Abstract
This study explores the coordination issues in the logistics service supply chain, which stem from the rapid development of live-streaming e-commerce (LSE). It also investigates the influences of a cost-sharing mechanism on the key decisions for live-streaming e-commerce logistics service supply chains (LSE-LSSC) with regard to the level of effort of logistics services. The motivation of the research is derived from the growing efficiency demands for logistics services in the LSE field. We consider an LSE-LSSC, which consists of one e-commerce shipper, one logistics service integrator, and one logistics service provider. On the basis of the game theoretic method, the performance of the logistics service supply chain is assessed and compared with four models among the participating entities of the LSE-LSSC. Some interesting results and key managerial insights are obtained in the modeling study. More importantly, the influences of cost-sharing on the key decisions of each player are discussed in detail, and the coordination contract of the level of effort of the logistics service is estimated.
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Acknowledgements
This study was supported by the National Natural Science Foundation of China (NSFC)-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (No. U1509221), National Social Science Foundation of China (SSFC) (No. 20&ZD129).
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Appendices
Appendix 1: Proofs for Proposition 1
The profit function for the whole LSE-LSSC with centralized decision in Model C can be written as
The first and the second derivatives of \(\pi _{{{ {SC}}}}^{{ {C}}}\) over \({ {p}}\) and \({ {s}}\) in Eq. (1) are expressed as follows:
The Hessian matrix is expressed as \(H = \left[ {\begin{array}{*{20}c} { - 2a} & b \\ b & { - k} \\ \end{array} } \right].\) Given that \(\left| {H_{1} } \right| = - 2a < 0\) and \(\left| {H_{2} } \right| =2ak - b^{2} > 0\), respectively, the Hessian matrix is a negative definite. Thus, a unique optimal (p, s) exists, which leads to the maximum of \(\pi _{{{ {SC}}}}^{{ {C}}} { {.}}\)
The first order condition is 0, i.e., \(\frac{{\partial \pi _{{SC}}^{C} }}{{\partial p}} = 0\) and \(\frac{{\partial \pi _{{{ {SC}}}}^{{ {C}}} }}{{\partial { {s}}}} = {0}\). The optimal product price and logistics level can be obtained.
Then, substituting Eqs. (6) and (7) into Eq. (1), the maximum profit of the whole LSE-LSSC is expressed as
Appendix 2: Proofs for Proposition 2
The profit functions of the e-shipper, the LSI, and the LSP in Model D are described as follows.
According to \({ {p}} \, = { {m}}\,+\,{ {m}}_{ {L}}\,+\,{ {w}}_{ {L}}\,+\,{ {c}}\), the above objective function can be changed by
The backward induction approach is used to solve the problem, and the steps are as follows.
First, we solve for the profit function of the LSP.
The first and second derivatives of \(\pi _{{ {C}}}^{{ {D}}}\) over \({ {w}}_{ {L}}\) in Eq. (17) are provided as follows:
For Eq. (19), \(\pi _{C}^{D}\) is a strictly concave function with respect to \({ {w}}_{ {L}}\). Equating Eq. (18) to 0, we obtain
Second, we substitute Eq. (20) into Eq. (14) and derive
The first and second derivatives of \(\pi _{B}^{D}\) over \({ {m}}_{ {L}}\) and \(s\) in Eq. (21) are written as follows:
According to Eqs. (24) and (25), \(\pi _{B}^{D}\) is a strictly concave function with respect to \({ {m}}_{ {L}}\) and \({ {s}}\).
Then, \(\frac{{\partial \pi _{B}^{D} }}{{\partial m_{L} }} = 0\) and \(\frac{{\partial \pi _{B}^{D} }}{{\partial s}} = 0\). We obtain
Third, we substitute Eqs. (20, 26 and 27) into Eq. (13) and derive
The first and second order conditions of \(\pi _{{ {A}}}^{{ {D}}}\) over \({ {m}}\) are
For Eq. (30), \(\pi _{A}^{D}\) is a strictly concave function with respect to \({ {m}}\). Equating Eq. (29) to 0, the optimal profit margin of the e-shipper is given by
Fourth, we substitute Eq. (31) into Eqs. (26 and 27). The response optimal profit margin and the logistics service level of the LSI are, respectively,
Then, we substitute Eqs. (31, 32 and 33) into Eq. (20). The optimal wholesale logistics price is
Thus, the optimal product price is expressed as
According to the equilibrium values above, the individual profits of the e-shipper, the LSI, and the LSP are obtained below, respectively.
At the same time, the total profit of the LSE-LSSC in the decentralized system is presented as
Appendix 3: Proofs for Proposition 3
In Model D-CS1, the profit functions of the e-shipper, the LSI, and the LSP are changed below:
Similarly, the backward induction approach is used to solve the issue, and the steps are as follows.
First, we solve for the profit function of the LSP.
The first and second derivatives of \(\pi _{C}^{D-CS1}\) over \({ {w}}_{ {L}}\) in Eq. (44) are expressed as
For Eq. (46), \(\pi _{C}^{D-CS1}\) is a strictly concave function with respect to \({ {w}}_{ {L}}\). Equating Eq. (45) to 0, we obtain
Second, we substitute Eq. (47) into Eq. (41) and derive
The first and second derivatives of \(\pi _{B}^{D-CS1}\) over \({ {m}}_{ {L}}\) and \({ {s}}\) in Eq. (48) are expressed as follows:
Given that the Hessian matrix \(H = \left[ {\begin{array}{*{20}c} { - a} & {b/2 } \\ { b/2} & {k\left( {\theta - 1} \right)} \\ \end{array} } \right]\), \(\left|{ {H}}_{1}\right|=-{ {a}} \, < 0\) and \(\left| {H_{2} } \right| = 4ak(1 - \theta ) - b^{2} > 0\). \(\pi _{B}^{{D - CS1}}\) is a strictly concave function with respect to \({ {m}}_{ {L}}\) and \({ {s}}\). Thus, \(\frac{{\partial \pi _{B}^{{D - CS1}} }}{{\partial m_{L} }} = 0\) and \(\frac{{\partial \pi _{B}^{{D - CS1}} }}{{\partial s}} =0\). We obtain
Third, we substitute Eqs. (47, 53 and 54) into Eq. (40) and derive
The first and second order conditions of \(\; \pi _{A}^{{D - CS1}}\) over \(m\) are
For Eq. (57), \(\pi _{A}^{{D - CS1}}\) is a strictly concave function with respect to \({ {m}}\). Equating Eq. (56) to 0, the optimal profit margin of the e-shipper is expressed as
Fourth, we substitute Eq. (58) into Eqs. (53 and54). The response optimal profit margin and the logistics service level of LSI are, respectively,
Then, we substitute Eqs. (58, 59 and 60) into Eq. (47). The optimal wholesale logistics price is
Thus, the optimal product price is expressed as
According to the equilibrium values above, the individual profits of the e-shipper, the LSI, and the LSP are obtained below, respectively.
At the same time, the total profit of the LSE-LSSC in Model D-CS1 is presented as
Appendix 4: Proofs for Proposition 4
In Model D-CS2, the profit functions of the e-shipper, the LSI, and the LSP are changed below.
Similar to "Appendix 3", the problems are addressed by the backward induction approach, and the steps are as follows.
First, we solve for the profit function of the LSP:
The first and second derivatives of \(\pi _{C}^{{D - CS2}}\) over \({ {w}}_{ {L}}\) in Eq. (71) are expressed as
For Eq. (73), \(\pi _{C}^{{D - CS2}}\) is a strictly concave function with respect to \({ {w}}_{ {L}}\). Equating Eq. (72) to 0, we obtain
Second, we substitute Eq. (74) into Eq. (68) and derive
The first and second derivatives of \(\pi _{B}^{{D - CS2}}\) over \({ {m}}_{ {L}}\) and \({ {s}}\) in Eq. (75) are expressed as follows:
Given that the Hessian matrix \(H = \left[ {\begin{array}{*{20}c} { - a} & {b/2} \\ {b/2} & {k\left( {\theta - 1} \right)} \\ \end{array} } \right]\), \(\left|{ {H}}_{1}\right|= -{ {a}} \, < 0\) and \(\left| {H_{2} } \right| = 4ak(1 - \theta ) - b^{2} > 0\). \(\pi _{B}^{{D - CS2}}\) is a strictly concave function with respect to \({ {m}}_{ {L}}\) and \({ {s}}\). Then, \(\frac{{\partial \pi _{B}^{{D - CS2}} }}{{\partial m_{L} }} =0\) and \(\frac{{\partial \pi _{B}^{{D - CS2}} }}{{\partial s}} = 0\). We obtain
Third, we substitute Eqs. (74, 80 and 81) into Eq. (67) and derive
The first and second order conditions of \(\pi _{A}^{{D - CS2}}\) over \({ {m}}\) are
For Eq. (84), \(\pi _{A}^{{D - CS2}}\) is a strictly concave function with respect to \({ {m}}\). Equating Eq. (83) to 0, the optimal profit margin of the e-shipper is expressed as
Fourth, we substitute Eq. (85) into Eqs. (80 and 81). The response optimal profit margin and the logistics service level of the LSI are, respectively,
Then, we substitute Eqs. (85, 86 and 87) into Eq. (74). The optimal wholesale logistics price is
Thus, the optimal product price is expressed as
According to the equilibrium values above, the individual profits of the e-shipper, the LSI, and the LSP are obtained below, respectively.
At the same time, the total profit of the LSE-LSSC in Model D-CS2 is presented by
Appendix 5: Proofs for Proposition 5
Thus,\({ {p}}^{ {C*}}<{ { p}}^{ {D*}}\), \({ {s}}^{ {C*}}> { {s}}^{ {D*}}\),\(\pi _{{SC{\kern 1pt} }}^{{C*}} > \pi _{{SC}}^{{D*}}\), and Proposition 5 is proven.
Appendix 6: Proofs for Proposition 6
Thus, \(\pi _{A}^{{D*}} = 2\pi _{B}^{{D*}}\) and \(\pi _{B}^{{D*}} > \pi _{C}^{{D*}}\), and Proposition 6 is proven.
Appendix 7: Proofs for Proposition 7
Thus, \(s^{D*} < s^{D - CS2*} < s^{D - CS1*}\) , and Proposition 7 is proven.
Appendix 8: Proofs for proposition 8
Thus, \(p^{D*} < p^{D - CS1*}\),\(p^{D*} < p^{D - CS2*}\),\(m_{L}^{D*} = m^{D - CS1*} < m^{D - CS2*}\), \(m_{L}^{D*} < m_{L}^{D - CS1*}\), and \(m_{L}^{D*} < m_{L}^{D - CS2*}\), and Proposition 8 is proven.
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Zhu, L., Liu, N. Game theoretic analysis of logistics service coordination in a live-streaming e-commerce system. Electron Commer Res 23, 1049–1087 (2023). https://doi.org/10.1007/s10660-021-09502-y
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DOI: https://doi.org/10.1007/s10660-021-09502-y