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Game theoretic analysis of logistics service coordination in a live-streaming e-commerce system

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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Notes

  1. 1.

    https://www.iimedia.cn/c400/65720.html.

  2. 2.

    https://en.wikipedia.org/wiki/Cainiao.

  3. 3.

    https://www.cainiao.com/aboutus.html?spm=cainiao.15076042.0.0.41985591T9qZ7N.

  4. 4.

    http://www.chinapostnews.com.cn/newspaper/epaper/html/2020-04/30/content_125664.htm.

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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

$$\max p_{{SC}}^{C} = \left( {p - c - c_{L} } \right)\left( {d_{0} - ap + bs} \right) - \frac{1}{2}ks^{2}$$
(1)

The first and the second derivatives of \(\pi _{{{ {SC}}}}^{{ {C}}}\) over \({ {p}}\) and \({ {s}}\) in Eq. (1) are expressed as follows:

$$\frac{{\partial \pi _{{SC}}^{C} }}{{\partial \ p}} = d_{0} + ac + ac_{L} - 2ap + bs$$
(2)
$$\frac{{\partial \pi _{{SC}}^{C} }}{{\partial \ s}} = - ks - bc - bc_{L} + bp$$
(3)
$$\frac{{\partial _{{\pi _{{SC}}^{C} }}^{2} }}{{\partial p^{2} }} = - 2a < 0$$
(4)
$$\frac{{\partial _{{\pi _{{SC}}^{C} }}^{2} }}{{\partial s^{2} }} = - k < 0$$
(5)

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.

$$P^{{C*}} = \frac{{d_{0} k + \left( {ak - b^{2} } \right)(c + c_{L} )}}{{2ak - b^{2} }}$$
(6)
$$S^{{C*}} = - \frac{{b[a(c + c_{L} ) - d_{0} ]}}{{2ak - b^{2} }}$$
(7)

Then, substituting Eqs. (6) and (7) into Eq. (1), the maximum profit of the whole LSE-LSSC is expressed as

$$\pi _{{SC}}^{{C*}} = \frac{{k(b^{2} + 2ak)[a(c + c_{L} ) - d_{0} ]^{2} }}{{2(2ak - b^{2} )^{2} }}$$
(8)

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.

$$\pi _{A}^{D} = (p - p_{L} - c)\left( {d_{0} - ap + bs} \right)$$
(9)
$$\pi _{B}^{D} = \left( {p_{L} - w_{L} } \right)\left( {d_{0} - ap + bs} \right) - \frac{1}{2}ks^{2}$$
(10)
$$\pi _{C}^{D} = \left( {w_{L} - c_{L} } \right)\left( {d_{0} - ap + bs} \right)$$
(11)
$$\pi _{{SC}}^{D} = \left( {p - c - c_{L} } \right)\left( {d_{0} - ap + bs} \right) - \frac{1}{2}ks^{2}$$
(12)

According to \({ {p}} \, = { {m}}\,+\,{ {m}}_{ {L}}\,+\,{ {w}}_{ {L}}\,+\,{ {c}}\), the above objective function can be changed by

$$\pi _{A}^{D} = m\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right]$$
(13)
$$\pi _{B}^{D} = m_{L} [d_{0} - a(m + m_{L} + w_{L} + c) + bs] - \frac{1}{2}ks^{2}$$
(14)
$$\pi _{C}^{D} = \left( {w_{L} - c_{L} } \right)\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right]$$
(15)
$$\pi _{{SC}}^{D} = \pi _{A}^{D} + \pi _{B}^{D} + \pi _{{C}}^{D}$$
(16)

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.

$$\max \pi _{C}^{D} = \left( {w_{L} - c_{L} } \right)\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right]$$
(17)

The first and second derivatives of \(\pi _{{ {C}}}^{{ {D}}}\) over \({ {w}}_{ {L}}\) in Eq. (17) are provided as follows:

$$\frac{{\partial \pi _{C}^{D} }}{{\partial w_{L} }} = d_{0} + bs + a\left( {c_{L} - 2w_{L} - m - m_{L} } \right)$$
(18)
$$\frac{{\partial _{{\pi _{C}^{D} }}^{2} }}{{\partial w_{L} ^{2} }} = - 2a < 0$$
(19)

For Eq. (19), \(\pi _{C}^{D}\) is a strictly concave function with respect to \({ {w}}_{ {L}}\). Equating Eq. (18) to 0, we obtain

$$w_{L}^{D} = \frac{{d_{0} + bs - a(c - c_{L} + m + m_{L} )}}{{2a}}$$
(20)

Second, we substitute Eq. (20) into Eq. (14) and derive

$$\max \pi _{B}^{D} = m_{L} \left[ {d_{0} - \frac{{d_{0} + bs + a\left( {c + c_{L} + m + m_{L} } \right)}}{2} + bs} \right] - \frac{1}{2}ks^{2}$$
(21)

The first and second derivatives of \(\pi _{B}^{D}\) over \({ {m}}_{ {L}}\) and \(s\) in Eq. (21) are written as follows:

$$\frac{{\partial \pi _{B}^{D} }}{{\partial m_{L} }} = \frac{{d_{0} + bs - a\left( {c + c_{L} + m + 2m_{L} } \right)}}{2}$$
(22)
$$\frac{{\partial \pi _{B}^{D} }}{{\partial s}} = \frac{{bm_{L} - 2ks}}{2}$$
(23)
$$\frac{{\partial _{{\pi _{B}^{D} }}^{2} }}{{\partial m_{L} ^{2} }} = - a < 0$$
(24)
$$\frac{{\partial _{{\pi _{B}^{D} }}^{2} }}{{\partial s^{2} }} = - k < 0$$
(25)

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

$$m_{L}^{D} =- \frac{{2k(ac + ac_{L} + am - d_{0} )}}{{4ak - b^{2} }}$$
(26)
$$S^{D} = - \frac{{b(ac + ac_{L} + am - d_{0} )}}{{4ak - b^{2} }}$$
(27)

Third, we substitute Eqs. (20, 26 and 27) into Eq. (13) and derive

$$\max \pi _{A}^{D} =\frac{{am(2c_{L} b^{2} - 3d_{0} k + 3ack - 5ac_{L} k + 3akm)}}{{4ak - b^{2} }}$$
(28)

The first and second order conditions of \(\pi _{{ {A}}}^{{ {D}}}\) over \({ {m}}\) are

$$\frac{{\partial \pi _{A}^{D} }}{{\partial m}} = - \frac{{ak(ac + ac_{L} + 2am - d_{0} )}}{{4ak - b^{2} }}$$
(29)
$$\frac{{\partial _{{\pi _{A}^{D} }}^{2} }}{{\partial m^{2} }} = - \frac{{2ka^{2} }}{{4ak - b^{2} }} < 0$$
(30)

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

$${ {m}}^{ {D*}}=-\frac{{ {a}}\left({ {c}}+{ {c}}_{ {L}}\right)-{ {d}}_{0}}{{2}{ {a}}}$$
(31)

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,

$${ {m}}_{ {L}}^{ {D*}}= -\frac{{ {k}}\left[{ {a}}\left({ {c}}+{ {c}}_{ {L}}\right)-{ {d}}_{0}\right]}{{4}{ {ak}}-{ {b}}^{2}}$$
(32)
$$s^{{D*}} = - \frac{{b[a\left( {c + c_{L} } \right) - d_{0} ]}}{{2(4ak - b^{2} )}}$$
(33)

Then, we substitute Eqs. (31, 32 and 33) into Eq. (20). The optimal wholesale logistics price is

$$w_{L}^{{D*}} = \frac{{k\left[ {a\left( {7c_{L} - c} \right) + d_{0} } \right] - 2c_{L} b^{2} }}{{2(4ak - b^{2} )}}$$
(34)

Thus, the optimal product price is expressed as

$$p^{{D*}} =m^{{D*}} + m_{L} ^{{D*}} + w_{L} ^{{D*}} + c = - \frac{{d_{0} (b^{2} - 7ak) + \left( {c + c_{L} } \right)(ab^{2} - a^{2} k)}}{{2a(4ak - b^{2} )}}$$
(35)

According to the equilibrium values above, the individual profits of the e-shipper, the LSI, and the LSP are obtained below, respectively.

$$\pi _{A}^{{D*}} = \frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{4(4ak - b^{2} )}}$$
(36)
$$\pi _{B}^{{D*}} = \frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{8(4ak - b^{2} )}}$$
(37)
$$\pi _{C}^{{D*}} = \frac{{ak^{2} [a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{4(4ak - b^{2} )^{2} }}$$
(38)

At the same time, the total profit of the LSE-LSSC in the decentralized system is presented as

$$\pi _{{SC}}^{{D*}} = \pi _{A}^{{D*}} + \pi _{B}^{{D*}} + \pi _{C}^{{D*}} = \frac{{k( - 3b^{2} + 14ak)[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{8(4ak - b^{2} )^{2} }}$$
(39)

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:

$$\pi _{A}^{{D - CS1}} = m[d_{0} - a(m + m_{L} + w_{L} + c) + bs]$$
(40)
$$\pi _{B}^{{D - CS1}} = m_{L} \left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right] - \frac{1}{2}(1 - \theta )ks^{2}$$
(41)
$$\pi _{C}^{{D - CS1}} = \left( {w_{L} - c_{L} } \right)\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right] - \frac{1}{2}\theta ks^{2}$$
(42)
$$\pi _{{SC}}^{{D - CS1}} = \pi _{A}^{{D - CS1}} + \pi _{B}^{{D - CS1}} + \pi _{C}^{{D - CS1}}$$
(43)

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.

$$\max \pi _{C}^{{D - CS1}} = \left( {w_{L} - c_{L} } \right)\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right] - \frac{1}{2}\theta ks^{2}$$
(44)

The first and second derivatives of \(\pi _{C}^{D-CS1}\) over \({ {w}}_{ {L}}\) in Eq. (44) are expressed as

$$\frac{{\partial \pi _{C}^{{D - CS1}} }}{{\partial w_{L} }} = d_{0} + bs + a(c_{L} - 2w_{L} - m - m_{L} )$$
(45)
$$\frac{{\partial _{{\pi _{C}^{{D - CS1}} }}^{2} }}{{\partial w_{L} ^{2} }} = - 2a < 0$$
(46)

For Eq. (46), \(\pi _{C}^{D-CS1}\) is a strictly concave function with respect to \({ {w}}_{ {L}}\). Equating Eq. (45) to 0, we obtain

$$w_{L}^{{D - CS1}} = \frac{{d_{0} + bs - a(c - c_{L} + m + m_{L} )}}{{2a}}$$
(47)

Second, we substitute Eq. (47) into Eq. (41) and derive

$$\max \pi _{B}^{{D - CS1}} = m_{L} \left[ {d_{0} - \frac{{d_{0} + bs + a\left( {c + c_{L} + m + m_{L} } \right)}}{2} + bs} \right] - \frac{1}{2}(1 - \theta )ks^{2}$$
(48)

The first and second derivatives of \(\pi _{B}^{D-CS1}\) over \({ {m}}_{ {L}}\) and \({ {s}}\) in Eq. (48) are expressed as follows:

$$\frac{{\partial \pi _{B}^{{D - CS1}} }}{{\partial m_{L} }} = \frac{{d_{0} + bs - a\left( {c + c_{L} + m + 2m_{L} } \right)}}{2}$$
(49)
$$\frac{{\partial \pi _{B}^{{D - CS1}} }}{{\partial s}} = \frac{{bm_{L} + 2ks(\theta - 1)}}{2}$$
(50)
$$\frac{{\partial _{{\pi _{B}^{{D - CS1}} }}^{2} }}{{\partial m_{L} ^{2} }} = - a< 0$$
(51)
$$\frac{{\partial _{{\pi _{B}^{{D - CS1}} }}^{2} }}{{\partial s^{2} }} = k(\theta - 1) <0$$
(52)

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

$$m_{L}^{{D - CS1}} = - \frac{{2k(\theta - 1)(ac + ac_{L} + am - d_{0} )}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(53)
$$S^{{D - CS1}} = \frac{{b(ac + ac_{L} + am - d_{0} )}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(54)

Third, we substitute Eqs. (47, 53 and 54) into Eq. (40) and derive

$$\max \pi _{A}^{{D - CS1}} = - \frac{{akm\left( {\theta - 1} \right)\left( {ac + ac_{L} + am - d_{0} } \right)}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(55)

The first and second order conditions of \(\; \pi _{A}^{{D - CS1}}\) over \(m\) are

$$\frac{{\partial \pi _{A}^{{D - CS1}} }}{{\partial m}} = - \frac{{ak(\theta - 1)(ac + ac_{L} + 2am - d_{0} )}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(56)
$$\frac{{\partial _{{\pi _{A}^{{D - CS1}} }}^{2} }}{{\partial m^{2} }} = - \frac{{2a^{2} k(\theta - 1)}}{{b^{2} - 4ak\left( {1 - \theta } \right)}} < 0$$
(57)

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

$$m^{{D - CS1*}} = - \frac{{a\left( {c + c_{L} } \right) - d_{0} }}{{2a}}$$
(58)

Fourth, we substitute Eq. (58) into Eqs. (53 and54). The response optimal profit margin and the logistics service level of LSI are, respectively,

$$m_{L}^{{D - CS1*}} = - \frac{{k(\theta - 1)[a\left( {c + c_{L} } \right) - d_{0} ]}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(59)
$$s^{{D - CS1*}} = \frac{{b[a\left( {c + c_{L} } \right) - d_{0} ]}}{{2[b^{2} - 4ak\left( {1 - \theta } \right)]}}$$
(60)

Then, we substitute Eqs. (58, 59 and 60) into Eq. (47). The optimal wholesale logistics price is

$$w_{L}^{{D - CS1*}} = \frac{{2c_{L} b^{2} + k(1 - \theta )\left[ {a\left( {c - 7c_{L} } \right) + d_{0} } \right]}}{{2[b^{2} - 4ak\left( {1 - \theta } \right)]}}$$
(61)

Thus, the optimal product price is expressed as

$${ {p}}^{ {D-CS}{1} {*}}= { {m}}^{ {D-CS}{1} {*}}{{+ { m}}_{ {L}}}^{ {D-CS}{1} {*}}+{{ { w}}_{ {L}}}^{ {D-CS}{1} {*}}+ { {c}}$$
$$= \frac{{b^{2} d_{0} - 7akd_{0} (1 - \theta ) + \left( {c + c_{L} } \right)[ab^{2} - a^{2} k\left( {1 - \theta } \right)]}}{{2a[b^{2} - 4ak\left( {1 - \theta } \right)]}}$$
(62)

According to the equilibrium values above, the individual profits of the e-shipper, the LSI, and the LSP are obtained below, respectively.

$$\pi _{A}^{{D - CS1*}} = \frac{{k(\theta - 1)[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{4[b^{2} - 4ak\left( {1 - \theta } \right)]}}$$
(63)
$$\pi _{B}^{{D - CS1*}} = \frac{{k(\theta - 1)[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{8[b^{2} - 4ak\left( {1 - \theta } \right)]}}$$
(64)
$$\pi _{C}^{{D - CS1*}} = \frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} [2ak\theta ^{2} - \left( {b^{2} + 4ak} \right)\theta + 2ak]}}{{8[b^{2} - 4ak\left( {1 - \theta } \right)]^{2} }}$$
(65)

At the same time, the total profit of the LSE-LSSC in Model D-CS1 is presented as

$$\pi _{{SC}}^{{D - CS1*}} = \pi _{A}^{{D - CS1*}} + \pi _{B}^{{D - CS1*}} + \pi _{C}^{{D - CS1*}} = \frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} [b^{2} (2\theta - 3) + 14ak\left( {\theta - 1} \right)^{2} ]}}{{8[b^{2} - 4ak\left( {1 - \theta } \right)]^{2} }}$$
(66)

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.

$$\pi _{A}^{{D - CS2}} = m[d_{0} - a(m + m_{L} + w_{L} + c) + bs] - \frac{1}{2}\theta ks^{2}$$
(67)
$$\pi _{B}^{{D - CS2}} = m_{L} \left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right] - \frac{1}{2}\left( {1 - \theta } \right)ks^{2}$$
(68)
$$\pi _{C}^{{D - CS2}} = \left( {w_{L} - c_{L} } \right)\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right]$$
(69)
$$\pi _{{{\text{SC}}}}^{{{\text{D - CS}}2}} = \pi _{{\text{A}}}^{{{\text{D-CS}}2}} {\text{ + }}\pi _{{\text{B}}}^{{{\text{D-CS}}2}} {\text{ + }}\pi _{{\text{C}}}^{{{\text{D-CS}}2}}$$
(70)

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:

$$\max \pi _{C}^{{D - CS2}} = \left( {w_{L} - c_{L} } \right)\left[ {d_{0} - a\left( {m + m_{L} + w_{L} + c} \right) + bs} \right] $$
(71)

The first and second derivatives of \(\pi _{C}^{{D - CS2}}\) over \({ {w}}_{ {L}}\) in Eq. (71) are expressed as

$$\frac{{\partial \pi _{C}^{{D - CS2}} }}{{\partial w_{L} }} =d_{0} + bs + a(c_{L} - 2w_{L} - m - m_{L} )$$
(72)
$$\frac{{\partial _{{\pi _{C}^{{D - CS2}} }}^{2} }}{{\partial w_{L} ^{2} }} = - 2a < 0$$
(73)

For Eq. (73), \(\pi _{C}^{{D - CS2}}\) is a strictly concave function with respect to \({ {w}}_{ {L}}\). Equating Eq. (72) to 0, we obtain

$$w_{L}^{{D - CS2}} = \frac{{d_{0} + bs - a(c - c_{L} + m + m_{L} )}}{{2a}}$$
(74)

Second, we substitute Eq. (74) into Eq. (68) and derive

$$\max \pi _{B}^{{D - CS2}} = m_{L} \left[ {d_{0} - \frac{{d_{0} + bs + a\left( {c + c_{L} + m + m_{L} } \right)}}{2} + bs} \right] - \frac{1}{2}(1 - \theta )ks^{2}$$
(75)

The first and second derivatives of \(\pi _{B}^{{D - CS2}}\) over \({ {m}}_{ {L}}\) and \({ {s}}\) in Eq. (75) are expressed as follows:

$$\frac{{\partial \pi _{B}^{{D - CS2}} }}{{\partial m_{L} }} = \frac{{d_{0} + bs - a\left( {c + c_{L} + m + 2m_{L} } \right)}}{2}$$
(76)
$$\frac{{\partial \pi _{B}^{{D - CS2}} }}{{\partial s}} =\frac{{bm_{L} + 2ks(\theta - 1)}}{2}$$
(77)
$$\frac{{\partial _{{\pi _{B}^{{D - CS2}} }}^{2} }}{{\partial m_{L} ^{2} }} = - a < 0$$
(78)
$$\frac{{\partial _{{\pi _{B}^{{D - CS2}} }}^{2} }}{{\partial s^{2} }} = k(\theta - 1) < 0$$
(79)

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

$$m_{L}^{{D - CS2}} = - \frac{{2k(\theta - 1)(ac + ac_{L} + am - d_{0} )}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(80)
$$S^{{D - CS2}} =\frac{{b(ac + ac_{L} + am - d_{0} )}}{{b^{2} - 4ak\left( {1 - \theta } \right)}}$$
(81)

Third, we substitute Eqs. (74, 80 and 81) into Eq. (67) and derive

$$\max \pi _{A}^{{D - CS2}}\, = - \frac{{k\left( {ac - ac_{L} + am - d_{0} } \right) + ab^{2} \theta \left( {c + c_{L} } \right) + 8a^{2} km\left( {1 + \theta^{2} } \right) + ab^{2} m\left( {3\theta - 2} \right) - (b^{2} d_{0} + 16a^{2} km)\theta }}{{2[b^{2} - 4ak\left( {1 - \theta } \right)]^{2} }}$$
(82)

The first and second order conditions of \(\pi _{A}^{{D - CS2}}\) over \({ {m}}\) are

$$\frac{{\partial \pi _{A}^{{D - CS2}} }}{{\partial m}}\, = - \frac{{ak\{b^{2} d_{0} \left( {1 - 2\theta } \right) - ab^{2} m\left( {2 - 3\theta } \right) + \left( {1 - \theta } \right)^{2} \left( {8a^{2} km - 4ad_{0} k} \right) + (c + c_{L} )[ab^{2} \left( {1 - \theta } \right) + 4a^{2} k\left( {1 - \theta } \right)^{2} ]\}}}{{[b^{2} - 4ak\left( {1 - \theta } \right)]^{2} }}$$
(83)
$$\frac{{\partial _{{\pi _{A}^{{D - CS2}} }}^{2} }}{{\partial m^{2} }} = - \frac{{a^{2} k[b^{2} (3\theta - 2) + 8ak\left( {\theta - 1} \right)^{2} }}{{[b^{2} - 4ak(1 - \theta )]^{2} }} < 0$$
(84)

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

$$m^{{D - CS2*}} = - \frac{{\left[ {a\left( {c + c_{L} } \right) - d_{0} } \right][b^{2} \left( {2\theta - 1} \right) + 4ak(\theta - 1)^{2} ]}}{{a[b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} ]}}$$
(85)

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,

$$m_{L}^{{D - CS2*}} = - \frac{{2k\left( {\theta - 1} \right)^{2} \left[ {a\left( {c + c_{L} } \right) - d_{0} } \right]}}{{b^{2} \left( {3\theta - 2} \right) + 8ak\left( {\theta - 1} \right)^{2} }}$$
(86)
$$s^{{D - CS2*}} = \frac{{b(\theta - 1)[a\left( {c + c_{L} } \right) - d_{0} ]}}{{b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} }}$$
(87)

Then, we substitute Eqs. (85, 86 and 87) into Eq. (74). The optimal wholesale logistics price is

$$w_{L}^{{D - CS2*}} = \frac{{b^{2} c_{L} \left( {3\theta - 2} \right) - k(\theta - 1)^{2} \left[ {a\left( {c - 7c_{L} } \right) + d_{0} } \right]}}{{b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} }}$$
(88)

Thus, the optimal product price is expressed as

$${ {p}}^{ {D-CS}{2} {*}}={ { m}}^{ {D-CS}{2} {*}}{{+ { m}}_{ {L}}}^{ {D-CS}{2} {*}}+{{ { w}}_{ {L}}}^{ {D-CS}{2} {*}}+ { {c}}$$
$$= \frac{{d_{0} \left( {2\theta - 1} \right)(b^{2} - 7ak) + (\theta - 1)[(ab^{2} - a^{2} k\left( {1 - \theta } \right)]}}{{a[b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} ]}}$$
(89)

According to the equilibrium values above, the individual profits of the e-shipper, the LSI, and the LSP are obtained below, respectively.

$$\pi _{A}^{{D - CS2*}} =\frac{{k(\theta - 1)^{2} [a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{2[b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} ]}}$$
(90)
$$\pi _{B}^{{D - CS2*}} = \frac{{k(\theta - 1)^{3} [a\left( {c + c_{L} } \right) - d_{0} ]^{2} [b^{2} + 4ak(\theta - 1)]}}{{2[b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} ]^{2} }}$$
(91)
$$\pi _{C}^{{D - CS2*}} = \frac{{ak^{2} (\theta - 1)^{4} [a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{[b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} ]^{2} }}$$
(92)

At the same time, the total profit of the LSE-LSSC in Model D-CS2 is presented by

$$\pi _{{SC}}^{{D - CS2*}} = \pi _{A}^{{D - CS2*}} + \pi _{B}^{{D - CS2*}} + \pi _{C}^{{D - CS2*}}$$
$$=\frac{{k(\theta - 1)^{2} [a\left( {c + c_{L} } \right) - d_{0} ]^{2} + (4\theta - 3) + 14ak\left( {\theta - 1} \right)^{2} }}{{2[b^{2} \left( {3\theta - 2} \right) + 8ak(\theta - 1)^{2} ]}}$$
(93)

Appendix 5: Proofs for Proposition 5

$$p^{{C*}} - p^{{D*}} = \frac{{(6ak - b^{2} )(ak - b^{2} )[a\left( {c + c_{L} } \right) - d_{0} ]}}{{2a(2ak - b^{2} )(4ak - b^{2} )}} < 0$$
$$\frac{{s^{{C*}} }}{{s^{{D*}} }} =\frac{{2(4ak - b^{2} )}}{{2ak - b^{2} }} > 1$$
$$\pi _{{SC}}^{{C*}} - \pi _{{SC}}^{{D*}} = \frac{{k\left( {18ak + 23b^{2} } \right)\left[ {a\left( {c + c_{L} } \right) - d_{0} } \right]^{2} }}{{32\left( {2ak - b^{2} } \right)^{2} }} > 0$$

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

$$\frac{{\pi _{A}^{{D*}} }}{{\pi _{B}^{{D*}} }} = \frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{4(4ak - b^{2} )}}/\frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{8(4ak - b^{2} )}} = 2$$
$$\frac{{\pi _{B}^{{D*}} }}{{\pi _{C}^{{D*}} }} = \frac{{k[a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{8(4ak - b^{2} )}}/\frac{{ak^{2} [a\left( {c + c_{L} } \right) - d_{0} ]^{2} }}{{4(4ak - b^{2} )^{2} }} > 1$$

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

$$s^{D*} - s^{D - CS1*} = - \frac{{2abk\theta [a\left( {c + c_{L} } \right) - d_{0} ]}}{{\left( {4ak - b^{2} } \right)[b^{2} - 4ak\left( {1 - \theta } \right)]}} < 0$$
$$s^{D*} - s^{D - CS2*} = \frac{{ - 2b\theta \left[ {a\left( {c + c_{L} } \right) - d_{0} } \right][b^{2} + 8ak(\theta - 1)]}}{{2\left( {4ak - b^{2} } \right)[b^{2} \left( {3\theta - 2} \right) + 8ak\left( {\theta - 1} \right)^{2} ]}} < 0$$
$$\frac{{s^{D - CS1*} }}{{s^{D - CS2*} }} = \frac{{b^{2} \left( {3\theta - 2} \right) + 8ak\left( {\theta - 1} \right)^{2} }}{{2b^{2} \left( {\theta - 1} \right) + 8ak\left( {\theta - 1} \right)^{2} }} > 1$$

Thus, \(s^{D*} < s^{D - CS2*} < s^{D - CS1*}\) , and Proposition 7 is proven.

Appendix 8: Proofs for proposition 8

$$p^{D*} - p^{D - CS1*} = - \frac{{3b^{2} k\theta [a\left( {c + c_{L} } \right) - d_{0} ]}}{{2(4ak - b^{2} )[b^{2} - 4ak(1 - \theta )]}} < 0$$
$$p^{D*} - p^{D - CS2*} = - \frac{{b^{2} \theta \left[ {a\left( {c + c_{L} } \right) - d_{0} } \right][(b^{2} - ak) + 6ak(\theta - 1)]}}{{2(\theta - 1)[b^{2} + 4ak(\theta - 1)] + b^{2} \theta }} < 0$$
$$m^{D - CS1*} - m^{D - CS2*} = \frac{{b^{2} \theta \left[ {a\left( {c + c_{L} } \right) - d_{0} } \right]}}{{2a[b^{2} (3\theta - 2) + 8ak\left( {\theta - 1} \right)^{2} ]}} < 0$$
$$m_{L}^{D*} - m_{L}^{D - CS1*} = - \frac{{b^{2} k\theta [a\left( {c + c_{L} } \right) - d_{0} ]}}{{(4ak - b^{2} )[b^{2} - 4ak(1 - \theta )]}} < 0$$
$$m_{L}^{D*} - m_{L}^{D - CS2*} = - \frac{{b^{2} k\theta (2\theta - 1)[a\left( {c + c_{L} } \right) - d_{0} ]}}{{(4ak - b^{2} )[b^{2} (3\theta - 2) + 8ak\left( {\theta - 1} \right)^{2} ]}} < 0$$

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 (2021). https://doi.org/10.1007/s10660-021-09502-y

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Keywords

  • Live-streaming e-commerce
  • Logistics service supply chain
  • Game theory
  • Cost-sharing mechanism