Delivery arrangement in online distribution channels under different power structures

Omega 126 (2024) 103070 Contents lists available at ScienceDirect Omega journal homepage: www.elsevier.com/locate/omega Delivery arrangement in online distribution channels under different power structures✰ Xiongwei Zhou a, Chaoqun Zhu a, Dan Cai b, * a b School of Business, Central South University, Changsha, 410083, PR China School of Economics and Management, Chongqing University of Posts and Telecommunications, Chongqing, 400065, PR China A R T I C L E I N F O A B S T R A C T Keywords: Supply chain management Delivery arrangement Online distribution channel Nonlinear delivery cost Power structure Logistics delivery is a critical but costly operation in e-commerce. The e-retailer who sells products online may ask the supplier to ship products to customers, and the supplier responds by accepting or rejecting such a delivery proposal. Motivated by these observations, we consider an online distribution channel with two competing suppliers and one common e-retailer and analytically characterize the equilibrium delivery arrangement based on the nonlinear delivery cost in the E-retailer-Stackelberg, Supplier-Stackelberg, and Vertical-Nash games. Our analysis reveals that the market follower might be more willing to deliver products than the market leader in each Stackelberg game. Counterintuitively, the e-retailer prefers undertaking the delivery when there is a large delivery diseconomy under each power structure but prefers asking both suppliers to deliver products when there is a delivery economy (a small delivery diseconomy) in the E-retailer-Stackelberg and Vertical-Nash games (Supplier-Stackelberg game). The suppliers agree to ship products only when the delivery economy is apparent in the E-retailer-Stackelberg game but reject to do so in the Vertical-Nash game; in the Supplier-Stackelberg game, they accept the delivery proposal even though there is a significant delivery diseconomy. Although both suppliers can benefit from refusing to undertake the delivery when there is fierce competition in each Stackelberg game, interestingly, they still agree to do so and trap themselves in a prisoner’s dilemma. In each Stackelberg game, only shifting the delivery responsibility to the suppliers might achieve a triple-win situation for the channel members, consumers, and society; in the Vertical-Nash game, however, this situation never occurs, and the consumer surplus and social welfare are invariant to the delivery arrangement. 1. Introduction Developments in payment and internet technologies have stimulated the rapid growth of e-commerce worldwide in the past decades. One salient feature of online retailing is that firms must deliver products to consumers once the payment is completed. It is well established that delivering products is a costly operation in online retailing (X. [1–6]). As per a recent report by the Council of Supply Chain Management Pro fessionals, logistics costs for businesses in the United States rose by 22.4 % to $1.85 trillion, accounting for 8 % of the country’s GDP in 2021.1 Similarly, the China Federation of Logistics and Purchasing reported that China spent approximately 335.2 trillion RMB on logistics in 2021, an increase of 9.2 % from the previous year.2 Given the significance of logistics delivery in online transactions, delivery arrangement, i.e., who is responsible for delivering products to consumers, has always been among the most critical issues for online distribution channel members. It is evident from practical observations that e-retailers who sell products online often take up the responsibility of product delivery. However, for some product categories, e-retailers may ask the suppliers to ship products to consumers directly. In response, the suppliers accept or reject this delivery proposal. For instance, JD.com, one of the largest e-retail giants in China, delivers substitutable products when selling ✰ Area - Supply Chain Management This manuscript was processed by Associate Editor Shen. * Corresponding author at: School of Economics and Management, Chongqing University of Posts and Telecommunications, Chongqing, 400065, PR China. E-mail address: caidancsu@126.com (D. Cai). 1 https://cscmp.org/ (accessed July 10, 2023). 2 http://www.chinawuliu.com.cn/ (accessed July 10, 2023). https://doi.org/10.1016/j.omega.2024.103070 Received 16 July 2023; Accepted 27 February 2024 Available online 8 March 2024 0305-0483/© 2024 Elsevier Ltd. All rights reserved. X. Zhou et al. Omega 126 (2024) 103070 products for competing suppliers such as Haier3 and Midea.4 Conversely, for suppliers like Li-Ning5 and ANTA,6 JD.com may ask them to deliver products, and both agree. The same commercial phenomenon also oc curs with other e-retailers such as Gome,7 Suning,8 Amazon (Direct Fulfillment Program) ,9 and Walmart (Dropship Vendor Program) .10 Another typical example is that, on the wholesale website 1688 in China, some suppliers accept the e-retailer’s delivery proposal while others reject it.11 The above anecdotal evidence implies that the attitudes of channel members towards undertaking the delivery are sometimes inconsistent, which, thus, also stimulates our interest in exploring the underlying mechanism that drives the delivery arrangement in online distribution channels. As mentioned above, product shipping is expensive for firms because it involves associated operations such as warehouse building or renting, package handling, and product transporting. Particularly when encountering scale diseconomies of delivery (G. [7]), channel members might be reluctant to undertake the delivery due to the increased costs. Suppliers usually consider these operational costs when deciding whether to accept the e-retailer’s delivery proposal. However, as is well documented in the literature, the delivery of products may also exhibit economies of scale ([8]; B. [9]). In this regard, JD.com12 and Haier13 have also publicly reported that delivering online orders may exhibit economies of scale. These two types of delivery scale effects are opposite and thus can profoundly affect the channel members’ preferences over undertaking the delivery. Despite these delivery scale effects, little is understood about their implications for the delivery arrangement in online distribution channels. Therefore, further investigation is neces sary to comprehend how and why these different scale effects shape firms’ delivery arrangement preferences. In addition to the delivery scale effect, power structure also mate rially matters because a dominant market power can ensure sufficient profits to cover the delivery cost (Y. [10]) and is essential in influencing channel members’ delivery willingness. When the e-retailer plays a dominant role in online distribution channels, she could claim retail markup at first, which narrows the wholesale price the supplier demands later and thus might reduce his willingness to be responsible for deliv ering products to customers. In contrast, a powerful supplier can sub stantially extract the wholesale price, leaving the e-retailer with inadequate retail markup to compensate for the associated costs when she undertakes the delivery. In recent years, with the expansion of on line sales scale and the continuous integration of related businesses, many e-retailers are becoming more formidable than their suppliers; e-retail giants such as Amazon, Walmart, and JD.com often exploit power to push their suppliers towards the situation where they have less bargaining power ([11]; Yujie [12,13]). In the vertical competition be tween channel members, well-known and influential suppliers such as Huawei, Siemens, Xiaomi, Nike, and Adidas may play a leading role when distributing products through downstream smaller e-retailers ([14]; G. [15]; T.-Y. [16]). In many cases, however, channel members may be involved in the vertical Nash competition (S. [17,18]). Although the existing literature has emphasized the critical role of the power structure in the supply chain’s operation behaviors ([19–23]; X. [24]), there is no existing theory that thoroughly explains how the power structure affects the delivery arrangement in online distribution channels. Both forces discussed above make the problem of delivery arrange ment more complex, and the equilibrium arrangement cannot be readily observed. To fill the research gap, we consider the nonlinear delivery cost reflecting delivery economy or diseconomy and develop a gametheoretic model to fully characterize the equilibrium delivery arrange ment under three power structures: E-retailer-Stackelberg, SupplierStackelberg, and Vertical-Nash games. In particular, we attempt in this paper to pursue the following research questions: (1) How would the power structure and nonlinear delivery cost shape firms’ delivery arrangement preferences? (2) Which delivery arrangement can be sus tained as an equilibrium outcome under different power structures? If so, under what conditions? (3) Is there a particular delivery arrangement that benefits the channel members involved and generates the highest consumer surplus and social welfare concurrently? To address these is sues, we consider an online distribution channel consisting of one common e-retailer and two competing suppliers. The e-retailer and suppliers are assumed to have the ability and funds to undertake the delivery. Initially, the e-retailer decides to ask none, one, or both of the suppliers to ship products to customers. After that, the supplier, condi tional upon receipt of a delivery proposal from the e-retailer, determines whether to undertake the delivery. Based on the interactions between the supplies and the e-retailer, we study three delivery arrangements: EE, SE/ES, and SS. The delivery arrangement EE means that the eretailer undertakes the delivery of two substitutable products, the hybrid delivery arrangement SE/ES denotes that only one of the suppliers de livers products to customers, and the delivery arrangement SS represents that both suppliers are responsible for shipping products to consumers. Our analysis reveals several noteworthy findings. (1) The market follower is more willing to undertake the delivery than the market leader when the delivery is economy in each Stackelberg game. (2) Channel members prefer undertaking the delivery when the delivery diseconomy is significant but may avoid it when there is a delivery economy. (3) The delivery arrangement EE might emerge in equilibrium under each power structure, irrespective of the delivery scale effect; however, the delivery arrangement SS emerges in equilibrium only when the delivery economy is large (the delivery diseconomy is not so small and the competition intensity is not too large) in the E-retailerStackelberg (Supplier-Stackelberg) game; even in a symmetric setting, the hybrid delivery arrangement SE/ES can emerge in equilibrium in the Supplier-Stackelberg game. (4) Interestingly, even if both suppliers can earn higher profits by rejecting the e-retailer’s delivery proposal when there is fierce competition in each Stackelberg game, they still accept it and eventually fall into a prisoner’s dilemma. (5) In each Stackelberg game, only shifting the delivery responsibility to the suppliers might benefit channel members and generate the highest consumer surplus and social welfare concurrently (i.e., a triple-win situation). In the Vertical-Nash game, however, such a situation never occurs, and the consumers and society are indifferent to who ships products. Our contribution is threefold. First, to the best of our knowledge, we are among the first to examine the delivery arrangement in an online distribution channel under different power structures and show that the equilibrium delivery arrangements under the three power structures are dramatically different. Moreover, the market follower who possesses less negotiation power might be more willing than the market leader to be responsible for delivering products. Therefore, our research provides novel insights into how power structure impacts the delivery arrange ment. Second, our analysis provides surprising insights that channel members might benefit from delivering the product with a significant delivery diseconomy but suffer from shipping the product with the apparent delivery economy. Third, our research sheds light on the implication of an e-retailer’s delivery proposal for the competing sup pliers and suggests that such a delivery proposal might lead to a 3 https://item.jd.com/100037217877.html (accessed July 10, 2023). https://item.jd.com/100022748756.html (accessed July 10, 2023). 5 https://item.jd.com/100033049869.html (accessed July 10, 2023). 6 https://item.jd.com/100044484654.html (accessed July 10, 2023). 7 https://item.gome.com.cn/9140420462-1131249887.html (accessed July 10, 2023). 8 https://product.suning.com/0000000000/12183309510.html? (accessed July 10, 2023). 9 https://vendorcentral.amazon.com/ (accessed July 10, 2023). 10 https://supplierhelp.walmart.com/s/ (accessed July 10, 2023). 11 https://www.1688.com (accessed July 10, 2023). 12 https://finance.sina.com.cn/stock/relnews/hk/2023-08-29/docimziwkqa1409310.shtml (accessed October 10, 2023). 13 https://www.sohu.com/a/16607610_115469 (accessed October 10, 2023). 4 2 X. Zhou et al. Omega 126 (2024) 103070 prisoner’s dilemma in each Stackelberg game. Overall, our findings uncover the underlying mechanism behind the delivery arrangement and provide concrete and valuable guidelines for channel members on who and when to undertake the delivery. We also consider three alternative extended scenarios in the Online Appendix, namely asymmetric suppliers with different market poten tials, differentiated delivery scale effects between channel members, and a lower linear delivery cost for the e-retailer, to verify the robustness of our main results. The remainder of the article proceeds as follows. We review the relevant literature in Section 2 and introduce the model in Section 3. Section 4 conducts the comparative analysis, and Section 5 derives the equilibrium delivery arrangement. Section 6 presents the welfare implications, and Section 7 concludes this study. All equilibrium outcomes and proofs are relegated to the Online Appendix. a proposal and ultimately fall into the prisoner’s dilemma. Therefore, on this point, our research sheds light on the dark side of the e-retailer’s delivery proposal. Our work also contributes to the growing literature investigating the impact of power structure on supply chain performance. Power structure usually refers to different timing rules involving demanding profit margins. For instance, S. C. Choi [36] investigates the implications of power structure for manufacturers’ decisions when distributing prod ucts through a retailer in a two-echelon supply chain. Nagarajan and Sošić [37] investigate the role of the power structure in supplier alli ances in a decentralized assembly system with the consideration of Supplier-Stackelberg, Vertical-Nash, and Assembler-Stackelberg game models. Pan, Lai, Leung, &Xiao [38] focus on a supply chain consisting of two competing manufacturers and a shared retailer or a single manufacturer and two competing retailers to demonstrate that the power structure influences the selection of two types of contracts. Using a game-theory-based framework, Shi, Zhang, &Ru [39] examine the impacts of the power structure and demand model on supply chain performance. Luo, Chen, Chen, &Wang [18] explore how the power structure affects pricing policies for differentiated brands by considering horizontal and vertical competition. Liao, Yano, &Trivedi [23] discuss how the power structure influences producer choice and store-brand quality optimization. With a dual-channel supply chain, Yujie Xiao, Niu, Zhang, &Xue [13] analyze how quality differentiation and power structure affect the retailer’s store-brand introduction incentive and profitability. Overall, these studies have discussed the impacts of power structure on the supply chain’s marketing strategy or operational behavior but neglect the implications of power structure for the delivery arrange ment. However, one exception is a study reported by Geng, Krishnan, &Queyranne [40], which explores the focal retailer’s internalization decision in logistics under different power structures. Specifically, considering the three power structures, Geng, Krishnan, &Queyranne [40] examine the conditions under which the focal retailer would benefit from internalizing the inbound logistics that raise its costs. Nevertheless, our work differs from theirs in the following ways. First, we allow two competing suppliers to respond by accepting or rejecting the e-retailer’s delivery proposal. In contrast, they allow only one focal retailer to decide whether or not to undertake the inbound logistics delivery. Importantly, our findings suggest that the competing suppliers might fall into the prisoner’s dilemma by accepting the e-retailer’s de livery proposal. Second, in their study, the delivery cost structure would differ due to the internalization decision, whereas, in our study, the delivery cost structure is consistent regardless of who undertakes the delivery. Under this assumption, each supplier might be willing to un dertake the delivery. Third, their work reveals that one retailer can benefit from undertaking the inbound logistics delivery because such a decision can negatively affect its rival profit; however, our analysis shows that the suppliers can benefit from accepting the e-retailer’s de livery proposal because of the higher demands they can obtain. 2. Literature review Our paper is related to the vast literature on logistics delivery-related operational problems in online distribution channels. As one of the critical operations for online sellers, logistics delivery has attracted much research interest (Y. [25,26]). One stream of the literature ex amines the logistics delivery-related operational problems from the perspective of demand-enhancing service. For example, X. Li, Li, Cai, &Shan [2] explore the service channel choice for supply chain players by considering four different service formats. By comparing three coordi nation models, W. Liu et al. [27] reveal the implications of demand disruption for logistics service supply chain (LSSC) coordination. Qin, Liu, &Tian [4] investigate the strategic and economic impacts of logis tics service sharing in a hybrid platform. Considering a retailer-led supply chain, Lou et al. [28] investigate logistics outsourcing choices and find that the retailer does not necessarily benefit from offering lo gistics services. Some recent works also examine firms’ logistics delivery strategies with other considerations including selling mode choice [29], service channel integration (Y. [10]), direct channel opening (L. [30]), channel structure selection (H. [31]), competition at the e-commerce platform [1] and platform-based co-opetition supply chain structure (P. [32]). Another stream of the literature investigates those problems by focusing on the delivery cost. Among them, Tian, Vakharia, Tan, &Xu [33] study the optimal selling mode for an e-commerce platform and reveal that the fulfillment (i.e., delivering physical goods to customers) costs and upstream competition intensity jointly play a critical role in the choice of selling mode. Niu et al. [34] consider the delivery cost gap between the fast-food restaurant and the online food-delivery platform and derive the optimal delivery arrangement for the fast-food restau rant. Likewise, B. He, Gupta, &Mirchandani [9] discuss the optimal selling and delivery strategy for a local Brick-and-Mortar store in the context of emerging online-to-offline (O2O) third-party platforms. Based on the linear logistics delivery cost, P. He et al. [35] explore the impacts of manufacturer encroachment and the e-commerce platform’s logistics integration strategy. Although the above papers have identified the significant role of logistics delivery in online distribution channels, one missing feature among all existing research is the delivery economy or diseconomy prevalent in real-life business settings. Our aim, however, is to under stand how and why these different scale effects shape firms’ delivery preferences. Counterintuitively, our analysis shows that the channel members prefer undertaking the delivery when there is a large delivery diseconomy but might prefer to refrain from undertaking the delivery when there is a delivery economy. Hence, this unexpected result chal lenges the conventional presumption that the channel members might be less willing to undertake the delivery when the delivery of products is more expensive. Furthermore, departing from the existing literature, we examine the equilibrium delivery arrangement in an online distribution channel with supplier competition. Interestingly, when there is fierce competition, the competing suppliers can achieve higher profits by rejecting the e-retailer’s delivery proposal but cannot help rejecting such 3. Model setup Consider an online distribution channel with two identical suppliers (indexed by 1 and 2) selling substitutable products through a nonex clusive e-retailer (indexed by e). The suppliers produce goods with a symmetric and constant marginal cost, which we normalize to zero without losing generality. We refer to either supplier as he and the eretailer as she. Table 1 summarizes all notations used in this study. In line with the extant literature [41,42], the consumer demand for each product is given by14 14 These demand functions are derived by maximizing the representative ∑ consumer utility U = (aqi − qi 2 /2 − pi qi ) − kq1 q2 . i=1,2 3 X. Zhou et al. Omega 126 (2024) 103070 Table 1 Notation. Notation Description ∧ ∼ − pi qi wsi mei a k cqi bq2i Δei Δsi Πe Πsi EE SE SS CS SW Superscript used to denote the E-retailer-Stackelberg game Superscript used to denote the Supplier-Stackelberg game Superscript used to denote the Vertical-Nash game Retail price of product i Demand of product i Supplier i’s wholesale price E-retailer’s retail markup for product i Market Potential Intensity of competition (degree of product differentiation) Linear part of delivery cost Nonlinear part of delivery cost E-retailer’s unit profit for product i Supplier i’s unit profit E-retailer’s payoff Supplier i’s payoff The delivery arrangement where the e-retailer undertakes the delivery of two substitutable products The hybrid delivery arrangement where supplier 1 undertakes the delivery of his product while the e-retailer undertakes the delivery of supplier 2′s product The delivery arrangement where both suppliers undertake the delivery of their products Consumer surplus Social welfare Fig. 1. Delivery arrangements: EE, SE, and SS. ( ) a p1 kp2 1− + 1+k 1− k 1− k ( ) ⎪ a p2 kp1 ⎪ ⎪ ⎩ q2 = 1− + 1+k 1− k 1− k ⎧ ⎪ ⎪ ⎪ ⎨ q1 = fulfill online orders for the products (e.g., sofa, bed, air conditioner, refrigerator, fresh food, and television) with characteristics such as large volume, heavy weight, perishable, or fragile because the increased sales quantities of these products need more expensive capacity or profes sional operation, and result in a lower delivery efficiency (G. [7]). However, when channel members fulfill online orders for products (e.g., electronic products, digital accessories, clothes, hats, and shoes) that are easy to handle, the delivery might be economy due to economies of scope ([8]; B. [9]). Therefore, to capture the essential feature of the delivery economy or diseconomy, we consider a nonlinear delivery cost cq + bq2 . Specifically, c represents the unit delivery cost, determined by delivery distance, product weight, volume, and other factors; b > 0 in dicates that the firms face the delivery diseconomy, while b < 0 implies that the firms encounter the delivery economy.15 This cost structure has been widely used in operations management literature to characterize the diseconomy or economy of scale [43–46]. Note that channel mem bers are indifferent to who undertakes the delivery in the case of linear delivery cost; thus, we omit the analysis of this trivial case. The delivery cost is assumed to be identical when the e-retailer or the supplier (1) The parameter k ∈ [0, 1) characterizes the degree of product differ entiation; a larger k implies more intense competition between the substitutable products. Here, p1 (p2 ) represents the retail price of prod uct 1 (2), which is the sum of the wholesale price ws1 (ws2 ) that supplier 1 (2) demands and the retail markup me1 (me2 ) that the e-retailer claims for product 1 (2). q1 (q2 ) is the quantity of product 1 (2). Suppliers are assumed to possess equal market potential, a, normalized to 1 without losing generality. We relax this assumption in the Online Appendix D and show that our main results hold qualitatively. In practice, product delivery can often be undertaken by different parties of the online distribution channel (X. [2,29]). In this study, we assume that both the supplier and the e-retailer have sufficient capital and capability to undertake the delivery. As highlighted previously, the delivery of online orders generally exhibits different scale effects. When the marginal delivery cost increases with the online order volume, we refer to it as the delivery diseconomy; conversely, when the marginal delivery cost decreases with the online order volume, we refer to it as the delivery economy. Delivery diseconomy might occur when merchants 15 When b < 0, we focus on −2bc > q so that the delivery is economy. A similar assumption was adopted by Shang, Ha, &Tong (35) and Zhao and Li (47). 4 X. Zhou et al. Omega 126 (2024) 103070 Fig. 2. Sequence of events. undertakes the delivery to clarify the implications of the power structure for the delivery arrangement. We relax this assumption in the Online Appendix E and F to check the robustness of our core model. Following industry practice, we consider three possible delivery ar rangements, depicted in Fig. 1. Under the delivery arrangement EE, the e-retailer undertakes the delivery of two substitutable products and in curs associated costs. Such an arrangement arises if the e-retailer asks neither of the suppliers to undertake the delivery or if the e-retailer asks at least one supplier to undertake the delivery and no suppliers agree to do it. Hence, the channel parties’ payoffs under this arrangement are ΠEE s1 EE EE EE EE EE EE EE EE EE EE EE = wEE s1 q1 , Πs2 = ws2 q2 , and Πe = me1 q1 − (c + bq1 )q1 + me2 q2 − EE (c + bqEE 2 )q2 . Under the hybrid delivery arrangement SE, product 1 is delivered by supplier 1, while product 2 is by the e-retailer; this arrangement occurs when the e-retailer asks at least one supplier to deliver products to customers, but only one supplier does. Thus, the channel parties’ payoffs under the delivery arrangement SE are ΠSE s1 = SE SE SE SE SE SE SE SE SE SE SE wSE s1 q1 − (c + bq1 )q1 , Πs2 = ws2 q2 , and Πe = me1 q1 + me2 q2 − SE (c + bqSE 2 )q2 . Since the two suppliers are symmetric, we omit the analysis of the hybrid delivery arrangement ES in the remainder of our study. Under the delivery arrangement SS, both suppliers undertake the delivery of their products and incur corresponding costs. This arrange ment happens when the e-retailer asks two suppliers to deliver products to consumers, and both agree. Accordingly, the channel parties’ payoffs SS SS SS SS SS under such an arrangement are ΠSS s1 = ws1 q1 − (c + bq1 )q1 , Πs2 = SS SS SS SS SS SS SS SS SS ws2 q2 − (c + bq2 )q2 , and Πe = me1 q1 + me2 q2 . We consider three different power structures: E-retailer-Stackelberg game (denoted by superscript “∧”), Supplier-Stackelberg game (denoted by superscript “∼”), and Vertical-Nash game (denoted by superscript “ − ”). In the E-retailer-Stackelberg game, the e-retailer initially claims her retail markups for two products, and then the suppliers set their wholesale prices. On the contrary, in the Supplier-Stackelberg game, the suppliers demand wholesale prices, followed by the e-retailer claiming her retail markups. In the Vertical-Nash game, the suppliers and the eretailer make decisions simultaneously. The sequence of events under each power structure is illustrated in Fig. 2. In the first stage, the eretailer decides to ask none, one, or both of the suppliers to deliver products to consumers.16 Then, the supplier, conditional on receiving the e-retailer’s delivery proposal, determines whether to accept this delivery proposal. After the product delivery arrangement, the suppliers and the e-retailer set their wholesale prices and retail markups in the second stage under different power structures, respectively. Notably, it is not the channel member with greater pricing power that can deter mine the delivery arrangement, but the e-retailer who acts as the product owner may ask the supplier to be responsible for delivering products (Yongbo [29,47]) and the supplier can respond by agreeing or rejecting to do so. For example, JD.com, Gome, Walmart, and Amazon have greater pricing power and ask the suppliers to be responsible for delivering products to customers, but some suppliers refuse to undertake the delivery. Another example is that numerous e-retailers working as market followers on the wholesale website 1688 in China ask the sup plier to deliver products, and the suppliers decide whether to undertake it. To ensure that the equilibrium outcomes are positive and unique and there exists the delivery economy, we assume that b > (k − 1)(k +2)/2 and max{̂c 1 (b, k), 0} < c < 1 (b > k − 1 and max{̃c1 (b, k), ̃c2 (b, k), 0} < c < 1) in the E-retailer-Stackelberg (Supplier-Stackelberg) game. Similarly, we assume that b > k2 − 1 and max{c1 (b, k), 0} < c < 1 in the Vertical-Nash game. More details about the expressions of ̂c 1 (b, k), ̃c1 (b, k), ̃c2 (b, k), and c1 (b, k) are presented in the Online Appendix A. 4. Comparison and analysis In this section, we compare the equilibrium outcomes to investigate the performance of the three arrangements and reveal how and why these comparative results vary with the power structure. All equilibrium outcomes and proofs appear in the Online Appendix A. 4.1. Retail markups and wholesale prices This subsection examines e-retailer’s retail markups and suppliers’ wholesale prices across the three delivery arrangements. The compara tive results of e-retailer’s retail markups are summarized in the following lemma. Lemma 1 (E-retailer’s Retail Markups). ̂ SS ̂ SE ̂ EE (1) For the E-retailer-Stackelberg game, max{ m ei , m e1 } < min{ m ei , ̂ SE m e2 }. SS SE EE SE ̃ ei , m ̃ e1 } < min{m ̃ e2 }. ̃ ei , m (2) For the Supplier-Stackelberg game, max{m SE max{mSS ei , me1 } SE min{mEE ei , me2 }. (3) For the Vertical-Nash game, < Lemma 1 (1) and (3) show that, compared with when the supplier undertakes the delivery, the e-retailer always claims a higher retail markup for the product when she undertakes the delivery in the Eretailer-Stackelberg and Vertica-Nash games. The intuition is that the eretailer incurs associated costs when delivering products to customers, which we call the cost-increasing effect, and she prefers to keep the retail markup high to compensate for these costs. Lemma 1 (2) replicates the results of Lemma 1 (1) and (3). However, the key rationale behind this is twofold: on the one hand, when the e-retailer undertakes the delivery, she prefers to keep the retail markup high owing to the cost-increasing effect; on the other hand, when the supplier (i.e., the market leader) undertakes the delivery, he might claim a higher wholesale price and thus leaves the e-retailer (i.e., the market follower) a lower retail markup, which we call the margin-squeezing effect. 16 Logistics delivery requires upfront operations, such as picking arrange ments, warehousing layout, and route planning. These operations stay mostly the same, while the price decision often changes. Therefore, we assume that the delivery arrangement occurs before the price decision. 5 X. Zhou et al. Omega 126 (2024) 103070 Lemma 2 presents the comparison results of the suppliers’ wholesale prices. Lemma 2 (Suppliers’ Wholesale Prices). derstand the above results, note that the demand for a product is negatively affected by its price but positively impacted by the substitute product price, and the retail prices across the three delivery arrange ments are equal when there is no delivery scale effect.17 When the eretailer (i.e., the market leader) undertakes the delivery, compared with without the delivery scale effect, the delivery economy directly moti vates her to reduce retail markup, which, however, indirectly forces the supplier (i.e., the market follower) to raise his wholesale price because the supplier’s wholesale price is negatively affected by the retail markup ̂ EE ̂ SS ̂ SE (1) For the E-retailer-Stackelberg game, max{ w si , w s2 } < min{ w si , ̂ SE w s1 }. SS SE EE SE ̃ s2 } < min{w ̃ s1 } ̃ si , w ̃ si , w (2) For the Supplier-Stackelberg game, max{w √̅̅̅ if ( 5 − 1)/2 < k < 1, 0 < b < k2 + k − 1 and 0 < c < ̃c3 (b, k); and ̃ EE ̃ SE ̃ SE ̃ SS max{w si , w s2 } < min{w si , w s1 } otherwise. SE SS SE (3) For the Vertical-Nash game, max{wEE si , ws2 } < min{wsi , ws1 }. The expression of ̃c3 (b, k) is provided in the Online Appendix A. Lemma 2 (1) indicates that, compared with when the e-retailer un dertakes the delivery, the supplier always claims a higher wholesale price when delivering products in the E-retailer-Stackelberg game. The supplier prefers to keep the wholesale price high due to the costincreasing effect when undertaking the delivery while earning a lower wholesale price because of the margin-squeezing effect when shifting the delivery to the market leader (i.e., the e-retailer). Because of the cost-increasing effect, the supplier in the Vertical-Nash game claims a higher wholesale price when he undertakes the delivery, as shown by Lemma 2 (3). Similar to Lemma 2 (1) and (3), Lemma 2 (2) suggests that, in the Supplier-Stackelberg game, compared with when the e-retailer un dertakes the delivery, the supplier always claims a higher wholesale price when undertaking the delivery in the case of delivery economy. However, somewhat surprisingly, Lemma 2 (2) reveals that when there is a delivery diseconomy, compared with under the delivery arrange ment EE, the supplier might even demand a lower wholesale price under the delivery arrangement SS. When the delivery is economy, compared with the delivery arrangement EE, the delivery arrangement SS not only motivates the supplier to demand a higher wholesale price due to the cost-increasing effect but also prevents the supplier from drastically ̃ EE reducing his wholesale price as competition becomes fierce (i.e., ∂w si ̂ EE ̂ EE ̂ SE ̂ SE the e-retailer demands (i.e., ∂ w si /∂ m ei < 0 and ∂ w s2 /∂ m e2 < 0) in such a case. However, when the product is delivered by the supplier, compared with without the delivery scale effect, such a delivery econ omy only directly motivates the supplier to reduce his wholesale price. Therefore, with the delivery economy, the double marginalization effect is alleviated when the supplier undertakes the delivery while aggravated when the e-retailer does it. Consequently, because of the dominant role the price of a product itself plays, shifting the delivery to the supplier generates a lower retail price (see Lemma A1 (1) in the Online Appendix A) and a higher demand for the product. On the contrary, when the delivery is diseconomy, shifting the delivery to the e-retailer can alle viate the double marginalization effect and thus generate a lower retail price (see Lemma A1 (1) in the Online Appendix A) and a higher demand. The results of Lemma 3 (2) are in stark contrast to those of Lemma 3 (1). That is because shifting the delivery responsibility to the market leader (follower) can alleviate the double marginalization effect in on line distribution channels when there is a delivery diseconomy (econ omy), and the price leadership in the Supplier-Stackelberg game is opposite to that in the E-retailer-Stackelberg game. However, different from parts (1) and (2), part (3) reveals that, in the Vertical-Nash game, the product demands are invariant to the delivery arrangement. When all channel members make decisions simultaneously, although the de livery arrangement makes the e-retailer (the supplier) claim different retail markups (wholesale prices), it does not change the sum of retail markup and wholesale price for a product. Consequently, the retail prices and product demands do not vary with the delivery arrangement (see Lemma A1 (3) in the Online Appendix A and Lemma 3 (3)). SS ̃ si /∂k < 0). As a consequence, compared with when not un /∂k < ∂w dertaking the delivery, the supplier always claims a higher wholesale price when undertaking the delivery in the case of a delivery economy. Differently, when the delivery is diseconomy, compared with the de livery arrangement EE, although the delivery arrangement SS motivates the supplier to demand a higher wholesale price due to the costincreasing effect, it forces the supplier to reduce his wholesale price ̃ SS ̃ EE more drastically as competition becomes intense (i.e., ∂w si /∂k < ∂w si /∂k < 0). Ultimately, compared to under the delivery arrangement EE, the supplier even demands a lower wholesale price under the delivery √̅̅̅ arrangement SS as competition becomes fierce (i.e., ( 5 − 1)/2 < k < 1), and the cost-increasing effect gets small (i.e., 0 < b < k2 +k − 1 and 0 < c < ̃c3 (b, k)). 4.3. Unit profits We compare firms’ unit profits across the three delivery arrange ments. The primary distinction between the unit profit and the retail markup (wholesale price) is that when the firm incurs the delivery cost, the former incorporates that cost while the latter does not. Let Δxei = mxei − Zxei (c +bqxi ) be the e-retailer’s unit profit for the product i and Δxsi = wxsi − Zxsi (c +bqxi ) be the supplier i’s unit profit under the delivery arrangement x ∈ {EE, SE, SS}, where Zeix , Zxsi ∈ {0,1}, and Zxsi = 1 − Zxei . The parameters Zxei and Zxsi assign the delivery cost to the e-retailer and the supplier, respectively. Zxei = 1 (i.e., Zxsi = 1 − Zxei = 0) implies that the e-retailer undertakes the delivery for the product i and Zsix = 1 (i.e., Zxei = 1 − Zxsi = 0) indicates that the supplier i undertakes the delivery under the delivery arrangement x. We first consider the e-retailer’s unit profits. Lemma 4 (E-retailer’s Unit Profits). 4.2. Product demands We now proceed to examine the product demands across the three delivery arrangements. Lemma 3 (Product Demands). EE SE SS SE (1) For the E-retailer-Stackelberg game, max{̂ qi , ̂ q 2 } < min{ ̂ qi , ̂ q1 } SS SE max{̂ qi , ̂ q1 } EE SE if (k − 1)(k +2)/2 0. qi , ̂ < min{ ̂ SS SE EE SE (2) For the Supplier-Stackelberg game, max{̃ qi , ̃ q1 } < min{̃ q2 } qi , ̃ EE SE SS SE q2 } < min{̃ qi , ̃ q1 } if b > 0. k − 1 < b < 0 and max{̃ qi , ̃ SE SS (3) For the Vertical-Nash game, qEE = qSE i 2 = q1 = qi . ̂ EE = Δ ̂ SE = Δ ̂ SE = Δ ̂ SS if (1) For the E-retailer-Stackelberg game, Δ ei e1 e2 ei (k − 1)(k +2)/2 0. ̃ SE < Δ ̃ EE < Δ ̃ SS < Δ ̃ SE if (2) For the Supplier-Stackelberg game, Δ e2 ei ei e1 √̅̅̅ ̃ SE < Δ ̃ SS < Δ ̃ EE < Δ ̃ SE if 0 < k < ( 5 − 1)/2 and k2 + k − 1 < b < 0; Δ e2 ei ei e1 if Part (1) of Lemma 3 indicates that, in the E-retailer-Stackelberg game, the demands of products delivered by the suppliers (both products under the delivery arrangement SS and product 1 under the delivery arrangement SE) are higher than those of products shipped by the eretailer (both products under the delivery arrangement EE and product 2 under the delivery arrangement SE) when the delivery is economy; however, the reverse is true when the delivery is diseconomy. To un ̃ SE < Δ ̃ SS < Δ ̃ EE < Δ ̃ SE if b > max{k2 + k − 1 < b < min{k2 +k − 1, 0}; Δ e1 ei ei e2 √ ̅̅̅ ̃ SE < Δ ̃ EE < Δ ̃ SS < Δ ̃ SE if ( 5 − 1)/2 < k < 1 and 0 < b < k − 1, 0}; and Δ e1 ei ei e2 k2 + k − 1. 17 6 This result can be verified with a simple algebraic operation. X. Zhou et al. Omega 126 (2024) 103070 SE EE SS compared with not undertaking delivery, the supplier will obtain a lower unit profit by undertaking the delivery for products with delivery diseconomy. Counterintuitively, Lemma 5 (1) illustrates that, in the Eretailer-Stackelberg game, both suppliers can achieve the highest unit profits via shifting the delivery to the e-retailer (i.e., the delivery arrangement EE) when the delivery is economy but can earn the highest unit profits through undertaking the delivery (i.e., the delivery arrangement SS) when the delivery is diseconomy. Before explaining this result, it is necessary to understand what factors and how they affect the suppliers’ unit profits under different delivery arrangements. On the downside, undertaking the delivery generates a cost-increasing effect that negatively affects the suppliers’ unit profits; on the upside, it also enables both suppliers to raise their wholesale prices, which alleviates the competition between them and thus generates the competitionmitigating effect, positively affecting the suppliers’ unit profits. Be sides, we learn from Lemma 2 (1) that when shifting the delivery re sponsibility to the e-retailer, the suppliers will suffer from the marginsqueezing effect that negatively affects their unit profits. When there is a delivery economy, undertaking the delivery generates a costincreasing effect, which dominates the competition-mitigating and margin-squeezing effects; thus, the suppliers earn the highest unit profits under the delivery arrangement EE in such a situation. Whereas in the case of the delivery diseconomy, both the competition-mitigating and margin-squeezing effects become more prominent. Therefore, the sup pliers earn the highest unit profits under the delivery arrangement SS even though the delivery is diseconomy. Different from Lemma 5 (1), Lemma 5 (2) demonstrates that, in the Supplier-Stackelberg game, both suppliers can achieve the highest unit profits through undertaking the delivery (i.e., the delivery arrangement SS) when the delivery is economy but can earn the highest unit profits via shifting the delivery to the e-retailer (i.e., the delivery arrangement EE) if the delivery is diseconomy. When the suppliers possess dominant market power, their unit profits will no longer be negatively affected by the margin-squeezing effect if they shift the delivery to the e-retailer. Therefore, which delivery arrangement can bring about the maximum unit profits hinges on the tradeoff between the cost-increasing effect and the competition-mitigating effect. Specifically, when there is a delivery economy, the delivery arrangement SS generates the most apparent EE ̃ SE ̃ SE ̃ SS wsi , w competition-mitigating effect (i.e., w si > max{̃ s1 , w s2 }) that overweighs the cost-increasing effect, thereby resulting the highest unit profits for them. Conversely, when the delivery is diseconomy, under taking the delivery might generate the competition-mitigating effect (shown by Lemma 2 (2)), but this effect is immaterial compared with the cost-increasing effect the suppliers can avoid by shifting the delivery to the e-retailer. Hence, the delivery arrangement EE eventually generates the highest unit profits for the suppliers in such a case. Lemma 5 (3) indicates that, in the Vertical-Nash game, the suppliers obtain the same and highest unit profits for products delivered by the eretailer (both products under the delivery arrangement EE and product 2 under the delivery arrangement SE) when there is a delivery economy; the reverse is true when there is a delivery diseconomy. The reason is that the cost-increasing effect avoided by shifting the delivery to the eretailer dominates (is dominated by) the competition-mitigating effect resulting from undertaking the delivery when the delivery is economy (diseconomy), and the suppliers obtain the same wholesale prices and the same demands for products delivered by the e-retailer (or themselves). To sum up, the differences in the comparison results of firms’ unit profits under the three power structures are driven by the fact that the power structure changes the number of driving forces affecting the unit profit and the relative magnitude of these driving forces. When the eretailer (supplier) does not undertake the delivery, the marginsqueezing effect does not influence her (his) unit profit in the Eretailer-Stackelberg and Vertical-Nash games (Supplier-Stackelberg and Vertical-Nash games) but by that effect when she (he) works as the SE (3) For the Vertical-Nash game,Δe2 = Δei < Δei = Δe1 if k2 − 1 0. Lemma 4 (1) shows that the e-retailer earns the same unit profits across the three delivery arrangements in the E-retailer-Stackelberg game. When the e-retailer acts as the market leader, she never suffers from the margin-squeezing effect when shifting the delivery to the market follower (i.e., the supplier), which implies that the dominant position enables her to obtain a reasonable retail markup when not undertaking the delivery. Such a dominant position, however, also en dows the e-retailer with the power to demand a sufficiently high retail markup to cover associated costs when undertaking the delivery. As a result, she achieves the same unit profits under the three delivery arrangements. However, in the Supplier-Stackelberg game, Lemma 4 (2) suggests that when there is a delivery economy, the e-retailer earns the highest unit profit for the product the supplier delivers under the hybrid arrangement SE. When shifting the delivery of products to both sup pliers, the e-retailer can avoid the cost-increasing effect but suffers from ̃ SS ̃ SE the margin-squeezing effect severely (i.e., m ei < m e1 ), and when un ̃ SE dertaking the delivery, the e-retailer can avoid that effect (i.e., m e1 < ̃ SE ̃ EE min{m ei , m e2 }) but it is overshadowed by the cost-increasing effect. As a result, when the delivery is economy, the e-retailer earns the highest unit profit for the product delivered by the supplier under the hybrid arrangement SE. Unlike when the delivery is economy, the e-retailer earns the highest unit profit for the product she delivers under the hybrid arrangement SE when the delivery is diseconomy, which arises because the e-retailer can avoid the margin-squeezing effect and de ̃ SE ̃ EE ̃ SE ̃ SS mands the highest retail markup (i.e.,m e2 > max{m ei , m e1 , m ei }) to compensate for the increased operational costs. Lemma 4 (2) further shows that, compared with the delivery arrangement SS, whether the delivery arrangement EE can bring about a higher unit profit for the eretailer depends on the competition intensity (k) and delivery scale ef fect (b). These results still hinge on the trade-off between the marginsqueezing effect caused by the delivery arrangement SS and the costincreasing effect arising under the delivery arrangement EE. Lemma 4 (3) indicates that, in the Vertical-Nash game, the e-retailer obtains the same and highest unit profits for products delivered by the suppliers (both products under the delivery arrangement SS and product 1 under the delivery arrangement SE) when there is a delivery economy; the opposite holds when there is a delivery diseconomy. Recall from Lemma 1 (3) that the cost-increasing effect stemming from undertaking the delivery stimulates the e-retailer to demand higher retail markups. Compared with such a positive impact, its negative impact on the unit profits is more (less) apparent when the delivery is economy (diseconomy). In addition, Lemma 1 (3) and Lemma 3 (3) show that the e-retailer obtains the same retail markups and the same demands for products delivered by suppliers (or herself). As such, the e-retailer ob tains the same and highest unit profits for products shipped by suppliers (herself) in the case of a delivery economy (diseconomy). The following lemma presents the comparison results of the sup plier’s unit profits. Lemma 5 (Supplier’s Unit Profits). ̂ SE < Δ ̂ SS < Δ ̂ SE < Δ ̂ EE if (1) For the E-retailer-Stackelberg game, Δ s2 si s1 si √ ̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ SS SE SE ̂ <Δ ̂ <Δ ̂ <Δ ̂ EE if (k − 1)(k +2)/2 0. < 0; and Δ si s2 s1 si ̃ EE < Δ ̃ SE < Δ ̃ SE < Δ ̃ SS if k − 1 (2) For the Supplier-Stackelberg game, Δ si s1 s2 si ̃ SS < Δ ̃ SE < Δ ̃ SE < Δ ̃ EE if b > 0. 0. Because of weak pricing power, one might intuitively believe that, 7 X. Zhou et al. Omega 126 (2024) 103070 market follower in the Supplier-Stackelberg (E-retailer-Stackelberg) game. (2) indicates that the e-retailer can obtain higher demands by asking the suppliers to deliver products; however, Lemma 4 (2) shows that she achieves much lower unit profits. Ultimately, when the delivery be comes more diseconomy, the decreased unit profits are higher than the increased demands as the e-retailer shifts the delivery to the suppliers. Hence, the e-retailer prefers to undertake the delivery in such a situa tion. However, it is worth mentioning that when there is a large delivery diseconomy, the force that drives the e-retailer to prefer undertaking the delivery in the E-retailer-Stackelberg game is the higher demand, but the one in the Supplier-Stackelberg or Vertical-Nash game is the higher unit profit. The suppliers’ delivery arrangement preferences are shown in the following proposition. Proposition 2 (Suppliers’ Delivery Arrangement Preferences). (1) For the E-retailer-Stackelberg game, when ̂ (k − 1)(k +2)/2 < b < b 1 (k), the suppliers prefer being the only supplier to 5. Equilibrium delivery arrangement In this section, we first investigate channel members’ delivery arrangement preferences and then derive the equilibrium delivery arrangement. All proofs are provided in the Online Appendix B. 5.1. Firms’ preferences of delivery arrangement The following proposition summarizes the e-retailer’s preference for delivery arrangement. Proposition 1 (E-retailer’s Delivery Arrangement Preference). (1) For the E-retailer-Stackelberg game, when (k − 1)(k +2)/2 max{ Π ̂ EE , Π ̂ SE , Π ̂ SS }; when undertake the delivery, i.e., Π s1 si s2 si ̂ b 1 (k) 0, the e-retailer prefers undertaking the de > max{ Π e e ̂ EE > max{ Π ̂ SE , Π ̂ SS }. livery by herself, i.e., Π e e e ̂ EE > max{ Π ̂ SE , Π ̂ SE , Π ̂ SS }; when b > 0, the suppliers prefer both of them to Π si s1 s2 si (2) For the Supplier-Stackelberg game, when b > ̃ b1 (k) or k − 1 max{ Π ̂ EE , Π ̂ SE , Π ̂ SE }. undertake the delivery, i.e., Π si si s1 s2 ̃ SS > max{Π ̃ EE , Π ̃ SE }. to deliver products, i.e., Π e e e (3) For the Vertical-Nash game, when k2 − 1 max{Π ̃ EE , Π ̃ SE , Π ̃ SS }; when ̃ undertake the delivery, i.e., Π k2 < k < 1 s2 si s1 si and k − 1 max{Π ̃ EE , Π ̃ SE , Π ̃ SE }; when b > max{0, ̃ take the delivery, i.e., Π b3 (k)}, (2) For the Supplier-Stackelberg game, when 0 < k < ̃ k1 and max{k − 1, ̃ b2 (k)} max{Π b1 (k), the e-retailer prefers both suppliers e e SS EE SE prefers both suppliers to deliver products, i.e., Πe > max{Πe , Πe }; when b EE si > 0, the e-retailer prefers undertaking the delivery by herself, i.e., Πe > SE si s1 s2 the suppliers prefer being the only supplier to undertake the delivery, i.e., ̃ SE > max{Π ̃ EE , Π ̃ SE , Π ̃ SS }; when ̃ Π k1 < k < 1 and 0 < b < ̃ b3 (k), the sup s1 si s2 si SS max{Πe , Πe }. The expression of ̃ b1 (k) is provided in the Online Appendix B. Proposition 1 (1) presents the e-retailer’s delivery arrangement preference in the E-retailer-Stackelberg game. Compared with when the delivery is economy, the delivery of products is more expensive when there is a delivery diseconomy. Hence, one may expect that the e-retailer will prefer the suppliers to deliver products when there is a delivery diseconomy but prefer to undertake the delivery when there is a delivery economy. Counterintuitively, as illustrated in Fig. 3(a), when the de livery is economy, the e-retailer prefers both suppliers to be responsible for delivery; however, when the delivery is diseconomy, she prefers undertaking the delivery for two substitutable products. These results happen because, as shown by Lemma 3 (1) and Lemma 4 (1), the eretailer achieves the highest demands by asking both suppliers to deliver products (undertaking the delivery for two substitutable products) when there is a delivery economy (diseconomy) and earns the same unit profits across the three delivery arrangements. Proposition 1 (3) repli cates the result of Proposition 1 (1). However, it is notable that the eretailer prefers both suppliers (herself) to deliver products only because of the highest unit profits in the Vertical-Nash game when there is a delivery economy (diseconomy). Nevertheless, Proposition 1 (2) reveals that, in the SupplierStackelberg game, the e-retailer prefers undertaking the delivery either when the delivery is economy or when the delivery diseconomy is apparent; however, she wants both suppliers to undertake the delivery if the delivery diseconomy is small (see Fig. 3(b)). The key driver is the tradeoff between the demands and the unit profits the e-retailer can obtain under different delivery arrangements. When there is a delivery economy, although Lemma 4 (2) tells us that the e-retailer might earn higher unit profits by asking the suppliers to deliver products, Lemma 3 (2) demonstrates that she achieves higher demands under the delivery arrangement EE. Consequently, the increases in product demands are overwhelmingly beneficial, and the e-retailer prefers to deliver products in such a case. By contrast, when there is a small delivery diseconomy, the e-retailer prefers asking both suppliers to undertake the delivery because the higher demands gained in this arrangement are appealing. Furthermore, as the delivery diseconomy becomes significant, Lemma 3 ̃ EE > max{Π ̃ SE , Π ̃ SE , Π ̃ SS }. pliers prefer the e-retailer to deliver products, i.e., Π si s1 s2 si (3) For the Vertical-Nash game, when k2 − 1 max{Πs1 , Πsi }; when SE SE b > 0, the suppliers prefer to undertake the delivery, i.e., Πs1 = Πs2 > EE SS max{Πsi , Πsi }. The expressions of ̃ k1 , ̃ k2 , ̂ b 1 (k), ̃ b2 (k), and ̃ b3 (k) are provided in the Online Appendix B. Proposition 2 (1) describes the suppliers’ delivery arrangement preferences in the E-retailer-Stackelberg game. As illustrated in Fig. 4 (a), when the delivery economy is large, the suppliers prefer being the only supplier to undertake the delivery. However, with a small delivery economy, they are better off when the e-retailer delivers products. The intuition behind these results still hinges on the impact of delivery arrangement on the product demands and the unit profits the suppliers can obtain. Specifically, when the delivery economy is apparent, Lemma 3 (1) and Lemma 5 (1) jointly demonstrate that the suppliers can achieve a higher demand and a relatively higher unit profit by becoming the only supplier to deliver the product. Therefore, the suppliers prefer being the unique supplier to undertake the delivery. As the delivery economy becomes small, although Lemma 3 (1) indicates that the suppliers can achieve a higher demand by undertaking the delivery, Lemma 5 (1) suggests that the suppliers obtain a much lower unit profit. Conse quently, the higher demand gained from undertaking the delivery cannot offset the loss of the unit profit in such a case; thus, the suppliers prefer the e-retailer to undertake the delivery. Surprisingly, Proposition 2 (1) further shows that when there is a delivery diseconomy, the sup pliers always prefer to undertake the delivery. In such a situation, un dertaking the delivery can make the suppliers avoid the salient marginsqueezing effect arising from the e-retailer’s delivering products and earn a higher unit profit because of the competition-mitigating effect. As a result, the increase in unit profit makes up for the decrease in demand when both suppliers switch from not undertaking the delivery to doing it. Proposition 2 (2) characterizes the suppliers’ delivery arrangement 8 X. Zhou et al. Omega 126 (2024) 103070 Fig. 3. E-retailer’s delivery arrangement preference. preferences in the Supplier-Stackelberg game. As shown in Fig. 4(b), when competition is weak, the suppliers prefer being the only supplier not to undertake the delivery if there is a delivery economy but prefer being the only supplier to undertake the delivery in the case of delivery diseconomy. These results can also be attributed to the trade-off be tween the demands and the unit profits under different delivery ar rangements. More specifically, compared with other delivery arrangements, Lemma 5 (2) and Lemma 3 (2) suggest that when competition is weak, the supplier who undertakes (does not undertake) the delivery under the hybrid delivery arrangement SE can obtain not only a relatively large unit profit but also a relatively high product de mand if the delivery is diseconomy (economy). Thus, the suppliers prefer being the only supplier to (not to) undertake the delivery when there is a diseconomy (economy). However, as competition becomes fierce, the suppliers prefer to undertake the delivery when there is a delivery economy but prefer the e-retailer to undertake the delivery when there is a delivery diseconomy (see Fig. 4(b)). The rationale lies in the fact that when there is a delivery economy (diseconomy), the increased unit profits are so high as to offset the loss of demands as the suppliers switch from other delivery arrangements to the delivery arrangement SS (EE). Lastly, Proposition 2 (3) reveals that, in the Vertical-Nash game, the suppliers prefer not to undertake the delivery when there is a delivery economy but prefer to undertake the delivery when there is a diseconomy. It is intuitive because, as indicated by Lemma 5 (3) and Lemma 3 (3), the suppliers obtain the highest unit profits through undertaking the delivery (shifting the delivery to the eretailer) when the delivery is diseconomy (economy) and achieve the same demands across the three arrangements. Therefore, we can learn from Proposition 2 that the suppliers’ de livery arrangement preferences vary with the power structure. Specif ically, when there is a small delivery economy (a small delivery diseconomy), the suppliers might prefer the e-retailer (both of them) to deliver products in the E-retailer-Stackelberg and Vertical-Nash games; however, the results reverse in the Supplier-Stackelberg game. The reason for these different outcomes is that the delivery arrangement yielding the highest unit profit varies with the power structure. Furthermore, when there is a large delivery economy, the suppliers might prefer being the only supplier to deliver the product in the Eretailer-Stackelberg game; conversely, in the Supplier-Stackelberg and Vertical-Nash games, they might like being the unique supplier not to ship products. The suppliers prefer to do so in the E-retailer-Stackelberg or Supplier-Stackelberg game because of the relatively higher unit profits and product demands. However, the suppliers prefer to do so in the Vertical-Nash game because of only the highest unit profits. It is also worth mentioning that when there is a large delivery diseconomy, the suppliers prefer undertaking the delivery due to the higher unit profits in the E-retailer-Stackelberg and Vertical-Nash games but do so owing to the higher demands in the Supplier-Stackelberg game. 9 Omega 126 (2024) 103070 X. Zhou et al. Fig. 4. Suppliers’ delivery arrangement preferences. We obtain three remarkable observations based on the firms’ de livery arrangement preferences. First, compared with the market leader, the market follower endowed with weaker pricing power might be more willing to deliver the product when the delivery is economy in each Stackelberg game. The main reason is that it can bring a higher product demand to the market follower. Second, channel members might benefit from undertaking the delivery when the delivery is diseconomy while suffering from undertaking the delivery when the delivery is economy. Specifically, in the E-retailer-Stackelberg and Vertical-Nash games, the e-retailer benefits from undertaking the delivery when the delivery is diseconomy but benefits from not undertaking the delivery when the delivery is economy. In the Supplier-Stackelberg and Vertical-Nash games, the suppliers might benefit from being the only supplier to (not to) undertake the delivery when the delivery is diseconomy (economy). Third, our work shows the non-negligible impact of power structure on channel members’ preferences regarding the delivery arrangement and also reveals that the drivers may change with the power structure. We next characterize the equilibrium delivery arrangement under each power structure. b 2 (k), and EE is (2) EE and SS are two possible equilibria if ̂ b 1 (k) 0. The expression of ̂ b 2 (k) is provided in the Online Appendix B. Part (1) of Proposition 3 indicates that when the delivery economy is large, the delivery arrangement SS emerges in equilibrium, as shown in the region I in Fig. 5(a). When there is a large delivery economy, given that the competitor rejects to undertake the delivery, the supplier ben efits from undertaking the delivery because he can obtain a higher de mand. However, when the competitor agrees to undertake the delivery, the supplier has no incentive to deviate from agreeing to undertake the delivery because of the higher demand loss. Therefore, the delivery arrangement SS emerges in equilibrium in such a case. However, in the region I in Fig. 5(a), a surprising result manifests: when competition becomes intense and the delivery economy gets small,18 the delivery arrangement SS remains a pure-strategy equilibrium even though it ̂ SS < Π ̂ EE ). Put differently, two sup hurts the suppliers’ profits (i.e., Π si si pliers are better off rejecting to undertake the delivery when there is fierce competition, yet they cannot help refusing to do so and eventually get involved in the prisoner’s dilemma. When the delivery economy is moderate, Proposition 3 (2) suggests that the delivery arrangements SS and EE are two possible equilibria (see the region II in Fig. 5(a)). It is natural to pick EE as the equilibrium outcome because both suppliers benefit more from this arrangement. As the delivery economy becomes 5.2. Equilibrium delivery arrangement The following proposition characterizes the equilibrium delivery arrangement in the E-retailer-Stackelberg game. Proposition 3 (Equilibrium Delivery Arrangement in the Eretailer-Stackelberg Game). (1) SS is the unique equilibrium if (k − 1)(k + 2)/2 < b < ̂ b 1 (k); 18 10 The expressions of ̂ b 3 (k) and ̂ b 4 (k) are provided in the Online Appendix B. X. Zhou et al. Omega 126 (2024) 103070 Fig. 5. Equilibrium delivery arrangement. small, Proposition 3 (3) reveals that the delivery arrangement EE emerges in equilibrium, shown by the region III in Fig. 5(a). The intui tion behind this result is that the increase in unit profit justifies the demand reduction when both suppliers switch from agreeing with the delivery proposal to rejecting it. Finally, because the e-retailer prefers delivering products when the delivery is diseconomy, Proposition 3 (3) indicates that the delivery arrangement EE emerges in equilibrium naturally (see the region IV in Fig. 5(a)). The following proposition shows the equilibrium delivery arrange ment in the Supplier-Stackelberg game. Proposition 4 (Equilibrium Delivery Arrangement in the Supplier-Stackelberg Game). (1) EE is the unique equilibrium if b > ̃ b1 (k), k − 1 < b < 0, or ̃ k1 < k < loss of unit profits because the small delivery diseconomy limits the competition-mitigating effect when there is fierce competition (shown by Lemma 5 (2)). Ultimately, the higher demands gained through un dertaking the delivery cannot make up for the higher loss of unit profits. Hence, both suppliers refuse to undertake the delivery, and delivery arrangement EE emerges in equilibrium. Furthermore, when the de livery diseconomy is not too small, whereas the competition is intense (see the region III in Fig. 5(b)), Proposition 4 (2) reveals that the hybrid delivery arrangement SE or ES emerges in equilibrium even if two competing suppliers are identical. It is worth noting that, compared with the delivery arrangement EE, the hybrid delivery arrangement benefits the supplier who agrees to undertake the delivery while harming the supplier who refuses to do it. Lastly, as the competitive intensity and delivery diseconomy fall into the region IV in Fig. 5(b), SS emerges in equilibrium because the increased demand gained by undertaking the delivery in such a case is so appealing that it enables the supplier to agree to deliver products regardless of which strategy the competitor adopts. Interestingly, there also exists a prisoner’s dilemma: when competition intensifies, and the delivery diseconomy gets small (i.e., ̃ k3 < k < 1 and max{0, ̃ b2 (k)} < b < ̃ b4 (k), see the region IV in Fig. 5(b)) 1 and 0 < b < ̃ b3 (k); (2) SE or ES is the unique equilibrium if ̃ k1 < k < 1 and ̃ b3 (k) < b ̃ < b2 (k); (3) SS is the unique equilibrium if max{0, ̃ b2 (k)} < b < ̃ b1 (k). From Proposition 1 (2), we know that the e-retailer prefers to un dertake the delivery either when there is a large delivery diseconomy or when the delivery is economy in the Supplier-Stackelberg game. Therefore, Proposition 4 (1) shows that the delivery arrangement EE emerges in equilibrium naturally in these two cases (see the regions I and V in Fig. 5(b)). Alternatively, Proposition 4 (1) demonstrates that the delivery arrangement EE can also be sustained as an equilibrium outcome when there is fierce competition and a relatively small delivery diseconomy (see the region II in Fig. 5(b)). Note that both suppliers can obtain higher product demands by undertaking the delivery in such a case (shown by Lemma 3 (2)); however, they suffer from a significant ,19 both suppliers agree to undertake the delivery but obtain lower profits compared with the case of rejecting to do it. The equilibrium delivery arrangement in the Vertical-Nash game is presented in the following proposition. Proposition 5 (Equilibrium Delivery Arrangement in the Vertical-Nash Game). EE is the unique equilibrium. 19 11 The expressions of ̃ k3 and ̃ b4 (k) are provided in the Online Appendix B. X. Zhou et al. Omega 126 (2024) 103070 Fig. 6. Consumer surplus-maximizing arrangements. may make the two suppliers unable to escape from this undesirable result in the E-retailer-Stackelberg game; nevertheless, in the SupplierStackelberg game, the decrease of delivery diseconomy (i.e., the de livery is less expensive) might make them unable to avoid this unfor tunate outcome. Our results might also provide a plausible explanation for the busi ness practices: some products are shipped by the e-retailer and others by the suppliers. In the E-retailer-Stackelberg game, online retail giants might prefer asking the competing suppliers to deliver products to consumers when the delivery is economy. A canonical example is that when selling home appliances for competing suppliers Haier and Midea, JD.com chooses to undertake the delivery because of the delivery diseconomy of these products. However, when selling products with a significant delivery economy (e.g., clothing and hats) online, JD.com may propose to let the competing suppliers Li-Ning and ANTA deliver products, and both agree to do it. By contrast, in the SupplierStackelberg game, numerous e-retailers might prefer asking the up stream suppliers to deliver products to customers when there is a de livery diseconomy and undertaking the delivery when the delivery is economy. For example, when selling home appliances for competing suppliers such as GREE and Hisense, the downstream e-retailers prefer asking them to ship products to customers because of the delivery diseconomy of these products, whereas, in the case of selling digital products for suppliers such as Xiaomi and OPPO, the downstream eretailers undertake the delivery due to the delivery economy of those items. Proposition 5 shows that the delivery arrangement EE constitutes the only equilibrium outcome in the Vertical-Nash game, which is entirely different from the equilibrium delivery arrangements in the other two games. Recall from Proposition 1 (3) that the e-retailer prefers deliv ering products when the delivery is diseconomy while prefers asking both suppliers to undertake the delivery when the delivery is economy. Thus, the delivery arrangement EE emerges in equilibrium naturally in the context of delivery diseconomy. When the delivery is economy, because of the lower unit profits the suppliers obtain by undertaking the delivery and the same demands the suppliers get among the three different delivery arrangements, both suppliers reject the e-retailer’s delivery proposal; thus, the delivery arrangement EE emerges in equi librium again (see Fig. 5(c)). Based on the above analysis, our research underscores the substantial implications of power structure for the delivery arrangement in online distribution channels. The delivery arrangement EE might emerge in equilibrium under certain conditions in each Stackelberg game. How ever, it is the unique equilibrium arrangement in the Vertical-Nash game. The delivery arrangement SS might emerge in equilibrium when there is a large delivery economy in the E-retailer-Stackelberg game, but only when the competition intensity is not too strong and the delivery diseconomy is not so small in the Supplier-Stackelberg game. That is because the power structure alters the conditions under which the e-retailer asks the suppliers to undertake the delivery and changes the conditions under which the suppliers can benefit from agreeing to undertake the delivery. Besides, our analysis shows that the hybrid de livery arrangement SE can emerge in equilibrium only in the SupplierStackelberg game. Moreover, a prisoner’s dilemma might arise in each Stackelberg game when competition becomes fierce. Notably, the decrease in the delivery economy (i.e., the delivery is more expensive) 6. Welfare implications In this section, we explore the impacts of delivery arrangements on 12 X. Zhou et al. Omega 126 (2024) 103070 Fig. 7. Social welfare-maximizing arrangements. consumer surplus and social welfare. The consumer surplus and social ∑ welfare are respectively given by CS = (aqi − qi 2 /2 − pi qi ) − kq1 q2 and thus bring about the highest consumer surplus. In contrast, when the delivery is diseconomy, Lemma A1 (1) in the Online Appendix A and Lemma 3 (1) reveal that, compared with the other delivery arrange ments, the delivery arrangement EE can mitigate the double marginal ization effect the most greatly and thus results in the highest consumer surplus. Following similar reasoning, in the Supplier-Stackelberg game, Proposition 6 (2) indicates that the delivery arrangement EE (SS) leads to the highest consumer surplus when there is a delivery economy (diseconomy), as shown in Fig. 6(b). Whereas in the Vertical-Nash game, because of the equal prices and the equal demands across the three different delivery arrangements (see Lemma A1 (3) in the Online Ap pendix A and Lemma 3 (3)), the consumer surplus is equal under the three delivery arrangements. Note that the mitigation of the double marginalization effect not only generates lower prices and higher demands but also leads to higher profits for the whole supply chain. Thus, Proposition 7 and Fig. 7 indi cate that the changing trend of the delivery arrangement that generates the highest social welfare is very similar to that of the delivery arrangement that leads to the highest consumer surplus. Proposition 7 (Social Welfare). i=1,2 and SW = CS + Πe + Πs1 + Πs2 . All proofs are provided in the Online Appendix C. The following proposition summarizes the consumer surplus. Proposition 6 (Consumer Surplus). SE EE SS ̂ > max{ CS ̂ , CS ̂ } if (1) For the E-retailer-Stackelberg game, CS SS EE SE ̂ ̂ ̂ ̂ ̂ (k − 1)(k +2)/2 max{ CS , CS } if b 5 (k) max{ CS ̂ SE , CS ̂ SS } if b > 0. and CS ̃ EE > max{CS ̃ SE , CS ̃ SS } if k − 1 (2) For the Supplier-Stackelberg game, CS ̃ SS > max{CS ̃ EE , CS ̃ SE } if b > 0. < b < 0 and CS EE SE SS (3) For the Vertical-Nash game, CS = CS = CS . The expression of ̂ b 5 (k) is provided in the Online Appendix C. As illustrated in Fig. 6(a), Proposition 6 (1) shows that, in the Eretailer-Stackelberg game, the hybrid delivery arrangement SE gener ates the highest consumer surplus when the delivery economy is large, and the delivery arrangement SS generates the highest consumer surplus when the delivery economy becomes small; nevertheless, when the de livery is diseconomy, only the delivery arrangement EE can benefit the consumers at most. Mitigating the double marginalization effect gen erates lower prices and higher demands, thus benefiting consumers. Along this line, the intuition behind the above results mainly hinges on the degree to which the delivery arrangement mitigates the double marginalization effect. Specifically, as shown in Lemma A1 (1) in the Online Appendix A and Lemma 3 (1), when the delivery economy is small (large), the delivery arrangement SS (SE) can alleviate the double marginalization effect compared with the other delivery arrangements ̂ SE > max{ SW ̂ EE , SW ̂ SS } if (1) For the E-retailer-Stackelberg game, SW ̂ SS > max{ SW ̂ EE , SW ̂ SE } (k − 1)(k +2)/2 max{ SW , SW } if b > 0. ̃ (2) For the Supplier-Stackelberg game, SW EE SE ̃SS > max{SW ̃EE , SW ̃SE } if b > 0. 1 max{SW SE SS (3) For the Vertical-Nash game, SW = SW = SW . The expression of ̂ b 6 (k) is provided in the Online Appendix C. To conclude, the delivery arrangements from which the consumer 13 X. Zhou et al. Omega 126 (2024) 103070 and society can benefit under the three power structures are funda mentally different. When there is a delivery economy (diseconomy), shifting the delivery to the suppliers (the e-retailer) generates the highest consumer surplus and social welfare in the E-retailer-Stackel berg game but hurts the consumers and society in the SupplierStackelberg game. The above results are different in that the delivery arrangement that can alleviate the double marginalization effect in the E-retailer-Stackelberg game is opposite to that in the SupplierStackelberg game. However, in the Vertical-Nash game, the consumers and society are indifferent to who delivers products. Moreover, in each Stackelberg game, when the delivery arrangement EE maximizes the consumer surplus and social welfare, it benefits the e-retailer only; in contrast, the delivery arrangement SS might benefit each channel member and generate the highest consumer surplus and social welfare simultaneously (i.e., a triple-win situation). However, this triple-win situation never happens in the Vertical-Nash game. Based on Figs. 5–7, we can see that such a triple-win situation might occur only when there is a delivery economy (i.e., max{(k − 1)(k + 2) /2, ̂ b 4 (k)} < outcome. Finally, although the delivery arrangement SS might maximize the profits of the entire supply chain, such an arrangement cannot be an equilibrium outcome under certain conditions. One key takeaway is that channel members can increase their profits and achieve a Pareto improvement through compensation. Concretely, when there is a small delivery economy in the E-retailer-Stackelberg game, it should be up to the e-retailer to compensate the suppliers to make them accept her de livery proposal. However, when there is a large delivery diseconomy in the Supplier-Stackelberg game, the supplier should make up for the eretailer so that she will shift the delivery to them. There are several limitations to this study. First, we focus on an online distribution channel with two suppliers and one e-retailer. It would be interesting to examine the delivery arrangement under different channel structures. Second, our study only considers the sym metric information. Therefore, it would be worth exploring how the information communication interacts with the delivery arrangement. Finally, we only consider the wholesale price contract prevalent in business practice. However, e-retailers like JD.com, Amazon, and Sun ing also offer logistics delivery services to suppliers under agency selling. Thus, examining the delivery arrangement under different selling modes would also be a promising research topic. b<̂ b 3 (k), see Fig. 5(a)) in the E-retailer-Stackelberg game but only when there is a delivery diseconomy (i.e., max{0, ̃ b4 (k)} < b < ̃ b1 (k), see Fig. 5(b)) in the Supplier-Stackelberg game. With Figs. 5–7, we also observe that the increased competition intensity shrinks the triple-win region in the E-retailer-Stackelberg game but generates a nonmonotonic effect on that region in the Supplier-Stackelberg game. CRediT authorship contribution statement Xiongwei Zhou: Conceptualization, Supervision, Validation, Writing – review & editing. Chaoqun Zhu: Conceptualization, Meth odology, Formal analysis, Writing – original draft. Dan Cai: Conceptu alization, Supervision, Resources, Writing – review & editing. 7. Concluding remarks This paper explores the delivery arrangement in an online distribu tion channel with two competing suppliers selling substitutable products through a common e-retailer. Our analysis highlights the substantial implications of power structure for the delivery arrangement. The eretailer’s delivering products (i.e., EE) might emerge in equilibrium in each Stackelberg game but constitutes the only equilibrium delivery arrangement in the Vertical-Nash game. The suppliers’ undertaking the delivery (i.e., SS) is the equilibrium outcome only if the delivery econ omy is significant (the competition is not too fierce and the delivery diseconomy is not too small) in the E-retailer-Stackelber (SupplierStackelberg) game. Only the Supplier-Stackelberg game can induce a hybrid delivery arrangement. Although both suppliers can achieve higher profits by refusing to undertake the delivery in each Stackelberg game, they might still agree to do so and fall into the prisoner’s dilemma. The rationale behind the above results is that the power structure alters the impacts of delivery arrangement on the unit profits and the demands the channel members obtain. These results might help us understand the business observations that some products are delivered by e-retailers while others by suppliers. Furthermore, we show that only shifting the delivery responsibility to the suppliers in each Stackelberg game might benefit channel members and generate the highest consumer surplus and social welfare simultaneously; in the Vertical-Nash game, however, this triple-win situation never occurs, and the consumer surplus and social welfare are invariant to the delivery arrangement. Our work also offers some valuable insights for online distribution channel members. First, compared with the market leader, the market follower might benefit more from undertaking the delivery when there is a delivery economy in each Stackelberg game. Second, counterintui tively, the supply chain players might benefit from delivering products with a significant delivery diseconomy but dislike undertaking the de livery of products with an apparent delivery economy. Hence, firms should distinguish types of products with different delivery scale effects when making delivery decisions. Furthermore, as competition becomes fierce in each Stackelberg game, the suppliers might trap themselves in a prisoner’s dilemma by accepting the e-retailer’s delivery proposal. Thus, our work uncovers the dark side of the e-retailer’s delivery proposal and suggests that the suppliers can reduce the intensity of competition by enhancing their product design differentiation to escape this undesirable Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability No data was used for the research described in the article. Acknowledgments The authors thank the editor and all reviewers for their valuable comments, which greatly improved the paper. This work was supported by the National Social Science Foundation of China [Grant No. 21AZD117] and the National Natural Science Foundation of China [Grant No. 71991463]. 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