi ves/76035 2.html. 3 Available at http://www.ec.com.cn/artic le/dssz/zhjg/20180 7/30337 _1.html.

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of commodities, promotions and information. What’s more, cross-sales is better meet the needs of consumers for diversification and convenience. Therefore, the phenomenon prevails that more and more companies are engaged in cross-sales nowadays.

However, unlike the flourishing of practice, there is not much research on cross-selling in academia. A few literature considers the cross-sales supply chain system which formed by competitive suppliers and competitive retailers, with a focus on the cooperation mechanism between upstream and downstream members under this structure, including information asymmetry [20], comparison of differ- ent cooperation contracts [6, 19], choice between exclusive sales and cross-sales [3, 27]. Different from them, this paper studies the problem of channel design for online retailers in the presence of a cross-sales structure. Combined with the situation in practice and related research, this paper aims to solve the following general questions:

1. In the online marketplace, which type of selling format should e-retailers rely on under horizontal competition for both suppliers and e-retailers?

2. What effect does the choice of selling format have on optimal wholesale prices, optimal retail prices, and total channel profits in a cross-sales supply chain?

3. How can a revenue-sharing mechanism be developed in a platform selling format to allow both e-retailers and suppliers to benefit more from the platform selling format than from the traditional reselling format?

4. How does the e-retailer respond to its competitor’s actions in shifting from the traditional reselling format to the platform selling format?

To answer these questions, we develop a game-theoretic model for a competi- tive supply chain with two competing suppliers and two competing e-retailers. As Stackelberg leaders, e-retailers can choose different online channels, either as an intermediary, serving both suppliers and customers, or as a reseller, order- ing from suppliers and also selling competing products through their own online platforms. We first analyze the case in which each supplier adopts an exclusive- sales strategy, selling products through an exclusive e-retailer, which serves as the theoretical benchmark in this study. Next, we examine the optimal strategy for a cross-sales supply chain in which each supplier sells products through two common e-retailers. Then, we investigate suppliers and e-retailers’ best responses by selling format in the online marketplace. Finally, we derive a Nash equilibrium for the e-retailers’ selling format game with a cross-sales supply chain and com- pare the different formats in terms of total supply chain profit and equilibrium prices.

The major contributions of this paper are as follows. First, although a number of studies have discussed the choice of selling format in the online marketplace, they focus on a monopoly supply chain [15, 24, 30], or a supply chain with down- stream competition[2, 26], or a supply chain with upstream competition [4, 13]. Instead of previous studies, our paper absorbs a supply chain setting with compet- itive suppliers and competitive retailers, which is more popular than others in the

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practise market. Second, a few studies related on cross-sales supply chain compe- tition, but they pay attention to information asymmetry [20], comparison of dif- ferent cooperation contracts [6, 19], choice between exclusive sales and cross- sales [3, 27]. Our focus is completely different because we provide insights into how retailers can design selling format in the online marketplace with a cross- sales structure. Third, most of literature about the cross-sales supply chain only consider a symmetric choice of suppliers, which means different suppliers always adopt the same format, such as Feng and Lu [6], or the suppliers always deal with different retailers thought same format, such as Li et al. [20]. In contrast to these studies, we allow two suppliers to choose different selling formats simultaneously and analyze the equilibrium selling format choice strategy. Fourthly, we develop a platform cooperation mechanism to help suppliers and e-retailers or the entire channel maximize their benefit, examine the benefit of platform selling through comparing traditional reselling, and the impact on channel choice of the profit sharing in present of cross-sales supply chain.

The remainder of this paper is organized as follows. Section 2 reviews the related literature. Section 3 describes the key elements of our basic model and introduces notations. Section 4 presents an exclusive-sales model and studies e-retailers’ and suppliers’ optimal selling format strategies in an exclusive-sales supply chain. In Sect. 5, we build a cross-sales model and compare and analyze e-retailers’ online equilibrium choice of selling format for a cross-sales supply chain. Section 6 offers concluding remarks and proposes some directions for future research.

2 Literature review

A significant portion of the literature on online selling formats addresses the choice between platform selling and reselling formats. Jiang et al. [15] study a e-retailer’s selling formats when there are two types of independent suppliers on a platform— one with high demand and the other with low demand. Zhang et  al. [30] analyze the impact of supplier’s product quality on platform contract selection and find that when product quality is exogenous, online retailer always prefers revenue sharing contracts. Otherwise, e-retailer maybe choose a fixed fee contract. These papers investigate the members’ strategic behavior in a one(supplier)-to-one(e-retailer) sup- ply chain.

Some papers study the trade-off between the traditional reselling and the platform selling format in a competitive environment, and they can be divided two aspects as following. One is to focus on competition among retailers, for example, Abhishek et  al. [2] consider a supplier selling products through one traditional retailer and two electronic retailers, and recognize the effects of selling format choice on supply chain members. They point out that e-retailers prefer to act only as intermediaries when the electronic channel is conducive to the demand of traditional channel, oth- erwise, vice versa. Shen et al. [26] discuss how a online retailer make a decision on selling formats in a supply chain consisting of one supplier and two retailers. They give the equilibrium selling formats of a Stackelberg game and a bargaining game, and characterize member’s sale quantities, prices and profits under each format.

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The other is considering the competition among suppliers, for example, Hagiu and Wright [12] assume there is information asymmetry between the supplier and the retailer due to uncertain demand, and provide the economic trade offs between the merchant mode and the two-sided platform mode in the market setting with two suppliers and one e-retailer. Chen et al. [4] believes that customer loyalty is a key factor in the long-term success of the retail platform. They model a supply chain system consisting of two suppliers and one online retailer, and study how selling for- mats affect online retailer’s profit, industry profit, consumer surplus and social ben- efits when customer have loyalty. The results show that the platform selling always leads to lower retail prices and higher consumer surplus. Tian et al. [24] consider a setting where two competing suppliers and one online retailer are in the market. They find that order fulfillment costs and the intensity of competition among sup- pliers are important factors influencing the choice of selling format. When they are low, platform selling format is best option for the retailer; as two factors increase, mixed mode is preferred; when they are quit high, the reselling format is the optimal choice.

Our paper is different from the above studies in following three aspects. First, the above studies main focus on supply chain structure with many-to-one (suppli- ers-to-retailer), or one-to-many (supplier-to-retailers). However, in reality, competi- tion usually exists not only among upstream suppliers but also among downstream retailers. By contrast, our paper investigates a two-to-two (suppliers-to-retailers) structure, i.e., the competing suppliers sell their products through two common (or exclusive) e-retailers, which consists of cross-sales (or exclusive-sales). Second, we believe that mixed selling format is an important format for suppliers, such as Haier adopts reselling format in JingDong.com, and uses platform selling format in Tao- bao.com. Therefore, we allow suppliers to choose one of selling formats, or two in this paper instead of only one in aforementioned studies. Third, we reveal the game equilibrium choices of the two selling formats under the cross-selling supply chain, and give the range of the revenue sharing proportion in the platform selling format.

Our work is also related to the extensive literature on the structure of supply chains with multiple suppliers. This literature mostly focuses on multiple suppli- ers trading with multiple exclusive e-retailers. McGuire and Staelin [22] investigate the effects of product substitutability on Nash equilibrium distribution structures in a duopoly in which each supplier distributes its goods through a single exclusive e-retailer. Coughlan [5], Moorthy [23], Gupta et al. [7, 8], Ha et al. [9, 10], Ai et al. [1] and Lin and Parlakturk [21] extend the McGuire model.

Some recent studies focus on competition between two supply chains. Zhu and He [29] assume that manufacturers can produce both green and traditional prod- ucts, it studies the equilibrium production strategy of manufacturers when there are two supply chain competitions in the market. Wu and Zhou [28] consider that manufacturers are engaged in recycling and remanufacturing of the product, and analyze the impact of competition between the two supply chains on the manu- facturer’s optimal recycling channel choice. Ha et  al. [11] discuss the problem of sharing demand information in two competing supply chains, each consisting of one manufacturer and one retailer. It compare and analyze the conditions for information sharing with quantity competition and price competition. Lee et  al.

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[18] study how competition between supply chains affects members’ incentives to implement CSR practices, and how market forces change the quality of the envi- ronment without allowing or allowing greening. Jin et al. [16] consider a setting with two supply chains, each consisting of one supplier and one manufacturer. It studies the interaction of supplier development and supplier integration in a com- petitive environment.

However, above studies focus on exclusive-sale of competitive supply chain structure. The core of our paper arises from the aforementioned market structure in that competing suppliers sell products to multiple e-retailers instead of to an exclusive e-retailer, and competing e-retailers also order products from multiple suppliers simultaneously. Wu and Mallik [27] establish a supply chain system consisting of two suppliers and two retailers, and allow suppliers to select exclu- sive sales or cross-sales. It identifies the suppliers’ equilibrium choice between integration and decentralized structure. However, in reality, the simple integra- tion structure is usually replaced by coordination and cooperation between the upstream and downstream enterprises under the decentralization. In contrast, our paper allow suppliers and retailers to share revenue and examine how online retailers set this sharing ratio to encourage or prevent suppliers from adopting a platform selling format.

Cai et  al. [3] assume that the upstream members can trade with one service provider exclusively, or trade with two service providers at the same time, which is similar to Wu and Mallik [27]. It analyzes the role of revenue sharing contract in exclusive transactions and contrasting the members’ profits in exclusive sup- ply chain with the one in cross-sales supply chain. Different from it, our paper focuses on the choice of the traditional reselling format and platform selling for- mat under the cross-sales structure.

Li et  al. [19] study contract choice game of two supply chains, each consist- ing of a supplier and a retailer. It allows suppliers to sell their products through two common retailers or exclusive retailers and adopt quantity discount or whole- sale price contracts. It finds out that there exists a large difference in the contract choice of game equilibrium between exclusive sales and cross-sales structure.

Feng and Lu [6] investigate the problem of contract design with a cross-sales supply chain, and it focus on the equilibrium contract choice by using two mod- els, a Stackelberg game in which a supplier gives a take-itor-leave-it choice to a retailer, and a bargaining game in which the contract terms are negotiated bilater- ally by upstream and downstream members. However, it only considers that two suppliers offers same selling format to retailers and retailers have the same reac- tion to it.

Li et al. [20] build a supply chain model of two suppliers and two retailers, which is the same as this article. It solves that how to design a coordination contract to motivate downstream companies to share their private demand information, and reveals the value of information sharing. Different from it, our research focuses on the choice of online selling format (the platform selling format and reselling format) under a cross-sales supply chain. We discuss the design of a revenue-sharing mecha- nism under a platform selling format and examine the effects of horizontal competi- tion and a revenue-sharing mechanism on the choice of online selling format.

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3 Model

Our model consists of two suppliers, denoted as S1 and S2, and two e-retailers, denoted as R1 and R2. The two suppliers produce substitutable products and sell through the two exclusive e-retailers or common e-retailers and thus form an exclu- sive-sales supply chain (as shown in Fig. 1a) or a cross-sales supply chain (as shown in Fig. 1b). Each e-retailer may choose a different format for its selling agreement, either (i) a traditional reselling agreement (D) in which the e-retailer orders the product from the supplier at a wholesale price, determines the retail price and sells the product to customers on its online platform or (ii) a platform selling agreement (P) in which the e-retailer offers an online platform to the supplier for a fraction r of supplier’s revenues, where the supplier decides the retail price and sells products to customers by means of the e-retailer’s online platform. We consider the e-retailers have the power to choose their cooperation agreement independently and simultane- ously because online retailers frequently have large customer bases and good market sense.

In addition, we assume that revenue-sharing rate is an exogenous variable and it is offered by online retailers before the transaction. Actually, online retailers do not set a separate revenue sharing ratio for each supplier, but rather design a sin- gle revenue sharing ratio for all manufacturers. For instance, in the e-book industry, e-book publisher Random House is collecting 70% of the profits for each book; APP developers working with Apple need to pay 30% of each transaction’s revenue as a mid-priced fee. This constant income sharing ratio is widely used in related research in previous studies [3, 4, 13, 14, 20].

The parameters and decision variables of our model are given in Table 1. To obtain the demand function of the competing supply chain, we follow Cai

et al. [3], and the utility function of the consumer can be written as follows:

b denotes substitutability of competing channels, and b ∈ (0, 1) . If b = 0 , the sup- ply chain is completely monopolistic; if b = 1 , the supply chains converge toward

(1)U = ∑ ij

(aijqij − q2 ij ∕2) − b

∑ ij≠mn

(qijqmn∕2) − ∑ ij

pijqij, i, j,m, n = {1, 2}

(a) an exclusive-sales supply chain

S1

R1

S2

R2

(b) a cross-sales supply chain

S1

R1

S2

R2

Fig. 1 Supply chain structure with two suppliers and two e-retailers

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complete substitutability. The term aij reflects the consumer’s preference for supply chain ij and can be considered the initial base demand when all prices equal zero.

Consistent with the previous literature [3, 14], maximizing Eq.  (1) produces the demand of supply chain ij as follows:

such that

where � represents the price coefficient, � represents the cross-price coefficient and Aij represents the “attractiveness” of supply chain ij . N is the total number of sup- ply chains [3, 14, 20]. We assume a11 = a12 = a21 = a22 = 1 . This setting has been widely employed in prior studies [19, 22, 25, 27], particularly for a complex model like ours. In addition, assume that the supplier’s production cost is zero. The pur- pose of this hypothesis is to simplify the derivation of the model. Even if the sup- plier’s production cost is not zero, it will not change the basic conclusions of this paper, and only increase the complexity of mathematical processing.

(2)qij = Aij − �pij + � ∑ mn≠ij

pmn, i, j,m, n = {1, 2}

Aij = (1 + (N − 2)b)aij − b

∑ mn≠ij amn

(1 − b)(1 + (N − 1)b)

� = 1 + (N − 2)b

(1 − b)(1 + (N − 1)b)

� = b

(1 − b)(1 + (N − 1)b) .

Table 1 Notation

i Suppler i or product i, for i = {1, 2}

j E-retailer j or electronic platform j, for j = {1, 2}

ij Supply chain ij, which consists of supplier i and e-retailer j, for i, j = {1, 2}

pij Retail price of supply chain ij, for i, j = {1, 2}

wij Wholesale price of supply chain ij qij Demand associated with supply chain ij, for i, j = {1, 2}

aij Consumer’s preference for supply chain ij, for i, j = {1, 2}

b Channel substitutability, for i, j = {1, 2}

� Price coefficient � Cross-price coefficient Aij The “attractiveness” of supply chain ij, for i, j = {1, 2}

N Total number of supply chains r E-retailer’s revenue-sharing rate of the supplier’s revenue SM i

Supplier i ’s profit under configuration M , where M = EDD,EPP,EDP

RM j

E-retailer j ’s profit under configuration M , where M = EDD,EPP,EDP

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4 Analysis of the exclusive‑sales supply chain

We define an exclusive-sales supply chain to be a supply chain consisting of two suppliers and two exclusive e-retailers, i.e., one e-retailer sells products from only one supplier. Based on two e-retailers’ decisions regarding the online selling format, there are thus 3 possible selling format configurations (as shown in Fig. 2):

1. EDD: for an exclusive-sales supply chain, both e-retailers buy products from suppliers and sell them through their online platforms

2. EPP: for an exclusive-sales supply chain, both e-retailers offer online platforms to allow suppliers to sell their products; and

3. EDP: for an exclusive-sales supply chain, one e-retailer buys products from one supplier and sells them through an online platform, and the other e-retailer offers an online platform that allows the other supplier to sell its product directly into the market.

4.1 The price game under a given selling format configuration

In configuration EDD, both e-retailers of the exclusive-sales supply chains order and distribute products. First, the two suppliers individually and simultaneously decide upon the wholesale price of products. Then, the two e-retailers set the retail prices based on their wholesale prices. Supplier i’s and e-retailer j’s profit functions can be written as follows:

Using backward induction to solve this subgame, we can obtain the unique opti- mal prices, which can be summarized in Lemma 1.

Lemma 1 Under an exclusive-sales supply chain set-up, if both e-retailers offer a reselling agreement to both suppliers, a unique price equilibrium can be found:

(3)SEDD i

= w ij qij,

(4)REDD j

= (pij − w ij )qij, i, j = {1, 2}; i = j

(a) EDD

S1

R1

S2

R2

(b) EPP

S1

R1

S2

R2

(c) EDP

S1

R1

S2

R2

Fig. 2 Selling format choice for an exclusive-sales supply chain

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In configuration EPP, both e-retailers of the exclusive-sales supply chains provide online platforms to allow suppliers to sell their products. First, two e-retailers simul- taneously announce the revenue-sharing rate, r . Then, the two suppliers set their retail prices. Supplier i’s and e-retailer j’s profit functions can be written as follows:

Using backward induction to solve this subgame, we can obtain the unique opti- mal prices, which can be summarized in Lemma 2.

Lemma 2 Under an exclusive-sales supply chain set-up, if both e-retailers offer a platform selling agreement to both suppliers, a unique price equilibrium can be found:

In configuration EDP, one of the e-retailers, (j) , orders products to distrib- ute, and the other e-retailer, (3 − j) , offers an online platform to his supplier. First, two e-retailers announce the operation agreement to their suppliers. In a reselling agreement, the wholesale price must be set by the supplier, i ; for a platform selling agreement, the revenue-sharing rate must be provided by e-retailer (3 − j) . Second, e-retailer j and supplier (3 − i) decide their retail prices simultaneously. Supplier i’s and e-retailer j’s profit functions can be written as follows:

Using backward induction to solve this subgame, we can obtain the unique opti- mal prices, which can be summarized in Lemma 3.

Lemma 3 Under an exclusive-sales supply chain, if one e-retailer offers a platform selling agreement to its supplier and the other e-retailer offers a reselling agreement to its suppler, a unique price equilibrium can be found:

Lemmas 1–3 show that with an exclusive-sales supply chain, both the wholesale price and the retail price in each configuration decrease in b . It is not difficult to understand that as the degree of channel substitutability increases, both e-retailers and suppliers may adopt low-price strategies to stoke market demand. Thus, in a competitive market, a high level of substitutability can intensify price competition.

(5)wEDD 11

= wEDD 22

= 2(1 − b)∕(4 − 3b),

(6)pEDD 11

= pEDD 22

= 3(1 − b)∕(4 − 3b).

SEDD i

= (1 − r)p ij qij, REDD

j = rp

ij qij, i, j = {1, 2}; i = j.

pEPP 11

= pEPP 22

= (1 − b)∕(2 − b).

SEDP i

= w ij qij, REDP

j = (pij − w

ij )qij.

SEDP 3−i

= (1 − r)p (3−i,3−j)

q (3−i,3−j)

, REDP 3−j

= rp (3−i,3−j)

q (3−i,3−j)

, i, j = {1, 2}; i = j

pEDP 11

= 3(1 − b)(2 + b)∕(8 − 3b2), pEDP 22

= (1 − b)(4 + 3b)∕(8 − 3b2)

wEDP 11

= 2(1 − b)(2 + b)∕(8 − 3b2).

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Proposition 1 pEDD 11

> pEDP 11

> pEPD 11

(pEDP 22

) > pEPP 11

.

Proposition 1 indicates that the retail price under a traditional reselling format is higher than that under a platform selling format (i.e., pEDD

11 > pEPD

11 , pEDP

11 > pEPP

11 and

pEDP 11

> pEDP 22

), as reselling yields double marginalization. Thus, consumers must pay a higher price because of a reselling agreement between the supplier and e-retailer. From the above analysis, we find that retail prices decrease with the number of plat- form selling formats.

Table 2 summarizes the profits of suppliers and e-retailers in an exclusive-sales supply chain, for the cases EDD,EPP,EDP(EPD) with Lemmas 1–3.

4.2 Selling format choice under an exclusive‑sales supply chain

To easily compare the profits of suppliers and e-retailers under two selling formats in an exclusive-sales supply chain, we first denote several thresholds, as follows.

Let r1 be the threshold of SEDD 1

= SEPP 1

, yielding,

Let r2 be the threshold of REDD 1

= REPP 1

, yielding,

Let r3 be the threshold of SEDD 1

= SEPD 1

, yielding,

Let r4 be the threshold of REDD 1

= REPD 1

, yielding,

Let r5 be the threshold of SEDP 1

= SEPP 1

, yielding,

Let r6 be the threshold of REDP 1

= REPP 1

, yielding,

From suppliers and e-retailers’ profit expressions of an exclusive-sales supply chain, we can obtain proposition 2, as follows.

r1 = (8 − 16b + 7b2)∕(3b − 4)2

r2 = (2 − b)2∕(4 − 3b)2

r3 = (128 − 192b2 + 63b4)∕((4 − 3b)2(4 + 3b)2)

r4 = (8 − 3b2)2∕((4 − 3b)2(4 + 3b)2)

r5 = (32 − 32b2 + 7b4)2∕(8 − 3b2)2

r6 = (2 − b)2(2 + b)2∕(8 − 3b2)2

Table 2 The profits of the suppliers and the e-retailers of exclusive-sales supply chain

Profit EDD EPP EDP EPD

S1 (1−r)(1−b)

(1+b)(2−b)2 2(1−b)

(1+b)(4−3b)2 2(1−b)(2+b)2

(1+b)(8−3b2)2 (1−r)(1−b)(4+3b)2

(1+b)(8−3b2)2

S2 (1−r)(1−b)

(1+b)(2−b)2 2(1−b)

(1+b)(4−3b)2 (1−r)(1−b)(4+3b)2

(1+b)(8−3b2)2 2(1−b)(2+b)2

(1+b)(8−3b2)2

R1 r(1−b)

(1+b)(2−b)2 (1−b)

(1+b)(4−3b)2 (1−b)(2+b)2

(1+b)(8−3b2)2 r(1−b)(4+3b)2

(1+b)(8−3b2)2

R2 r(1−b)

(1+b)(2−b)2 (1−b)

(1+b)(4−3b)2 r(1−b)(4+3b)2

(1+b)(8−3b2)2 (1−b)(2+b)2

(1+b)(8−3b2)2

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Proposition 2

(i) SEDD i

≤ SEPP i

,REDD j

≤ REPP j

, for i, j = {1, 2}; if r2(b) ≤ r ≤ r1(b).

(ii) SEDD 1

≤ SEPD 1

,REDD 1

≤ REPD 1

, if r4(b) ≤ r ≤ r3(b).

(iii) SEPD 2

≤ SEPP 2

,REPD 2

≤ REPP 2

, if r6(b) ≤ r ≤ r5(b).

Proposition 2 shows that (i) when two e-retailers in an exclusive supply chain have the same choice of selling format, all members (i.e., S1, S2, R1 and R2) can obtain more profit under a platform selling agreement (i.e., SEDD i

≤ SEPP i

,REDD j

≤ REPP j

, where i, j = {1, 2} ) as long as the revenue-sharing rate falls within a certain range (i.e. r2 ≤ r ≤ r1 ); (ii) when one e-retailer (e.g., R1) adopts a reselling agreement, for any given r4 ≤ r ≤ r3 , a platform selling agreement becomes attractive to R2 and S2; and (iii) likewise, when one e-retailer (e.g., R1) chooses a platform selling agreement as in Lemma 1, the platform selling agreement is more profitable than a reselling agreement for R2 and S2, as long as r6 ≤ r ≤ r5 . Thus, the supplier and e-retailer can benefit from a platform selling agreement as long as the revenue-sharing rate falls within a certain range, no matter which agree- ment the competitors choose.

Then, we can posit the following corollary.

Corollary 1 If 0 < b < 0.4226, then (r2, r1) ∩ (r4, r3) ∩ (r6, r5) = (r2, r1).

From Corollary 1, we obtain the relationship of the range (r2, r1) , (r4, r3) and (r6, r5) . In other words, with 0 < b < 0.4226 , if and only if r ⊂ (r2, r1) , then r ⊂ (r4, r3) , r ⊂ (r6, r5) , which indicates that with 0 < b < 0.4226 , a range (r2, r1) exists such that SEDD

i ≤ SEPP

i ,REDD

j ≤ REPP

j , SEDD

1 ≤ SEPD

1 ,REDD

1 ≤ REPD

1 ,

SEPD 2

≤ SEPP 2

,REPD 2

≤ REPP 2

as long as r2 < r < r1.

Fig. 3 Thresholds rx, x = 1, 2, 3, 4, 5, 6 change with b

405

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A choice of selling format in the online marketplace with…

The results we have analyzed, including r1, r2, r3, r4, r5, r6 , are shown in Fig.  3, which shows that in the shaded region A (i.e., the range, r2 < r < r1 ), the suppliers and the e-retailers in an exclusive-sales supply chain would prefer a platform selling agreement. As b increases, the range shrinks. In the shaded region B, which is the range (r5, r6) , the suppliers and the e-retailers would prefer a reselling agreement. Otherwise, suppliers and e-retailers cannot maximize their profits at the same time.

Theorem  1 Under an exclusive-sales supply chain, (i) for any given 0 < b < 0.4226 , r1(b) and r2(b) exist such that a platform selling format is the unique Nash equilibrium for all supply chain members, as r2(b) < r < r1(b); and (ii) for any given 0.9194 < b < 1 , r5(b) and r6(b) exist such that a reselling format is the unique Nash equilibrium for all supply chain members, as r5(b) < r < r6(b).

Theorem 1 demonstrates that when channel substitutability is low, entering into a platform selling agreement can be an equilibrium for both e-retailers and both suppli- ers. A supplier has the incentive to use a platform selling format because, in so doing, the demand for the supplier’s product significantly increases because of the lower mar- ket price; furthermore, an e-retailer can obtain a reasonable amount of revenue trans- ferred from the supplier, such that both the supplier and e-retailer prefer to use platform selling. However, when the channel substitutability is sufficiently high, a platform sell- ing format intensifies price competition, which could hurt its benefits and even lead to less profit than reselling. The range of r2(b) < r < r1(b) provides a guideline for such an equilibrium revenue-sharing rate, if all members in an exclusive-sales supply chain decide to enter into a platform selling agreement.

We also find that configuration EDD can be an equilibrium when channel substi- tutability is high because both e-retailers and suppliers are better off using a reselling format when channels are sufficiently monopolistic. Notably, no equilibrium result can be found for configurations EDP and EPD , which is reasonable because the disadvan- taged e-retailers would shift from one channel to the other. In addition, this arrange- ment might benefit upstream members as well.

Theorem 1 is a good way to reveal the current status of clothing categories. Tmall. com and VIP.com, two of the largest B2C online retailers to sell clothing in China. Some clothing suppliers choose exclusive sale, such as JackJones and Only, famous brands for men’s and men’s clothing respectively, which sell their products only on Tmall.com. Tmall.com has been acting as an intermediary to help clothing suppliers match the right consumers. Different from it, Vip.com has profited from reselling the discounted clothes for the season. When the competition is weak, as long as Tmall.com sets the revenue sharing rate within a certain range, both the suppliers on its platform and the platform can obtain higher profits through this platform selling. Because Vip. com has a lower substitutability for Tmall, the reselling format is not very beneficial to Vip.com. It has faced greater pressure in recent years, which can be proved by joining the platform of Jingdong.com.

406 X. Li, X. Ai

1 3

5 Analysis of the cross‑sales supply chain

We define a cross-sales supply chain to be a supply chain consisting of two compet- ing e-retailers and two competing suppliers in which each supplier’s product must go through two common e-retailers. We first provide the optimal solution for the price game and then analyze the selling format choice when facing a cross-sales supply chain. There are four possible format configurations (as shown as Fig. 4).

1. CDD : for a cross-sales supply chain, both e-retailers buy products from suppliers and sell them into the market by themselves;

2. CPP : for a cross-sales supply chain, both e-retailers offer the online platform to allow suppliers to sell their products to market; and

3. CDP : for a cross-sales supply chain, one e-retailer buys products from suppliers and sells them into the market. The other e-retailer offers the online platform to allow suppliers to sell their product into the market.

4. C(DP)(DP) : for a cross-sales supply chain, two e-retailers not only buy products from supplier i and sell them but also offer the online platforms to allow supplier 3 − i to sell its product into the market, where i = 1, 2.

5.1 Price game under given configurations

We denote ST ij as supplier i ’s profit from trading with e-retailer j under configuration

T , RN ij

as e-retailer j ’s profit from trading with supplier i under configuration N , where T = CDD,CPP,CDP . Let ST

i be supplier i’s total profit and let RT

j be the

e-retailer j ’s total profit under configuration N, then ST i = ST

ij + ST

i,3−j ,

RN j = RN

ij + RN

3−i,j , where i, j = 1, 2.

In our model, we suppose the e-retailer is dominant, which is common in prac- tice, such as JingDong and TaoBao. To preserve a good teamwork relationship with their suppliers, e-retailers frequently focus on suppliers’ benefits when they choose the cooperation agreement. We consider that each e-retailer and his two suppliers form a group, that is, R1, S1, S2 form channel1 or group1, and R2, S1, S2 form channel2 or group2. Consequently, the profit of e-retailer j ’s channel under configuration T is GT

j = RT

j + ST

i,j + ST

3−i,j , where i, j = 1, 2 ,

(a) CDD

S1

R1

S2

R2

(b) CPP

S1

R1

S2

R2

(c) CDP

S1

R1

S2

R2

(d) C(DP)(DP)

S1

R1

S2

R2

Fig. 4 Selling format choice for a cross-sales supply chain

407

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A choice of selling format in the online marketplace with…

T = CDD,CPP,CDP . When the e-retailer chooses the cooperation agreement, it will maximize the channel’s profit as its objective.

5.1.1 Configuration CDD

In this configuration, both e-retailers of the cross-sales supply chain order prod- ucts to distribute. First, two suppliers individually and simultaneously decide the wholesale price of products. Then, two e-retailers set retail prices based on their wholesale prices. Supplier i’s and e-retailer j’s profit functions can be written as follows:

Using backward induction to solve this subgame, we can obtain the unique opti- mal prices, which can be summarized in Lemma 4.

Lemma 4 Under a cross-sales supply chain, if both e-retailers offer a reselling agreement to both suppliers, a unique price equilibrium can be found:

5.1.2 Configuration CPP

In this configuration, both e-retailers in the cross-sales supply chain provide online platforms to allow suppliers to sell their products. First, two e-retailers simultane- ously announce the revenue-sharing rate r . Then, two suppliers set their retail prices. Supplier i ’s and e-retailer j ’s profit functions can be written as follows:

Using backward induction to solve this subgame, we can obtain the unique opti- mal prices, which are summarized in Lemma 5.

Lemma 5 Under a cross-sales supply chain, if both e-retailers offer a platform sell- ing agreement to both suppliers, a unique price equilibrium can be found:

(7)SCDD i

= w ij qij + w

i,3−j qi,3−j,

(8)RCDD j

= (pij − w ij )qij + (p3−i,j − w

3−i,j )q3−i,j, i, j = {1, 2}; i = j

(9)pCDD 11

= pCDD 12

= pCDD 21

= pCDD 22

= 3(1 − b)

2(2 − b) ,

(10)wCDD 11

= wCDD 12

= wCDD 21

= wCDD 22

= 1 − b

2 − b .

SCDD i

= (1 − r)(p ij qij + p

i,3−j qi,3−j),

RCDD j

= r(p ij qij + p

3−i,j q3−i,j), i, j = {1, 2}; i = j.

408 X. Li, X. Ai

1 3

5.1.3 Configuration CDP

In this configuration, one of the e-retailers, (j) , orders products to distribute, and the other e-retailer, (3 − j) , offers an online platform to his suppliers. First, the two e-retailers announce the operations agreement to their suppliers. For a resell- ing agreement, the wholesale price must be offered by supplier i and (3-i); for a platform selling agreement, the revenue-sharing rate must be provided by e-retailer (3 − j) . Second, supplier (i) and (3 − i) choose their retail prices of platform (3-j), and e-retailer (j) choose their retail prices of platform (j) simultaneously. Supplier i ’s and e-retailer j ’s profit functions can be written as follows:

We can obtain the unique optimal prices, which can be summarized in Lemma 6.

Lemma 6 Under cross-sales supply chains, if one e-retailer offers a platform sell- ing agreement to two suppliers, and the other e-retailer offers a reselling agreement to two suppliers, a unique price equilibrium can be found:

5.1.4 Configuration C(DP)(DP)

In this configuration, two e-retailers buy products from supplier i and sell them by themselves, but they also offer online platforms to allow supplier 3−i sell its product into the market. First, the two e-retailers announce the operations agreement to their suppliers. For a reselling agreement, the wholesale price must be offered by the sup- pliers; for a platform selling agreement, the revenue-sharing rate must be provided by the e-retailers. Second, e-retailers and suppliers decide upon their retail prices. Supplier i ’s and e-retailer j ’s profit functions can be written as follows:

pCPP 11

= pCPP 12

= pCPP 21

= pCPP 22

= (1 − b)∕2.

SCDP i

= w ij qij + (1 − r)p

i,3−j qi,3−j, RCDP

j = (pij − w

ij )qij + (p3−i,j − w

3−i,j )q3−i,j.

SCDP 3−i

= w 3−i,j

q3−i,j + (1 − r)p 3−i,3−j

q3−i,3−j, RCDP 3−j

= rpi,3−jqi,3−j + rp3−i,3−jq3−i,3−j.

i, j = {1, 2}; i = j.

pCDP 11

= pCDP 21

= 4 + (11 + r)b − 2b2 − (r + 13)b3

2(2 + 8b + (1 − r + 6)b2 − (1 + r)b3) ,

pCPD 11

= pCPD 21

= 2 + 7b − 9b3

2(2 + 8b + (1 − r + 6)b2 − (1 + r)b3) ,

wCDP 11

= wCDP 21

= (1 − b)(2 + (7 + r)b + (5 + 2r)b2)

2 + 8b + (1 − r + 6)b2 − (1 + r)b3 .

409

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A choice of selling format in the online marketplace with…

We can obtain the unique optimal prices, which can be summarized in Lemma 7.

Lemma 7 Under a cross-sales supply chain, if both e-retailers offer a platform sell- ing agreement to S1 and a reselling agreement to S2, a unique price equilibrium can be found:

Lemmas 4–7 explore suppliers’ and retailers’ optimal decisions in each con- figuration of the cross-competing supply chain. As with Lemmas 1–3, both the wholesale prices and retail prices in each configuration are reduced as b increases. However, for configurations CDP and C(DP)(DP) , the product prices and wholesale prices not only depend on channel substitutability b but also are affected by the revenue-sharing rate r. As the revenue-sharing rate increases, both wholesale prices and retail prices in configurations CDP and C(DP)(DP) decrease.

From Lemmas 4–7, we find that the traditional reselling format leads to higher retail prices for consumers.

Proposition 3

Proposition 3 implies that only the prices of CDD and CDP depend on chan- nel substitutability and the revenue-sharing rate. In other words, if the chan- nels are relatively more monopolistic (0 < b < (

√ 20 − 4r + r2 + r)∕(10 − 2r)) ,

the supplier offers a lower product price when entering into a platform selling agreement (i.e., pCDD

11 > pCDP

11 ). Otherwise, the opposite is true because stronger

channel substitutability leads to more intense competition. The range of chan- nel substitutability (b ⊂ [0, (

√ 20 − 4r + r2 + r)∕(10 − 2r)) increases with rev-

enue-sharing rate r. Additionally, we find that a platform selling agreement often leads to a lower price than a reselling agreement (e.g., pCDD

11 > pCPP

11 ,

S C(DP)(DP)

i = w

ij qij + w

i,3−j qi,3−j, S

C(DP)(DP)

3−i = (1 − r)(p

(i,3−j) q (i,3−j)

+ p (3−i,3−j)

q (3−i,3−j)

)

R C(DP)(DP)

j = (pij − w

ij )qij + rp3−i,jq3−i,j, R

C(DP)(DP)

3−j = (pi,3−j − w

i,3−j )qi,3−j + rp3−i,3−jq3−i,3−j

i, j = {1, 2}; i = j

p C(DP)(DP)

11 = p

C(DP)(DP)

12 =

6 + (16 + r)b − 2b2 − (20 + r)b3

2(4 + 15b + (12 − r)b2 − (3 + r)b3) ,

p C(DP)(DP)

21 = p

C(DP)(DP)

22 =

4 + 13b − 17b3

2(4 + 15b + (12 − r)b2 − (3 + r)b3) ,

w C(DP)(DP)

11 = w

C(DP)(DP)

12 =

2 + (6 − r)b − 2rb2 + (3r − 8)b3

4 + 15b + (12 − r)b2 − (3 + r)b3 .

⎧ ⎪⎨⎪⎩

pCDD 11

> pCDP 11

> p C(DP)(DP)

11 > p

C(PD)(PD)

11 > pCPD

11 > pCPP

11 , if 0 < b < (

√ 20 − 4r + r2 + r)∕(10 − 2r)

pCDP 11

≥ pCDD 11

> p C(DP)(DP)

11 > p

C(PD)(PD)

11 > pCPD

11 > pCPP

11 , if (

√ 20 − 4r + r2 + r)∕(10 − 2r) ≤ b < 1

410 X. Li, X. Ai

1 3

pCDP 11

> pCPP 11

, pCDD 11

> p C(PD)(PD)

11 ), and furthermore, the competitor’s shift from a

reselling agreement to a platform selling agreement can also cause a price drop (e.g., pCPD

11 > pCPP

11 , pC(DP)(DP)

11 > p

C(PD)(PD)

11 , pC(PD)(PD)

11 > pCPP

11 ).

We summarize the profits of the suppliers and the e-retailers of a cross-sales supply chain for the cases CDD,CPP,CDP,C(DP)(DP) with Lemmas 4–7 in Table 2.

5.2 Selling format choice under a cross‑sales supply chain

In this section, we first examine the case in which both e-retailers are treated symmetri- cally, which allows us to compare the single and hybrid formats. Finally, we explore the role of asymmetric e-retailers in Sect. 5.2.2, which not only allows us to analyze the choice between a pure “two-sided platform” and a pure “reseller” but also offers an explanation of how the e-retailer responds to its competitor’s action in shifting from one format to the other. We consider that each e-retailer and his two suppliers form a group, that is, R1, S1, S2 form channel1 or group1, and R2, S1, S2 form channel2 or group2. Consequently, the profit of e-retailer j ’s channel under configuration T is GT

j = RT

j + ST

i,j + ST

3−i,j , where i, j = 1, 2 , T = CDD,CPP,CDP . When the e-retailer

chooses the cooperation agreement, it will maximize the channel’s profit as its objec- tive (Table 3).

5.2.1 With symmetric e‑retailers

We consider that both e-retailers always have the same choices regarding the selling format. They can offer a pure “two-sided platform” (i.e., CPP ), a pure “reseller” (i.e., CDD ), or a hybrid format (i.e., C(DP)(DP) ). We compare these configurations in Prop- osition 4.

Proposition 4 For a cross-sales supply chain with symmetric e-retailers, two thresholds r7(b) and r8(b) can be found such that

Table 3 The profits of the suppliers and the e-retailers of cross-sales supply chain

L1 = 72 + 451r − 3r2 , L2 = 80 + 636r − 16r2 , L3 = 32 + 333r − 21r2 , L4 = 881 − 86r + r 2 ,

L5 = 11708 − 198r + r 2 , L6 = 577 − 148r + 15r2

profit DD PP DP (DP)(DP)

S1 1−b2

(1+3b)(2−b)2 (1−r)(1−b2)

2(1+3b)

(1−r)(1−b2)(1+3b)(4+16b+(16−r)b2)

4(2+8b+(1−r+6)b2−(1+r)b3)2 (1−r)(1−b2)(4+17b+17b2)

2(1+3b)(4+15b+(12−r)b2−(3+r)b3)2

S2 1−b2

(1+3b)(2−b)2 (1−r)(1−b2)

2(1+3b)

(1−r)(1−b2)(1+3b)(4+16b+(16−r)b2)

4(2+8b+(1−r+6)b2−(1+r)b3)2 (1−b2)(2+(8−r)b+(8−3r)b2)

(1+3b)(4+15b+(12−r)b2−(3+r)b3)2

R1 1−b2

2(1+3b)(2−b)2 r(1−b2)

2(1+3b)

(1−r)2(1+3b)(1−b2)b2

2(2+8b+(1−r+6)b2−(1+r)b3)2 (1−b2)(4+16r+4(35r+7)b+L1b

2+L2b 3+L3b

4)

4(1+3b)(4+15b+(12−r)b2−(3+r)b3)2

R2 1−b2

2(1+3b)(2−b)2 r(1−b2)

2(1+3b)

r(1+3b)(2+3b)(2+5b)(1−b2)b2

2(2+8b+(1−r+6)b2−(1+r)b3)2 (1−b2)(4+16r+4(35r+7)b+L1b

2+L2b 3+L3b

4)

4(1+3b)(4+15b+(12−r)b2−(3+r)b3)2

G1 3(1−b2)

2(1+3b)(2−b)2 1−b2

2(1+3b)

(1−r)2(1+3b)(1−b2)b2

(2+8b+(1−r+6)b2−(1+r)b3)2 (1−b2)(36+4(73−3r)b+L4b

2+L5b 3+L6b

4)

4(1+3b)(4+15b+(12−r)b2−(3+r)b3)2

G2 3(1−b2)

2(1+3b)(2−b)2 1−b2

2(1+3b)

(1+3b)(2+3b)(2+5b)(1−b2)

2(2+8b+(1−r+6)b2−(1+r)b3)2 (1−b2)(36+4(73−3r)b+L4b

2+L5b 3+L6b

4)

4(1+3b)(4+15b+(12−r)b2−(3+r)b3)2

411

1 3

A choice of selling format in the online marketplace with…

where

Figure  5 offers graphically the results of proposition 4. The rectangular area {(b, r)|(0 < b < 1, 0 < r < 1)} is divided into four regions (IV, V, VI and VII) by three curves: (i) GCDD

1 (b, r) = GCPP

1 (b, r) , (ii) GCDD

1 (b, r) = G

C(DP)(DP)

1 (b, r) , (iii)

GCPP 1

(b, r) = G C(DP)(DP)

1 (b, r) . In region IV, the profits of two e-retailers’ groups

under a platform selling format is the highest of the three formats. In regions V and VI, using a hybrid format (i.e., configuration C(DP)(DP) ) is the best option for the two e-retailers’ groups. In region VII, traditional reselling brings more profits for two e-retailers’ groups than the other two formats. The above

⎧ ⎪⎪⎨⎪⎪⎩

G CDD

1 > G

C(DP)(DP)

1 > G

CPP

1 if b ∈ (0, 0.2341), or b ∈ (0.2341, 0.2564)andr ∈ (r7, 1)

G CDD

1 > G

CPP

1 > G

C(DP)(DP)

1 if b ∈ (0.2564, 0.2679), or b ∈ (0.2341, 0.2564) and r ∈ (0, r7)

G CPP

1 > G

CDD

1 > G

C(DP)(DP)

1 if b ∈ (0.2679, 0.2781), or b ∈ (0.2781, 0.2951) and r ∈ (0, r8)

G CPP

1 > G

C(DP)(DP)

1 > G

CDD

1 if b ∈ (0.2951, 1), or b ∈ (0.2781, 0.2951) and r ∈ (r8, 1)

.

r7(b) =

√ Δ1 − 44b6 + 123b5 + 183b4 + 32b3 − 28b2 − 8b

9b6 + 4b5 + 3b4 + 12b3 + 4b2 ,

r8(b) = −6b5 + 52b4 + 79b3 + 28b2 − 3b − 2

2b5 + 4b4 + 5b3 + 4b2 + b ,

Δ1 = 5491b12 − 13294b11 − 15045b10 + 31620b9 + 31482b8

− 16372b7 − 27526b6 − 8616b5 + 1064b4 + 960b3 + 128b2.

GCPP(b, r)=GC(DP)(DP)(b, r) GCDD(b, r)=GCPP(b, r) GCDD(b, r)=GC(DP)(DP)(b, r)

VII:GCPP<GC(DP)(DP)<GCDD

VI:GCPP<GCDD<GC(DP)(DP)

V:GCDD<GCPP<GC(DP)(DP)

IV:GCDD< GC(DP)(DP)<GCPP

1 1

1

11

1

1 1 1

11

1

1 1 1

1 1 1

Fig. 5 Selling format choice of a cross-sales supply chain with symmetric e-retailers

412 X. Li, X. Ai

1 3

results suggest that two e-retailers’ groups can benefit from a platform selling format when channel substitutability is low. However, as channel substitutability rises, the groups under a platform selling format cannot benefit from this format because of price competition. When channel substitutability is high, the groups will prefer a traditional reselling format to a platform selling format.

5.2.2 With asymmetric e‑retailers

Above, we pay attention to symmetrical e-retailers and find that two e-retailers that could trade with two suppliers through each format would not want to do so simulta- neously. In reality, e-retailers that adopt different selling formats are quite prevalent, such as Taobao and JingDong. Taobao is one of the most common “two-sided plat- forms”, and JingDong is one of the most common “resellers” in China. In the fol- lowing sections, we analyze the case of the online retailers’ choice between the pure “two-sided platform” and the pure “reseller”. We compare the three configurations CDD,CPP,CDP and show the equilibrium selling formats.

Theoretically, a cooperative agreement can be a mutually beneficial choice for both an e-retailer and two suppliers only if this channel or group is more Pareto-effi- cient; otherwise, there is no merit in forming this agreement. The following proposi- tions regarding overall channel efficiency provide a guideline for potential coopera- tion via revenue-sharing.

Comparing GDD 1

to GPP 1

, we find:

Proposition 5 When two e-retailers in the cross-sales supply chain are offer- ing traditional reselling format, if 0 < b < 0.2679, both e-retailers’ group have an incentive to adopt a platform selling format because GCDD

1 < GCPP

1 .

Proposition 5 indicates that if online retailers observe that the degree of substitu- tion (b) is below 0.2679, cooperation in two e-retailers’ groups to move together to use a platform selling format is an optimal strategy for both suppliers and e-retail- ers because a cooperative agreement can be a mutually beneficial choice for both the e-retailer and the suppliers only if this group is more Pareto-efficient. As long as the revenue-sharing rate setting is at a certain level, both suppliers and e-retail- ers can capture more profit using a platform selling format. Otherwise, when 0.2679 < b < 1 , a traditional reselling is a better strategy for either suppliers or e-retailers, or for both.

The following proposition compares GDD 1

to GCPD 1

.

Proposition 6 When one e-retailer in the cross-sales supply chain is using a tra- ditional reselling format, if (i) 0 < b < 0.7624, or (ii) 0.7624 < b < 0.9574 and 1 > r > r9(b), the other e-retailer is incentivized to move from a traditional reselling format to a platform selling format,

where r9 =

√ b(16 − 5Δ2) + (3b2 − 2)

√ Δ2 − 2b3 + 14b2 + 4

2b2(1 + b) , Δ2 = 4(15b2 + 16b + 4)∕3.

413

1 3

A choice of selling format in the online marketplace with…

Proposition 6 shows that when R2 in the cross-sales supply chain is using a tradi- tional reselling format, R1’s choice not only depends on b but also is influenced by r. If the degree of substitution is low, or the degree of substitution is relatively high and the revenue-sharing rate of R1 is in the (r9, 1) range, a platform selling format is more profitable than a traditional reselling format for R1’s group. In addition, this proposition suggests to the managers of R2, that when their competitor R1 is mov- ing to a platform selling format and b is below 0.7624 or 0.7624 < b < 0.9574 and 1 > r > r9(b) , it should reevaluate its current selling format.

We summarize the results in proposition 7 by comparing GCDP 1

and GCPP 1

Proposition 7 When one e-retailer in the cross-sales supply chain is using a plat- form selling format, the other e-retailer is incentivized to move from a traditional reselling format to a platform selling format, as GCDP

1 < GCPP

1 .

Proposition 7 implies that contrary to propositions 6 and 7, when R2 is currently adopting a platform selling format, a platform selling format is always the best option for R1’s group. In other words, as long as the revenue-sharing rate is at a cer- tain level, a platform selling format is a win–win strategy for both suppliers and R1.

Theorem 2 Under a cross-sales supply chain, b can be found such that platform selling is the unique Nash equilibrium for all groups as 0 < b < 0.2679.

Figure  6 illustrates the results of Theorem  2 in the rectangular area {(b, r)|(0 < b < 1, 0 < r < 1)} . It shows that in region I, the CPP is not only a

GCDD(b, r)=GCPP(b, r) GCDD(b, r)=GCDP(b, r)

I: CPP

1

22

1

II: CPP

III: CPP/CDD

Fig. 6 Selling format choice of a cross-sales supply chain with asymmetric e-retailers

414 X. Li, X. Ai

1 3

unique equilibrium but also the best option for both e-retailers and suppliers, as long as the revenue-sharing rate is at a certain level, because two e-retailers’ groups can earn more profits under a platform selling format than under a traditional reselling format. In region II, the CPP is also a unique equilibrium but not the best option for both e-retailers and suppliers because the profits of the two e-retailers’ groups are higher under a traditional reselling format than under a platform selling format, leading to the classical prisoner’s dilemma. In region III, CPP and CDD are the Nash equilibrium, CDD is the best option, and CPP is the prisoner’s dilemma.

The above results indicate that as the degree of channel substitutability increases, the equilibrium selling format of two e-retailers will gradually move from a platform selling format to a reselling format with. In other words, the two e-retailers’ channel can earn more profits from a platform selling format for channels of low substitut- ability and from a reselling format for channels of high substitutability for the fol- lowing reasons. When channel competition is not intense, a platform selling format can induce suppliers to set lower retail prices and generate more demand, leading the e-retailers to earn more profit sharing. Conversely, when channel competition intensifies, a platform selling format intensifies the price competition, which hurts group benefits, and the two e-retailers will thus move to a reselling format.

This theorem can be well explained by the case of JingDong.com and Taobao. com, two of the largest B2C online retailers in the world, and we pay more atten- tion to food, household appliances and electronic product on two platforms. At the beginning of the establishment of JingDong.com, it mainly made profits from the resale of suppliers’ household appliances and electronic products. Because of its large difference with Taobao.com’s development model, the competition between the two is weak and the degree of substitution is low. In this case, Taobao.com’s household appliances and electronic product have always adopted the platform sales model. However, with the continuous growth of JingDong.com’s scale and increas- ing market share, this situation has been broken. In recent years, the competition between the two platforms has been very fierce, and Taobao.com has also begun to absorb the same model of household appliances and electronic product as JingDong. com, through Suning. It is worth mentioning that for the food category, Taobao.com changed from the initial platform sales to the Tmall supermarket in the resale mode, which is similar to JingDong.com. This is the embodiment of the format “CPP” and format “CDD” as the Nash equilibrium in region III.

6 Conclusions

In this paper, we consider a cross-sales supply chain model consisting of two com- peting manufacturers and two competing e-retailers. As Stackelberg leaders, the e-retailers can choose different selling formats, either that of a “two-sided platform”, serving both suppliers and customers, or a reseller, ordering from suppliers and also selling competing products through their own online platforms. Each supplier adopts an exclusive-sales strategy, selling products through one exclusive e-retailer, or a cross-sales strategy, selling products through two common e-retailers. We presented the members’ and entire channels’ strategy decisions for different selling format

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A choice of selling format in the online marketplace with…

configurations and developed the impact of channel competition on the e-retailers’ equilibrium choice to provide a new perspective on product selling.

Our analysis shows that consumers must pay a higher price under a reselling for- mat, and the retail prices reduce with the number of platform selling formats. It also shows that the selling format choice of e-retailers depends on channel competition and revenue sharing. With an exclusive-sales supply chain, when channel substitutability is low and the revenue-sharing rate is in a certain level, a platform selling format is the unique Nash equilibrium for all supply chain members; otherwise, a reselling format is preferred by either suppliers, or e-retailers, or all members of the supply chain. With a cross-sales supply chain, as channel substitutability increases, two symmetric e-retail- ers are incentivized to move from a platform selling format to a reselling format. For two asymmetric e-retailers, only platform selling is the unique Nash equilibrium for all channels. Furthermore, we demonstrate the impact of such action on competing supply chains when an e-retailer has an incentive to move from one format to the other.

There are several interesting directions for future research on this issue. First, we only discuss the impact of revenue-sharing cooperation under a platform selling format. In reality, suppliers using an online platform to sell their products always pay fixed cost (i.e., the platform fee) plus the revenue sharing. Second, it would be interesting to ana- lyze what occurs when one e-retailer offers a single selling format and the other offers a hybrid selling format. Finally, we consider a supply chain structure of two suppliers and two e-retailers, so a natural extension would be to study a competing supply chain consisting of multiple suppliers and multiple e-retailers.

Acknowledgements The authors gratefully acknowledge financial support from the National Natural Sci- ence Foundation of China (Grants Nos. 71372140, 71531003 and 71432003), the Humanities and Social Science Planning Youth Fund of the Ministry of Education of China (Grants Nos. 16YJC630057)

Appendix

Proof of demand function for Eq. (2)

First, use the first order conditions (FOCs) of the the utility function of the consumer Eq. (1):

we can find the optimal values of qij:

Then, solving Eqs. (11)–(14), we can get:

�U / �qij = aij − qij − b(q3−i,j + qi,3−j + q3−i,3−j) − pij = 0

(11)q11 = a11 − bq12 − bq21 − bq22 − p11,

(12)q12 = a12 − bq11 − bq21 − bq22 − p12,

(13)q21 = a21 − bq12 − bq11 − bq22 − p21, and

(14)q22 = a22 − bq12 − bq21 − bq11 − p22.

416 X. Li, X. Ai

1 3

Simplifying (15), we can get:

Proof of Lemma 1–3 First, we insert the (inverse) demand Eq. (2) into the retailers’ profit functions (4). That gives us the following updated retailers’ profit functions:

Then use the first order conditions (FOCs) of the profit functions (16) to find the optimal values of pij , and it can easily be proven that the FOCs guarantee optimality.

Second, inserting Eq.  (17) into the suppliers’ profit functions (3) and using the first order conditions (FOCs), we can find the optimal values of wij:

It can easily be proven that the FOCs guarantee optimality. Third, inserting Eq. (5) into Eq. (17), we can derive the optimal values of pij:

The Proof of Lemma 1 is completed. □

The Proof of Lemmas 2 and 3 is similar to the Proof of Lemma 3.

Proof of Proposition 1

Then, we can derive that pEDD 11

> pEDP 11

> pEPD 11

(pEDP 22

) > pEPP 11

. The proof of Proposition 1 is completed. □

Proof of  Proposition 2 (i) SEDD i

∕SEPP i

= (1−r)(1−b)

(1+b)(2−b)2 ∕

2(1−b)

(1+b)(4−3b)2 =

(1−r)(4−3b)2

2(2−b)2 , and we

can get

(15)qij = (a − ab − pij − 2bpij + bpi,3−j + bp3−i,j + bp3−i,3−j)∕(−3b 2 + 2b + 1).

qij = Aij − �pij + � ∑ mn≠ij

pmn, andAij = a∕(1 + 3b),

� = (1 + 2b)∕[(1 − b)(1 + 3b)], � = b∕[(1 − b)(1 + 3b)].

(16)REDD j

= (pij − w ij )(aij − ba3−i,3−j − pij + bp3−i,3−j)∕(1 − b2), i, j = {1, 2}

(17)p ij = wij∕2 − b∕2 + bp3−i,3−j∕2 + 1∕2 i, j = {1, 2}

(5)wEDD 11

=wEDD 22

=2(1 − b)∕(4 − 3b)

(6)pEDD 11

= pEDD 22

= 3(1 − b)∕(4 − 3b)

pEDD 11

pEDP 11

= 3(1 − b)∕(4 − 3b)

3(1 − b)(2 + b)∕(8 − 3b2) =

(8 − 3b2)

(2 + b)(4 − 3b) =

8 − 3b2

8 − 3b2 − 2b > 1,

pEDP 11

pEPD 11

= 3(1 − b)(2 + b)∕(8 − 3b2)

(1 − b)(4 + 3b)∕(8 − 3b2) =

3(2 + b)

(4 + 3b) > 1, and

pEDP 22

pEPP 11

= (1 − b)(4 + 3b)∕(8 − 3b2)

(1 − b)∕(2 − b) =

(4 + 3b)(2 − b)

(8 − 3b2) =

8 − 3b2 + 2b

8 − 3b2 > 1.

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A choice of selling format in the online marketplace with…

if r = r1 = (8 − 16b + 7b2)∕(3b − 4)2 , then SEDD i

=SEPP i

; if r > r1 = (8 − 16b + 7b2)∕(3b − 4)2 , then SEDD

i > SEPP

i ;

if r < r1 = (8 − 16b + 7b2)∕(3b − 4)2 , then SEDD i

> SEPP i

REDD i

∕REPP i

= r(1−b)

(1+b)(2−b)2 ∕

(1−b)

(1+b)(4−3b)2 =

r(4−3b)2

(2−b)2 , and we can get

if r = r2 = (2 − b)2∕(4 − 3b)2 , then REDD j

=REPP j

;

if r < r2 = (2 − b)2∕(4 − 3b)2 , then REDD j

> REPP j

;

if r > r2 = (2 − b)2∕(4 − 3b)2 , then REDD j

< REPP j

.

Therefore, SEDD i

≤ SEPP i

, REDD j

≤ REPP j

, for i, j = {1, 2}; if r2(b) ≤ r ≤ r1(b).

Similarly, we can derive (ii) SEDD

1 ≤ SEPD

1 , REDD

1 ≤ REPD

1 , if r4(b) ≤ r ≤ r3(b).

(iii) SEPD 2

≤ SEPP 2

, REPD 2

≤ REPP 2

, if r6(b) ≤ r ≤ r5(b).

The Proof of Proposition 2 is completed. □

Proof of Corollary 1

And by the expression of r1, r2, r3, r4, r5, r6 , we can attain

If 0 < b < 0.4226 , then r1(b) > r2(b) ; otherwise, r1(b) ≤ r2(b). If 0 < b < 0.7506 , then r3(b) > r4(b) ; otherwise, r3(b) ≤ r4(b). If 0 < b < 0.9194 , then r5(b) > r6(b) ; otherwise, r5(b) ≤ r6(b). If 0 < b < 0.5021 , then r1(b) > r4(b) ; otherwise, r1(b) ≤ r4(b). If 0 < b < 0.5144 , then r1(b) > r6(b) ; otherwise, r1(b) ≤ r6(b). If 0 < b < 0.6074 , then r3(b) > r2(b) ; otherwise, r3(b) ≤ r2(b). If 0 < b < 0.7899 , then r3(b) > r6(b) ; otherwise, r3(b) ≤ r6(b). If 0 < b < 0.6518 , then r5(b) > r2(b) ; otherwise, r5(b) ≤ r2(b). If 0 < b < 0.8312 , then r5(b) > r4(b) ; otherwise, r5(b) ≤ r4(b).

Therefore, if b ∈ (0, 0.4026) , then r6(b) ≤ r4(b) ≤ r2(b) ≤ r1(b) ≤ r3(b) ≤ r5(b) . Based on the above results, we can derive the first Corollary 1.

The Proof of Corollary 1 is completed. □

Proof of Theorem 1 From Corollary 1, we can derive that:

(1) If b ∈ (0, 0.4226) , then SEDD i

< SEPP i

, SEDD 2

< SEDP 2

, and SEDP 1

< SEPP 1

; REDD 1

< REPP 1

, REDD 1

< REDP 1

, and REDP 1

< REPP 1

; REDD 2

< REPP 2

, REDD 2

< REDP 2

, and REDP

2 < REPP

2 .

r 3 − r

1 = (128 − 192b2 + 63b4)∕((4 − 3b)2(4 + 3b)2) − (8 − 16b + 7b2)∕(3b − 4)2 > 0,

r 5 − r

3 = (32 − 32b2 + 7b4)2∕(8 − 3b2)2 − (128 − 192b2 + 63b4)∕((4 − 3b)2(4 + 3b)2) > 0,

r 6 − r

4 = (2 − b)2(2 + b)2∕(8 − 3b2)2 − (8 − 3b2)2∕((4 − 3b)2(4 + 3b)2) > 0,

r 4 − r

2 = (8 − 3b2)2∕((4 − 3b)2(4 + 3b)2) − (2 − b)2∕(4 − 3b)2 > 0.

418 X. Li, X. Ai

1 3

(2) If b ∈ (0, 0.9194) , then SEDD i

> SEPP i

, SEDD 2

> SEDP 2

, and SEDP 1

> SEPP 1

; REDD 1

> REPP 1

, REDD 1

> REDP 1

, and REDP 1

> REPP 1

; REDD 2

> REPP 2

, REDD 2

> REDP 2

, and REDP

2 > REPP

2 .

Therefore, we can derive that

(i) for any given b ∈ (0, 0.2037) , there exist r 1 and r

2 such that a platform selling

format is the unique Nash equilibrium for all supply chain members, as long as r

2 < r < r

1 .

(ii) for any given b ∈ (0.9194, 1) , there exists r 5 and r

6 such that a reselling for-

mat is the unique Nash equilibrium for all supply chain members, as long as r 5 < r < r

6 .

The Proof of Theorem 1 is completed. □

Proof of Lemmas 4–7 For the case CDD, take the first derivative of Eq. (8) w.r.t. p 11

and p

12 :

The Hessian matrix is:

With (16) and (17), we have:

Inserting Eqs. (18) and (19) into the suppliers’ profit functions (7) and using the first order conditions (FOCs), we can get the optimal values of wCDD

ij ,which is shown

as Eq. (10). Putting Eq. (10) into Eqs. (18) and (19), and then solving them, we can attain the optimal values of pCDD

ij shown as Eq. (9).

(16)

�RCDD 1

�pCDD 11

= a − �(2p11 − w11) + �(p21 − w21) + �(p12 + p21 + p22) = 0, and

(17) �RCDD

2

�pCDD 12

= a − �(2p12 − w12) + �(p22 − w22) + �(p11 + p21 + p22) = 0.

H =

⎡⎢⎢⎣

𝜕2RCDD 1

𝜕pCDD2 11

𝜕2RCDD 1

𝜕pCDD 11

𝜕pCDD 21

𝜕2RCDD 1

𝜕pCDD 21

𝜕pCDD 11

𝜕2RCDD 1

𝜕pCDD2 21

⎤⎥⎥⎦ =

� −2𝛽 𝜃

𝜃 −2𝛽

� , and then �H� > 0.

(18) p11 = (a − w11 − ab + bp12 + 2bp21 + bp22 + 2bw11 − bw21)∕(2b + 2), and

(19)p12 = (a − w12 − ab + bp11 + 2bp22 + bp21 + 2bw12 − bw22)∕(2b + 2),

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A choice of selling format in the online marketplace with…

The Hessian matrix is following as:

The Proof of Lemma 4 is completed. □

The Proof of Lemma 5–7 is similar to the Proof of Lemma 4.

Proof of Proposition 3 From pCDD 11

and pCDD 11

’s expression, we can get

If 0 < b < ( √ 20 − 4r + r2 + r)∕(10 − 2r) , and then p

CDD 11

pCDP 11

= 3(1−b)

2(2−b) ∕

4+(11+r)b−2b2−(r+13)b3

2(2+8b+(1−r+6)b2−(1+r)b3) > 1;

if ( √ 20 − 4r + r2 + r)∕(10 − 2r) ≤ b < 1 , and then p

CDD 11

pCDP 11

= 3(1−b)

2(2−b) ∕

4+(11+r)b−2b2−(r+13)b3

2(2+8b+(1−r+6)b2−(1+r)b3) ≤ 1;

In addition,

Then, we can derive Proposition 3. The proof of Proposition 3 is completed. □

Proof of Proposition 4 Let r7 be the threshold of GCPP 1

= G C(DP)(DP)

1 , yielding,

Let r8 be the threshold of GCDD 1

= G C(DP)(DP)

1 , yielding,

H =

⎡ ⎢⎢⎣

𝜕2SCDD 1

𝜕wCDD2 11

𝜕2SCDD 1

𝜕wCDD 11

𝜕wCDD 21

𝜕2SCDD 1

𝜕wCDD 21

𝜕wCDD 11

𝜕2SCDD 1

𝜕wCDD2 21

⎤ ⎥⎥⎦ , and

�H� = (3311b6 + 10736b5 + 14204b4 + 9856b3

+ 3792b2 + 768b + 64)∕[64(1 + 2b)4(1 + 2b − 3b2)] > 0.

pCDP 11

p C(DP)(DP)

11

= (1 − b)

( 15b + br + b2r + 13b2 + 4

)( 30b − 2b2r − 2b3r + 24b2 − 6b3 + 8

) ( − 17b3 + 13b + 4

)( 16b − 2b2r − 2b3r + 14b2 − 2b3 + 4

) > 1,

p C(DP)(DP)

11

p C(PD)(PD)

11

=

( 22b + br + b2r + 20b2 + 6

) ( 4 − 17b3 + 17b

) > 1,

p C(DP)(DP)

11

pCPD 11

=

( − 17b3 + 13b + 4

)( 16b − 2b2r − 2b3r + 14b2 − 2b3 + 4

) ( −9b3 + 7b + 2

)( 30b − 2b2r − 2b3r + 24b2 − 6b3 + 8

) > 1,

pCPD 11

pCPP 11

=

( 9b3 + 9b + 2

) ( 8b − b2r − b3r + 7b2 − 6b3 + 2

) > 1.

r7(b)=

√ Δ1 − 44b6 + 123b5 + 183b4 + 32b3 − 28b2 − 8b

9b6 + 4b5 + 3b4 + 12b3 + 4b2 .

r8(b)= −6b5+52b4+79b3+28b2 − 3b − 2

2b5 + 4b4 + 5b3 + 4b2 + b ,

420 X. Li, X. Ai

1 3

where

The rectangular area {(b, r)|(0 < b < 1, 0 < r < 1)} is divided into four regions (IV, V, VI and VII) by three curves: GCDD

1 (b, r) = GCPP

1 (b, r) ,

GCDD 1

(b, r) = G C(DP)(DP)

1 (b, r) and GCPP

1 (b, r) = G

C(DP)(DP)

1 (b, r) . then, we can get,

If (b, r) ∈ IV , and then GCDD 1

(b, r) < G C(DP)(DP)

1 (b, r) < GCPP

1 (b, r);

If (b, r) ∈ V , and then GCDD 1

(b, r) < GCPP 1

(b, r) < G C(DP)(DP)

1 (b, r);

If (b, r) ∈ VI , and then GCPP 1

(b, r) < GCDD 1

(b, r) < G C(DP)(DP)

1 (b, r);

If (b, r) ∈ VII , and then GCPP 1

(b, r) < G C(DP)(DP)

1 (b, r) < GCDD

1 (b, r).

Therefore, we can derive Proposition 4. The proof of Proposition 4 is completed. □

Proof of Proposition 5

Then, we can get that when 0 < b < 0.2679 , there exists GCDD 1

< GCPP 1

. The proof of Proposition 5 is completed. □

Proof of Proposition 6

Let r9 be the threshold of GCPD 1

= GCDD 1

, yielding,

In addition,

when 0 < b < 0.7624 , there exists 0 > r9(b); when 0.7624 < b < 0.9574 , there exists 1 > r > r9(b).

Therefore, when one e-retailer in the cross-sales supply chain is using a tradi- tional reselling format, if (i) 0 < b < 0.7624 , or (ii) 0.7624 < b < 0.9574 and

Δ1 = 5491b12 − 13294b11 − 15045b10 + 31620b9+31482b8 − 16372b7

− 27526b6 − 8616b5 + 1064b4 + 960b3 + 128b2

GCDD 1

− GCPP 1

= 3(1 − b2)

2(1 + 3b)(2 − b)2 −

(1 − b2)

2(1 + 3b) =

(1 − b2)[3 − 2(2 − b)2]

2(1 + 3b)(2 − b)2 ,

GCDD 1

− GCPD 1

= 3(1 − b2)

2(1 + 3b)(2 − b)2 −

(1 + 3b)(2 + 3b)(2 + 5b)(1 − b2)

2(2 + 8b + (1 − r + 6)b2 − (1 + r)b3)2

r9 =

√ b(16 − 5Δ2) + (3b2 − 2)

√ Δ2 − 2b3 + 14b2 + 4

2b2(1 + b) , Δ2 = 4(15b2 + 16b + 4)∕3.

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A choice of selling format in the online marketplace with…

1 > r > r9(b) , the other e-retailer is incentivized to move from a traditional reselling format to a platform selling format,

The proof of Proposition 6 is completed. □

Proof of Proposition 7

Then, when one e-retailer in the cross-sales supply chain is using a platform selling format, the other e-retailer is incentivized to move from a traditional reselling for- mat to a platform selling format, as GCDP

1 < GCPP

1 .

The proof of Proposition 7 is completed. □

Proof of  Theorem  2 The rectangular area {(b, r)|(0 < b < 1, 0 < r < 1)} is divided into four regions (I, II and III) by two curves: GCDD

1 (b, r) = GCPP

1 (b, r) ,

GCDD 1

(b, r) = GCDP 1

(b, r) and GCPP 1

(b, r) = GCDP 1

(b, r). Then, we can get if (b, r) ∈ I (i.e. 0 < b < 0.2679 ), then

GCDD 1

(b, r) < GCDP 1

(b, r) < GCPP 1

(b, r);

If (b, r) ∈ II , (i.e. 02679 < b < 0.7624 , or 0.7624 < b < 0.9574 and 1 > r > r9(b)), then GCDD

1 (b, r) < GCPP

1 (b, r) < GCDP

1 (b, r);

If (b, r) ∈ III,(i.e. 0.9574 < b < 1 , or 0.7624 < b < 0.9574 and 0 < r < r9(b)), then GCDD

1 (b, r) < G

C(DP)(DP)

1 (b, r),GCPP

1 (b, r) < G

C(DP)(DP)

1 (b, r).

In the other words, under a cross-sales supply chain, b can be found such that platform selling is the unique Nash equilibrium for all groups as 0 < b < 0.2679.

The proof of Theorem 2 is completed. □

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where r9 =

√ b(16 − 5Δ2) + (3b2 − 2)

√ Δ2 − 2b3 + 14b2 + 4

2b2(1 + b) , Δ2 = 4(15b2 + 16b + 4)∕3.

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(1 − r)2(1 + 3b)(1 − b2)b2

(2 + 8b + (1 − r + 6)b2 − (1 + r)b3)2 > 0,

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