Ship space sharing strategies with different rental modes: How does NVOCCs cooperate with booking platform?

Abstract

The development of sharing economy has effectively changed the gap between market participants and promoted the emergence of new business modes, which has also brought enlightenment to the operation of shipping e-commerce. In this paper, we develop a liner shipping system consisting of one ship space booking platform and two Non-Vessel Operating Common Carriers (NVOCCs). Under two rental modes proposed by platform, the NVOCCs separately determine the order quantity of ship spaces and then consider whether to share the remaining idle ship spaces on the platform before the start of voyage. Introducing multi-party game theory, the optimal sharing strategy and ship space rental mode are obtained. The results reveal that both NVOCCs choosing sharing strategy will achieve a win–win situation, which is also beneficial for the platform. Furthermore, we find the most appropriate operation mode of the platform is to provide differentiated rentals, and this choice is independent of NVOCCs’ sharing strategies. The differential rentals mode is also more beneficial to the NVOCC with larger potential demand, but adverse for the NVOCC with less potential demand, so the latter is more inclined to share remaining spaces under the same rental mode. We also extend the model to multiple NVOCCs and present numerical analysis to verify the validity of the conclusions.

Appendices

Appendix 1. Proofs

When the random factor ${\upxi }_{i}$ in demand function is assumed to follow the uniform distribution, i.e. $\xi_{i} \sim U\left( {0,B_{i} } \right)$, the equilibriums of liner shipping system under three different sharing strategies with same rental level are described in Table

Table 5 Equilibriums of different sharing strategies under same rental

5.

In order to ensure that the liner company is profitable and the order quantity is not negative, we require $p > w$, so we can get $pB_{1} \left( {p + h_{2} } \right) + pB_{2} \left( {p + h_{1} } \right) - 2\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right) > 0$ under SN strategy, $B_{1} \left( {p + h_{2} } \right) + B_{2} p - 2\left( {a - bp} \right)\left( {p + h_{2} } \right) > 0$ under SO strategy and $B_{1} + B_{2} - 2\left( {a - bp} \right) > 0$ under SB strategy.

Proof of Lemma 2 From Table 5, we can get the derivatives of unit rental of booking platform with respect to idle costs (1) $\frac{{\partial w^{SN} }}{{\partial h_{1} }} = \frac{{\left( {a - bp} \right)B_{1} \left( {p + h_{2} } \right)^{2} }}{{\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]^{2} }} > 0$, $\frac{{\partial w^{SO} }}{{\partial h_{1} }} = \frac{{\partial w^{SB} }}{{\partial h_{1} }} = 0$, $\frac{{\partial w^{SN} }}{{\partial h_{2} }} = \frac{{\left( {a - bp} \right)B_{2} \left( {p + h_{1} } \right)^{2} }}{{\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]^{2} }} > 0$, $\frac{{\partial w^{SO} }}{{\partial h_{2} }} = \frac{{p^{3} \left[ {4p\left( {a - bp} \right) + cB_{1} } \right]B_{2} }}{{\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]^{2} }} > 0$, $\frac{{\partial w^{SB} }}{{\partial h_{2} }} = 0$. (2) Taking the derivative with respect to price-sensitivity coefficient, it is obtained as $\frac{{\partial w^{SN} }}{{\partial {\text{b}}}} = - \frac{{p\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)}}{{B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{1} } \right)}} < 0$, $\frac{{\partial w^{SO} }}{{\partial {\text{b}}}} = - \frac{{2p^{3} \left( {p + h_{2} } \right)}}{{2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)}} < 0$, $\frac{{\partial w^{SB} }}{{\partial {\text{b}}}} = - \frac{{2p^{3} }}{{\left( {c + 2p} \right)\left( {B_{1} + B_{2} } \right)}} < 0$. Furthermore, we can prove that $\frac{{\partial w^{SN} }}{{\partial u_{1} }} = - \frac{{\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)^{2} }}{{4\left[ {u_{2} \left( {p + h_{1} } \right) + u_{1} \left( {p + h_{2} } \right)} \right]^{2} }} < 0$. $\frac{{\partial w^{SO} }}{{\partial u_{1} }} = - \frac{{p^{2} \left( {p + h_{2} } \right)\left[ {\left( {c + 2p} \right)\left( {a - bp} \right)\left( {p + h_{2} } \right) - cpu_{2} } \right]}}{{\left[ {2p^{2} u_{2} + \left( {c + 2p} \right)u_{1} \left( {p + h_{2} } \right)} \right]^{2} }}$, where $\left( {c + 2p} \right)\left( {a - bp} \right)\left( {p + h_{2} } \right) - cpu_{2} = 2p\left( {a - bp} \right)\left( {p + h_{2} } \right) + c\left[ {\left( {p + h_{2} } \right)\left( {a - bp} \right) - u_{2} p} \right]$, owing that $pB_{1} \left( {p + h_{2} } \right) + pB_{2} \left( {p + h_{1} } \right) - 2\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right) > 0$ and $B_{1} \left( {p + h_{2} } \right) + B_{2} p - 2\left( {a - bp} \right)\left( {p + h_{2} } \right) > 0$, so $\left( {p + h_{2} } \right)\left( {a - bp} \right) - u_{2} p > 0$, thus $\frac{{\partial w^{SO} }}{{\partial u_{1} }} < 0$. $\frac{{\partial w^{SB} }}{{\partial u_{1} }} = - \frac{{p^{2} \left( {a - bp} \right)}}{{\left( {c + 2p} \right)\left( {u_{1} + u_{2} } \right)^{2} }} < 0$. According to Table 5, the derivative of rental with respect to $u_{2}$ is similar, so we omit the derivative results here.

Proof of Lemma 3 Taking the derivative with respect to idle costs gives $\frac{{\partial Q_{1}^{SN} }}{{\partial h_{1} }} = - \frac{{B_{1} \left\{ {2pB_{1} B_{2} \left( {p + h_{1} } \right)\left( {p + h_{2} } \right) + pB_{1}^{2} \left( {p + h_{2} } \right)^{2} + B_{2} \left( {p + h_{1} } \right)^{2} \left[ {pB_{2} - 2\left( {a - bp} \right)\left( {p + h_{2} } \right)} \right]} \right\}}}{{2\left( {p + h_{1} } \right)^{2} \left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]^{2} }}$, owing that $B_{1} p\left( {p + h_{2} } \right) + B_{2} p\left( {p + h_{1} } \right) - 2\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)\left( {a - bp} \right) > 0$, so $pB_{1} B_{2} \left( {p + h_{1} } \right)\left( {p + h_{2} } \right) + pB_{2}^{2} \left( {p + h_{1} } \right)^{2} - 2B_{2} \left( {p + h_{1} } \right)^{2} \left( {a - bp} \right)\left( {p + h_{2} } \right) = B_{2} \left( {p + h_{1} } \right)\left[ {pB_{1} \left( {p + h_{2} } \right) + pB_{2} \left( {p + h_{1} } \right) - 2\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)} \right] > 0$, thus $\frac{{\partial Q_{1}^{SN} }}{{\partial h_{1} }} < 0.$ $\frac{{\partial Q_{1}^{SO} }}{{\partial h_{1} }} = \frac{{\partial Q_{1}^{SB} }}{{\partial h_{1} }} = \frac{{\partial Q_{1}^{SB} }}{{\partial h_{2} }} = 0$, $\frac{{\partial Q_{1}^{SN} }}{{\partial h_{2} }} = - \frac{{\left( {a - bp} \right)B_{1} B_{2} \left( {p + h_{1} } \right)}}{{\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]^{2} }} < 0$, $\frac{{\partial Q_{1}^{SO} }}{{\partial h_{2} }} = - \frac{{p^{2} B_{1} B_{2} \left[ {4p\left( {a - bp} \right) + cB_{1} } \right]}}{{\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]^{2} }} < 0$. The derivatives of $Q_{1}^{S*}$ and $Q_{2}^{S*}$ with respect to the idle cost are symmetrical, and will not be repeated here. (2) Taking the derivative with respect to price-sensitivity gives $\frac{{\partial Q_{1}^{SN} }}{{\partial {\text{b}}}} = - \frac{{pB_{2} \left( {p + h_{1} } \right)}}{{B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)}} < 0$, $\frac{{\partial Q_{1}^{SO} }}{{\partial {\text{b}}}} = - \frac{{p\left[ {2p^{2} B_{2} + cB_{1} \left( {p + h_{2} } \right)} \right]}}{{2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)}} < 0$, $\frac{{\partial Q_{1}^{SB} }}{{\partial {\text{b}}}} = - \frac{{p\left[ {c\left( {B_{1} + B_{2} } \right) + 2pB_{2} } \right]}}{{\left( {c + 2p} \right)\left( {B_{1} + B_{2} } \right)}} < 0$. Moreover, taking the derivative with respect to the potential demand gives us $\frac{{\partial Q_{1}^{SN} }}{{\partial u_{1} }} = \frac{{2pu_{1} u_{2} \left( {p + h_{1} } \right)\left( {p + h_{2} } \right) + pu_{1}^{2} \left( {p + h_{2} } \right)^{2} + u_{2} \left( {p + h_{1} } \right)^{2} \left[ {pu_{2} - \left( {a - bp} \right)\left( {p + h_{2} } \right)} \right]}}{{\left( {p + h_{1} } \right)\left[ {u_{2} \left( {p + h_{1} } \right) + u_{1} \left( {p + h_{2} } \right)} \right]^{2} }} > 0$, it’s prove is the same as $\frac{{\partial Q_{1}^{SN} }}{{\partial h_{1} }}$. Under the SO strategy, $\frac{{\partial Q_{1}^{SO} }}{{\partial u_{1} }} = \frac{{8p^{3} u_{1} u_{2} \left( {p + h_{2} } \right) + 2p\left( {c + 2p} \right)u_{1}^{2} \left( {p + h_{2} } \right)^{2} + 4p^{3} u_{2} \left[ {pu_{2} - \left( {a - bp} \right)\left( {p + h_{2} } \right)} \right]}}{{\left[ {2p^{2} u_{2} + \left( {c + 2p} \right)u_{1} \left( {p + h_{2} } \right)} \right]^{2} }}$, according to $8p^{3} u_{2} \left[ {u_{1} \left( {p + h_{2} } \right) + pu_{2} - \left( {a - bp} \right)\left( {p + h_{2} } \right)} \right] > 0$, we can get $\frac{{\partial Q_{1}^{SO} }}{{\partial u_{1} }} > 0$. $\frac{{\partial Q_{1}^{SB} }}{{\partial u_{1} }} = \frac{{2p\left[ {\left( {u_{1} + u_{2} } \right)^{2} - \left( {a - bp} \right)u_{2} } \right]}}{{\left( {c + 2p} \right)\left( {u_{1} + u_{2} } \right)^{2} }}$, it is known that $\left( {u_{1} + u_{2} } \right)^{2} > u_{2} \left( {u_{1} + u_{2} } \right)$, so $\left( {u_{1} + u_{2} } \right)^{2} - \left( {a - bp} \right)u_{2} > u_{2} \left[ {u_{1} + u_{2} - \left( {a - bp} \right)} \right] > 0$, thus $\frac{{\partial Q_{1}^{SB} }}{{\partial u_{1} }} > 0$. Similarly, we can easily obtain $\frac{{\partial Q_{1}^{SN} }}{{\partial u_{2} }} = \frac{{\left( {a - bp} \right)u_{1} \left( {p + h_{1} } \right)\left( {p + h_{2} } \right)}}{{\left[ {u_{2} \left( {p + h_{1} } \right) + u_{1} \left( {p + h_{2} } \right)} \right]^{2} }} > 0$, $\frac{{\partial Q_{1}^{SO} }}{{\partial u_{2} }} = \frac{{2p^{2} u_{1} \left[ {2p\left( {a - bp} \right) + cu_{1} } \right]\left( {p + h_{2} } \right)}}{{\left[ {2p^{2} u_{2} + \left( {c + 2p} \right)u_{1} \left( {p + h_{2} } \right)} \right]^{2} }} > 0$, $\frac{{\partial Q_{1}^{SB} }}{{\partial u_{2} }} = \frac{{2p\left( {a - bp} \right)u_{1} }}{{\left( {c + 2p} \right)\left( {u_{1} + u_{2} } \right)^{2} }} > 0$. The derivative of $Q_{2}^{S*}$ with respect to $b$ and $B_{i}$ is similar to the above.

Proof of Proposition 2 (1) According to the difference between the optimal rental under the SN strategy and the SO strategy in Table 5, we can get $w^{SN} - w^{SO} = \frac{{B_{1} \left( {p + h_{2} } \right)\left\{ {2\left( {a - bp} \right)\left( {p + h_{2} } \right)\left[ {cp + \left( {c + 2p} \right)h_{1} } \right] - cpB_{2} \left( {p + h_{1} } \right) - cpB_{1} \left( {p + h_{2} } \right)} \right\}}}{{2\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$, where $2\left( {a - bp} \right)\left( {p + h_{2} } \right)\left[ {cp + \left( {c + 2p} \right)h_{1} } \right] - cpB_{2} \left( {p + h_{1} } \right) - cpB_{1} \left( {p + h_{2} } \right) = 2\left( {a - bp} \right)\left( {p + h_{2} } \right)\left[ {c\left( {p + h_{1} } \right) + 2ph_{1} } \right] - c\left[ {pB_{2} \left( {p + h_{1} } \right) + pB_{1} \left( {p + h_{2} } \right)} \right] \ge 2\left( {a - bp} \right)\left( {p + h_{2} } \right)\left( {p + h_{1} } \right) - pB_{2} \left( {p + h_{1} } \right) - pB_{1} \left( {p + h_{2} } \right) > 0$, thus $w^{SN} > w^{SO}$. Similarly, we can get the difference $w^{SO} - w^{SB} = \frac{{p^{2} B_{2} \left\{ {2\left( {a - bp} \right)\left[ {cp + \left( {c + 2p} \right)h_{2} } \right] - cpB_{1} - cpB_{2} } \right\}}}{{\left( {c + 2p} \right)\left( {B_{1} + B_{2} } \right)\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$, we have obtained $2\left( {a - bp} \right)\left( {p + h_{2} } \right) - pB_{1} - pB_{2} > 0$, thus $w^{SO} > w^{SB}$. Therefore, the relationship of rental under different sharing strategies is as follows $w^{SN} > w^{SO} > w^{SB}$.

According to the difference between the optimal rental under the SN strategy and the SO strategy in Table 5, we can get $Q_{1}^{SO} - Q_{1}^{SN} = \frac{{B_{1} \left\{ {pB_{1}^{2} \left( {2h_{1} - c} \right)\left( {p + h_{2} } \right)^{2} + 2pB_{2} h_{1} \left( {p + h_{1} } \right)\left[ {pB_{2} - 2\left( {a - bp} \right)\left( {p + h_{2} } \right)} \right] - B_{1} \left( {p + h_{2} } \right)\left\{ {pB_{2} \left[ {cp + \left( {c - 4p - 2h_{1} } \right)h_{1} } \right] - 2c\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)} \right\}} \right\}}}{{2\left( {p + h_{1} } \right)\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$, owing that $pB_{1}^{2} \left( {2h_{1} - c} \right)\left( {p + h_{2} } \right)^{2} + 2pB_{2} h_{1} \left( {p + h_{1} } \right)\left[ {pB_{2} - 2\left( {a - bp} \right)\left( {p + h_{2} } \right)} \right] - B_{1} \left( {p + h_{2} } \right)\left\{ {pB_{2} \left[ {cp + \left( {c - 4p - 2h_{1} } \right)h_{1} } \right] - 2c\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)} \right\} = 2pB_{1} h_{1} \left( {p + h_{2} } \right)\left[ {p\left( {B_{1} + B_{2} } \right) + B_{1} h_{2} } \right] + cB_{1} \left( {p + h_{2} } \right)\left[ {pB_{1} \left( {p + h_{2} } \right) + pB_{2} \left( {p + h_{1} } \right) - 2\left( {a - bp} \right)\left( {p + h_{1} } \right)\left( {p + h_{2} } \right)} \right] + 2pB_{2} h_{1} \left( {p + h_{1} } \right)\left[ {pB_{2} + B_{1} \left( {p + h_{2} } \right) - 2\left( {a - bp} \right)\left( {p + h_{2} } \right)} \right] > 0$, so we get $Q_{1}^{SO} > Q_{1}^{SN}$. $Q_{1}^{SB} - Q_{1}^{SO} = \frac{{pB_{1} B_{2} \left\{ {2\left( {a - bp} \right)\left[ {cp + \left( {c + 2p} \right)h_{2} } \right] - cpB_{1} - cpB_{2} } \right\}}}{{\left( {c + 2p} \right)\left( {B_{1} + B_{2} } \right)\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$. The comparison of $w^{SO}$ and $w^{SB}$ shows that $2\left( {a - bp} \right)\left[ {cp + \left( {c + 2p} \right)h_{2} } \right] - cpB_{1} - cpB_{2} > 0$, so we can get $Q_{1}^{SB} > Q_{1}^{SO}$. Therefore, the relationship of NVOCC $I$’s order quantity under different sharing strategies is as follows $Q_{1}^{SN} < Q_{1}^{SO} < Q_{1}^{SB}$.

$Q_{2}^{SO} - Q_{2}^{SN} = \frac{{B_{1} B_{2} \left\{ {2\left( {a - bp} \right)\left( {p + h_{2} } \right)\left[ {cp + \left( {c + 2p} \right)h_{1} } \right] - cpB_{2} \left( {p + h_{1} } \right) - cpB_{1} \left( {p + h_{2} } \right)} \right\}}}{{2\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$. The comparison of $w^{SN}$ and $w^{SO}$ shows that $2\left( {a - bp} \right)\left( {p + h_{2} } \right)\left[ {cp + \left( {c + 2p} \right)h_{1} } \right] - cpB_{2} \left( {p + h_{1} } \right) - cpB_{1} \left( {p + h_{2} } \right) > 0$, so we can get $Q_{2}^{SO} > Q_{2}^{SN}$. Similarly, the difference of order quantity $Q_{2}^{SB} - Q_{2}^{SO} =$

$\frac{{pB_{2} \left\{ {p^{2} B_{2}^{2} \left( {2h_{2} - c} \right) + B_{2} \left\{ {2cp\left( {a - bp} \right)\left( {p + h_{2} } \right) + B_{1} \left[ {\left( {c + 4p} \right)ph_{2} + \left( {c + 2p} \right)h_{2}^{2} - cp^{2} } \right]} \right\} - \left( {c + 2p} \right)B_{1} \left( {2a - 2bp - B_{1} } \right)h_{2} \left( {p + h_{2} } \right)} \right\}}}{{\left( {c + 2p} \right)\left( {B_{1} + B_{2} } \right)\left( {p + h_{2} } \right)\left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$, where $p^{2} B_{2}^{2} \left( {2h_{2} - c} \right) + B_{2} \left\{ {2cp\left( {a - bp} \right)\left( {p + h_{2} } \right) + B_{1} \left[ {\left( {c + 4p} \right)ph_{2} + \left( {c + 2p} \right)h_{2}^{2} - cp^{2} } \right]} \right\} - \left( {c + 2p} \right)B_{1} \left( {2a - 2bp - B_{1} } \right)h_{2} \left( {p + h_{2} } \right) = 2p^{2} B_{2} h_{2} \left( {B_{1} + B_{2} } \right) + cpB_{2} \left[ {2\left( {a - bp} \right)\left( {p + h_{2} } \right) - pB_{1} - pB_{2} } \right] + B_{1} h_{2} \left( {c + 2p} \right)\left( {p + h_{2} } \right)\left[ {B_{1} + B_{2} - 2\left( {a - bp} \right)} \right] > 0$, thus $Q_{2}^{SB} > Q_{2}^{SO}$. Therefore, the relationship of NVOCC $II$’s order quantity under different sharing strategies is as follows $Q_{2}^{SN} < Q_{2}^{SO} < Q_{2}^{SB}$.

(2) According to the profit function of the NVOCC, $\pi_{c1}^{SO} - \pi_{c1}^{SN} = \left( {p - w^{SO} } \right)Q_{1}^{SO} - \left( {p + h_{1} } \right)\frac{{z_{1}^{SO2} }}{{2B_{1} }} - \left( {p - w^{SN} } \right)Q_{1}^{SN} + \left( {p + h_{1} } \right)\frac{{z_{1}^{SN2} }}{{2B_{1} }} = \left[ {p - \frac{p}{{2B_{1} }}\left( {z_{1}^{SO} + z_{1}^{SN} } \right)} \right]\left( {z_{1}^{SO} - z_{1}^{SN} } \right) + \frac{{h_{1} }}{{2B_{1} }}z_{1}^{SN2} + w^{SN} Q_{1}^{SN} - w^{SO} Q_{1}^{SO}$, where $\frac{{h_{1} }}{{2B_{1} }}z_{1}^{SN2} + w^{SN} Q_{1}^{SN} - w^{SO} Q_{1}^{SO} \ge \frac{{h_{1} }}{{2B_{1} }}z_{1}^{SN2} + w^{SO} \left( {z_{1}^{SN} - z_{1}^{SO} } \right) \ge \frac{{h_{1} }}{{2B_{1} }}\left[ {z_{1}^{SN2} + 2B_{1} \left( {z_{1}^{SN} - z_{1}^{SO} } \right)} \right] = \frac{{h_{1} }}{{2B_{1} }}\left[ {z_{1}^{SN} \left( {z_{1}^{SN} + 2B_{1} } \right) - 2B_{1} z_{1}^{SO} } \right]$. Known that $B_{1} - z_{1}^{SN} = \frac{{B_{1} \left\{ {B_{1} \left( {p + 2h_{1} } \right)\left( {p + h_{2} } \right) + \left( {p + h_{1} } \right)\left[ {B_{2} \left( {p + 2h_{1} } \right) + 2\left( {a - bp} \right)\left( {p + h_{2} } \right)} \right]} \right\}}}{{2\left( {p + h_{1} } \right)\left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]}} > 0$, $B_{1} - z_{1}^{SO} = \frac{{B_{1} \left\{ {\left( {c + p} \right)B_{1} \left( {p + h_{2} } \right) + p\left[ {pB_{2} + 2\left( {a - bp} \right)\left( {p + h_{2} } \right)} \right]} \right\}}}{{2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)}} > 0$, so $2B_{1} - \left( {z_{1}^{SO} + z_{1}^{SN} } \right) > 0$. Thus, we can obtain $\pi_{c1}^{SO} > \pi_{c1}^{SN}$. $\pi_{c1}^{SB} - \pi_{c1}^{SO} = \left[ {p - \frac{p}{{2B_{1} }}\left( {z_{1}^{SB} + z_{1}^{SO} } \right)} \right]\left( {z_{1}^{SB} - z_{1}^{SO} } \right) + w^{SO} Q_{1}^{SO} - w^{SB} Q_{1}^{SB} \ge \left[ {p - w^{SB} - \frac{p}{{2B_{1} }}\left( {z_{1}^{SB} + z_{1}^{SO} } \right)} \right]\left( {z_{1}^{SB} - z_{1}^{SO} } \right)$, and $B_{1} - z_{1}^{SB} = \frac{{B_{1} \left[ {2p\left( {a - bp} \right) + \left( {c + p} \right)\left( {B_{1} + B_{2} } \right)} \right]}}{{\left( {c + 2p} \right)\left( {B_{1} + B_{2} } \right)}} > 0$, so $\pi_{c1}^{SB} > \pi_{c1}^{SO}$. Therefore, the relationship of NVOCC $I$’s profit under different sharing strategies is $\pi_{c1}^{SN} < \pi_{c1}^{SO} < \pi_{c1}^{SB}$. Moreover, since the two liner companies are symmetrical and the rentals payable are the same, the conclusion of NVOCC $II$ is similar.

$\pi_{p}^{SO} - \pi_{p}^{SN} = w\left( {Q_{1}^{SO} + Q_{2}^{SO} } \right) - c\frac{{z_{1}^{SO2} }}{{2B_{1} }} - w\left( {Q_{1}^{SN} + Q_{2}^{SN} } \right) = w\left( {z_{1}^{SO} - z_{1}^{SN} + z_{2}^{SO} - z_{2}^{SN} } \right) - c\frac{{z_{1}^{SO2} }}{{2B_{1} }}$, where $w\left( {z_{1}^{SO} - z_{1}^{SN} } \right) - c\frac{{z_{1}^{SO2} }}{{2B_{1} }} > c\left( {z_{1}^{SO} - z_{1}^{SN} - \frac{{z_{1}^{SO2} }}{{2B_{1} }}} \right) = \frac{c}{{2B_{1} }}\left[ {2B_{1} \left( {z_{1}^{SO} - z_{1}^{SN} } \right) - z_{1}^{SO2} } \right]$ and it has been proved that $B_{1} > z_{1}^{SO} > z_{1}^{SN}$, thus $\pi_{p}^{SO} > \pi_{p}^{SN}$. $\pi_{p}^{SB} - \pi_{p}^{SO} = w\left( {Q_{1}^{SB} + Q_{2}^{SB} } \right) - c\left( {\frac{{z_{1}^{SB2} }}{{2B_{1} }} + \frac{{z_{2}^{SB2} }}{{2B_{2} }}} \right) - w\left( {Q_{1}^{SO} + Q_{2}^{SO} } \right) + c\frac{{z_{1}^{SO2} }}{{2B_{1} }} = \left[ {w - \frac{c}{{2B_{1} }}\left( {z_{1}^{SB} + z_{1}^{SO} } \right)} \right]\left( {z_{1}^{SB} - z_{1}^{SO} } \right) + w\left( {z_{2}^{SB} - z_{2}^{SO} } \right) - c\frac{{z_{2}^{SB2} }}{{2B_{2} }}$, where $\left( {z_{2}^{SB} - z_{2}^{SO} } \right) - c\frac{{z_{2}^{SB2} }}{{2B_{2} }} > 0$ is similar to the previous and $w - \frac{c}{{2B_{1} }}\left( {z_{1}^{SB} + z_{1}^{SO} } \right) = \frac{1}{{2B_{1} }}\left[ {2B_{1} w - c\left( {z_{1}^{SB} + z_{1}^{SO} } \right)} \right] > \frac{c}{{2B_{1} }}\left[ {2B_{1} - \left( {z_{1}^{SB} + z_{1}^{SO} } \right)} \right] > 0$, so $\pi_{p}^{SB} > \pi_{p}^{SO}$. Therefore, the relationship of ordering platform’s profit under different sharing strategies is $\pi_{p}^{SN} < \pi_{p}^{SO} < \pi_{p}^{SB}$.

Accordingly, the equilibriums of liner shipping system under three different sharing strategies with different rental levels are described in Table

Table 6 Equilibriums of different sharing strategies under different rentals

6.

In order to ensure that the liner company $i \left( {i = I,II} \right)$ is profitable and the order quantity is not negative, we require $p > w_{i}$, so we can get $pB_{i} - \left( {a - bp} \right)\left( {p + h_{i} } \right) > 0$ under DN strategy, $pB_{1} - \left( {a - bp} \right) > 0$ and $pB_{2} - \left( {a - bp} \right)\left( {p + h_{2} } \right) > 0$ under DO strategy and $B_{i} - \left( {a - bp} \right) > 0$ under DB strategy.

Proof of Lemma 5 From Table 6, we can get the derivative about idle cost $\frac{{\partial w_{1}^{DN} }}{{\partial h_{1} }} = \frac{a - bp}{{2B_{1} }} > 0$, $\frac{{\partial w_{1}^{DO} }}{{\partial h_{1} }} = \frac{{\partial w_{1}^{DB} }}{{\partial h_{1} }} = 0$. Taking the derivative about price-sensitivity coefficient give us $\frac{{\partial w_{1}^{DN} }}{\partial b} = - \frac{{p\left( {p + h_{1} } \right)}}{{2B_{1} }} < 0$, $\frac{{\partial w_{1}^{DO} }}{\partial b} = \frac{{\partial w_{1}^{DB} }}{\partial b} = - \frac{{p^{3} }}{{\left( {c + 2p} \right)B_{1} }} < 0$. Similarly, we can get $\frac{{\partial w_{1}^{DN} }}{{\partial u_{1} }} = - \frac{{\left( {a - bp} \right)\left( {p + h_{1} } \right)}}{{8u_{1}^{2} }} < 0$, $\frac{{\partial w_{1}^{DO} }}{{\partial u_{1} }} = \frac{{\partial w_{1}^{DB} }}{{\partial u_{1} }} = - \frac{{p^{2} \left( {a - bp} \right)}}{{4\left( {c + 2p} \right)u_{1}^{2} }} < 0$. When the ordering platform charges different rentals for the two NVOCCs respectively, the decision of $w_{i}^{D*}$ is only related to the liner company $i$, so the change of $w_{2}^{D*}$ is symmetrical to $w_{1}^{D*}$, which is omitted here.

Proof of Lemma 6 Taking the derivative with respect to idle cost gives $\frac{{\partial Q_{1}^{DN} }}{{\partial h_{1} }} = - \frac{{pB_{1} }}{{2\left( {p + h_{1} } \right)^{2} }} < 0$, $\frac{{\partial Q_{1}^{DO} }}{{\partial h_{1} }} = \frac{{\partial Q_{1}^{DB} }}{{\partial h_{1} }} = 0$. Taking the derivative with respect to price-sensitivity coefficient give us $\frac{{\partial Q_{1}^{DN} }}{\partial b} = - \frac{p}{2} < 0$, $\frac{{\partial Q_{1}^{DO} }}{\partial b} = \frac{{\partial Q_{1}^{DO} }}{\partial b} = - \frac{{p\left( {p + {\text{c}}} \right)}}{c + 2p} < 0$. Similarly, $\frac{{\partial Q_{1}^{DN} }}{{\partial u_{1} }} = \frac{p}{{2\left( {p + h_{1} } \right)}} > 0$, $\frac{{\partial Q_{1}^{DO} }}{{\partial u_{1} }} = \frac{{\partial Q_{1}^{DB} }}{{\partial u _{1} }} = \frac{p}{c + 2p} > 0$. The decisions of $Q_{1}^{D*}$ and $Q_{2}^{D*}$ are symmetrical, and will not be repeated here.

Proof of Proposition 4 (1) According to the difference between the optimal order quantity under the DN strategy and the DO strategy in Table 6, $Q_{1}^{DO} - Q_{1}^{DN} = \frac{{c\left( {a - bp} \right)\left( {p + h_{1} } \right) + pB_{1} \left( {2h_{1} - c} \right)}}{{2\left( {c + 2p} \right)\left( {p + h_{1} } \right)}} > 0$. Meanwhile, $Q_{1}^{DO} - Q_{1}^{DB} = 0$. Thus, we can obtain $Q_{1}^{DN} < Q_{1}^{DO} = Q_{1}^{DB}$. The decisions of $Q_{1}^{D*}$ and $Q_{2}^{D*}$ are symmetrical, and will not be repeated here.

(2) $\pi_{c1}^{DO} - \pi_{c1}^{DN} = \left( {p - w_{1}^{DO} } \right)Q_{1}^{DO} - p\frac{{z_{1}^{DO2} }}{{2B_{1} }} - \left( {p - w_{1}^{DN} } \right)Q_{1}^{DN} + \left( {p + h_{1} } \right)\frac{{z_{1}^{DN2} }}{{2B_{1} }} = p\left[ {z_{1}^{DO} - z_{1}^{DN} - \frac{1}{{2B_{1} }}\left( {z_{1}^{DO2} - z_{1}^{DN2} } \right)} \right] + w_{1}^{DN} Q_{1}^{DN} - w_{1}^{DO} Q_{1}^{DO} + h_{1} \frac{{z_{1}^{DN2} }}{{2B_{1} }} \ge \frac{p}{{2B_{1} }}\left( {z_{1}^{DO} - z_{1}^{DN} } \right)\left[ {2B_{1} - \left( {z_{1}^{DO} + z_{1}^{DN} } \right)} \right]$ $+ \frac{{h_{1} }}{{2B_{1} }}\left[ {2B_{1} \left( {z_{1}^{DN} - z_{1}^{DO} } \right) + z_{1}^{DN2} } \right]$. Similar to the same rental situation, $B_{1} - z_{1}^{DN} = \frac{1}{2}\left[ {a - bp + \frac{{B_{1} \left( {p + 2h_{1} } \right)}}{{p + h_{1} }}} \right] > 0$ and $B_{1} - z_{1}^{DO} = \frac{{p\left( {a - bp} \right) + \left( {c + p} \right)B_{1} }}{c + 2p} > 0$, so $2B_{1} - \left( {z_{1}^{DO} + z_{1}^{DN} } \right) > 0$, thus we can obtain $\pi_{c1}^{DO} > \pi_{c1}^{DN}$. Meanwhile, $\pi_{c1}^{DO} - \pi_{c1}^{DB} = 0$. Therefore, the relationship of NVOCC $I$’s profit under different sharing strategies is $\pi_{c1}^{DN} < \pi_{c1}^{DO} = \pi_{c1}^{DB}$.

Proof of Proposition 5 Similarly, the difference between the optimal rental under the DN strategy and the DO strategy gives us $w_{1}^{DN} - w_{1}^{DO} = \frac{{\left( {a - bp} \right)\left[ {cp + \left( {c + 2p} \right)h_{1} } \right] - cpB_{1} }}{{2\left( {c + 2p} \right)B_{1} }} > 0$ and $w_{1}^{DO} - w_{1}^{DB} = 0$. Thus, we can obtain $w_{1}^{DN} > w_{1}^{DO} = w_{1}^{DB}$. The decisions of $w_{1}^{D*}$ and $w_{2}^{D*}$ are symmetrical, and will not be repeated here. (2) $\pi_{p}^{DO} - \pi_{p}^{DN} = w_{1}^{DO} Q_{1}^{DO} + w_{2}^{DO} Q_{2}^{DO} - c\frac{{z_{1}^{DO2} }}{{2B_{1} }} - \left( {w_{1}^{DN} Q_{1}^{DN} + w_{2}^{DN} Q_{2}^{DN} } \right) = w_{1}^{DO} Q_{1}^{DO} - w_{1}^{DN} Q_{1}^{DN} - c\frac{{z_{1}^{DO2} }}{{2B_{1} }} > c\left( {z_{1}^{DO} - z_{1}^{DN} - \frac{{z_{1}^{DO2} }}{{2B_{1} }}} \right)$, where $z_{1}^{DO} - z_{1}^{DN} - \frac{{z_{1}^{DO2} }}{{2B_{1} }} > 0$ is similar to the same rental situation, so $\pi_{p}^{DO} > \pi_{p}^{DN}$. Meanwhile, $\pi_{p}^{DB} - \pi_{p}^{DO} = w_{1}^{DB} Q_{1}^{DB} + w_{2}^{DB} Q_{2}^{DB} - c\left( {\frac{{z_{1}^{DB2} }}{{2B_{1} }} + \frac{{z_{2}^{DB2} }}{{2B_{2} }}} \right) - \left( {w_{1}^{DO} Q_{1}^{DO} + w_{2}^{DO} Q_{2}^{DO} } \right) + c\frac{{z_{1}^{DO2} }}{{2B_{1} }} = w_{2}^{DB} Q_{2}^{DB} - w_{2}^{DO} Q_{2}^{DO} - c\frac{{z_{2}^{DB2} }}{{2B_{2} }} > c\left( {z_{1}^{DB} - z_{1}^{DO} - \frac{{z_{1}^{DB2} }}{{2B_{1} }}} \right)$, so $\pi_{p}^{DB} > \pi_{p}^{DO}$. Therefore, the relationship of ordering platform’s profit under different sharing strategies is $\pi_{p}^{DN} < \pi_{p}^{DO} < \pi_{p}^{DB}$.

Proof of Proposition 6 It can be known from the assumption that $\frac{{B_{1} }}{{h_{1} }} > \frac{{B_{2} }}{{h_{2} }}$, so $w_{2}^{DN} - w^{SN} = \frac{{\left( {a - bp} \right)\left( {p + h_{2} } \right)\left[ {B_{1} \left( {p + h_{2} } \right) - B_{2} \left( {p + h_{1} } \right)} \right]}}{{2B_{2} \left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]}} > 0$, $w^{SN} - w_{1}^{DN} = \frac{{\left( {a - bp} \right)\left( {p + h_{1} } \right)\left[ {B_{1} \left( {p + h_{2} } \right) - B_{2} \left( {p + h_{1} } \right)} \right]}}{{2B_{1} \left[ {B_{2} \left( {p + h_{1} } \right) + B_{1} \left( {p + h_{2} } \right)} \right]}} > 0$. Therefore, when neither one NVOCC chooses sharing, the relationship of ordering platform’s optimal rentals is $w_{2}^{DN} > w^{SN} > w_{1}^{DN}$.

$w_{2}^{DO} - w^{SO} = \frac{{\left( {p + h_{2} } \right)\left\{ {B_{1} \left[ {\left( {c + 2p} \right)\left( {a - bp} \right)\left( {p + h_{2} } \right) - cpB_{2} } \right] - 2p^{2} \left( {a - bp} \right)B_{2} } \right\}}}{{2B_{2} \left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}}$, owing that $B_{1} \left[ {\left( {c + 2p} \right)\left( {a - bp} \right)\left( {p + h_{2} } \right) - cpB_{2} } \right] - 2p^{2} \left( {a - bp} \right)B_{2} = cB_{1} \left[ {\left( {a - bp} \right)\left( {p + h_{2} } \right) - pB_{2} } \right] + 2p\left( {a - bp} \right)\left[ {p\left( {B_{1} - B_{2} } \right) + B_{1} h_{2} } \right] > 0$, so we can obtain $w_{2}^{DO} > w^{SO}$. Meanwhile, the difference $w^{SO} - w_{1}^{DO} = \frac{{p^{2} \left\{ {B_{1} \left[ {\left( {c + 2p} \right)\left( {a - bp} \right)\left( {p + h_{2} } \right) - cpB_{2} } \right] - 2p^{2} \left( {a - bp} \right)B_{2} } \right\}}}{{\left( {c + 2p} \right)B_{1} \left[ {2p^{2} B_{2} + \left( {c + 2p} \right)B_{1} \left( {p + h_{2} } \right)} \right]}} > 0$, thus $w_{2}^{DO} > w^{SO} > w_{1}^{DO}$.

Similarly, $w_{2}^{DB} - w^{SB} = \frac{{p^{2} \left( {a - bp} \right)\left( {B_{1} - B_{2} } \right)}}{{\left( {c + 2p} \right)B_{2} \left( {B_{1} + B_{2} } \right)}} > 0$, $w^{SB} > w_{1}^{DB} = \frac{{p^{2} \left( {a - bp} \right)\left( {B_{1} - B_{2} } \right)}}{{\left( {c + 2p} \right)B_{1} \left( {B_{1} + B_{2} } \right)}} > 0$, thus $w_{2}^{DB} > w^{SB} > w_{1}^{DB}$.

Proof of Proposition 8 The proof process is similar to Proposition 6 and is omitted here.

Proof of Proposition 7, 9 According to the relationship of rentals and order quantities obtained in the foregoing, the relationship among profits of NVOCC and ordering platform under different rental modes can be derived, and the proofs are similar to Propositions 4 and 5. Therefore they are omitted here.

Appendix 2. Proofs for extension

The superscripts NSN, NSO, and NSB are used to represent three sharing strategies in the following models considering $n$ NVOCCs.

Under the NSN strategy, all of $n$ NVOCCs choose the non-sharing strategy. After the booking platform determines the rental $w$, each NVOCC decides its own booking quantity $Q_{i} \left( {i = 1,2, \ldots ,n} \right)$. In this case, the profit functions are as follows

$$ \pi_{ci} \left( {Q_{i} |Q_{1} ,Q_{2} , \ldots ,Q_{i - 1} ,Q_{i + 1} , \ldots ,Q_{n} } \right) = p \cdot \min \left( {Q_{i} ,D_{i} } \right) - wQ_{i} - h_{i} \left( {Q_{i} - D_{i} } \right)^{ + } $$

$$ \pi_{p} \left( {w|Q_{1} ,Q_{2} , \ldots ,Q_{i} , \ldots ,Q_{n} } \right) = w \cdot \mathop \sum \limits_{i = 1}^{n} Q_{i} $$

$$ i = 1,2, \ldots ,n $$

The solution process is similar to Sect. 4.1, and the optimal rental and sharing levels of $n$ NVOCCs are obtained

$$ \overline{z}_{i} \left( w \right) = F_{i}^{ - 1} \left( {\frac{p - w}{{p + h_{i} }}} \right) $$

$$ w^{NSN} = \frac{{\left[ {n\left( {a - bp} \right) + \mathop \sum \nolimits_{i = 1}^{n} \overline{z}_{i} \left( {w^{NSN} } \right)} \right]}}{{ \mathop \sum \nolimits_{i = 1}^{n} \left[ {f_{i} \left( {z_{i} } \right)\left( {p + h_{i} } \right)} \right]^{ - 1} }} $$

Under the NSO strategy, assuming that the first $k$ NVOCCs choose sharing strategy, and the next $n - k$ NVOCCs choose non-sharing strategy. In this case, the profit functions obtained are as follows

$$ \pi_{ci} \left( {Q_{i} |Q_{1} ,Q_{2} , \ldots ,Q_{i - 1} ,Q_{i + 1} , \ldots ,Q_{k} } \right) = p \cdot \min \left( {Q_{i} ,D_{i} } \right) - wQ_{i} , \quad i = 1,2, \ldots ,k $$

$$ \pi_{cj} \left( {Q_{j} |Q_{k + 1} , \ldots ,Q_{j - 1} ,Q_{j + 1} , \ldots ,Q_{n} } \right) = p \cdot \min \left( {Q_{j} ,D_{j} } \right) - wQ_{j} - h_{j} \left( {Q_{j} - D_{j} } \right)^{ + } ,\quad j = k + 1, \ldots ,n $$

$$ \pi_{p} \left( {w|Q_{1} , \ldots ,Q_{i} , \ldots ,Q_{j} \ldots ,Q_{n} } \right) = w \cdot \left( {\mathop \sum \limits_{i = 1}^{k} Q_{i} + \mathop \sum \limits_{j = k + 1}^{n} Q_{j} } \right) - c \cdot \mathop \sum \limits_{i = 1}^{k} \left( {Q_{i} - D_{i} } \right)^{ + } $$

The optimal rental of the booking platform and the sharing level of the NVOCC $i$ that chooses the sharing strategy and the NVOCC $j$ that chooses the non-sharing strategy are obtained

$$ \overline{z}_{i} \left( w \right) = F_{1}^{ - 1} \left( {\frac{p - w}{p}} \right) i = 1,2, \ldots ,k $$

$$ \overline{z}_{j} \left( w \right) = F_{2}^{ - 1} \left( {\frac{p - w}{{p + h_{j} }}} \right) j = k + 1, \ldots ,n $$

$$ w^{NSO} = \frac{{p\left\{ {c\mathop \sum \nolimits_{i = 1}^{k} f_{i} \left( {z_{i} } \right)^{ - 1} + p\left[ {n\left( {a - bp} \right) + \mathop \sum \nolimits_{i = 1}^{n} \overline{z}_{i} \left( {w^{NSO} } \right)} \right]} \right\}}}{{p^{2} \mathop \sum \nolimits_{j = k + 1}^{n} \left[ {f_{j} \left( {z_{j} } \right)\left( {p + h_{j} } \right)} \right]^{ - 1} + \left( {p + c} \right)\mathop \sum \nolimits_{i = 1}^{k} f_{i} \left( {z_{i} } \right)^{ - 1} }} $$

Under the NSB strategy, all $of n$ NVOCCs choose the sharing strategy. The profit functions of the booking platform and the NVOCCs are as follows

$$ \pi_{ci} \left( {Q_{i} |Q_{1} ,Q_{2} , \ldots ,Q_{i - 1} ,Q_{i + 1} , \ldots ,Q_{n} } \right) = p \cdot \min \left( {Q_{i} ,D_{i} } \right) - wQ_{i} $$

$$ \pi_{p} \left( {w|Q_{1} ,Q_{2} , \ldots ,Q_{i} , \ldots ,Q_{n} } \right) = w \cdot \mathop \sum \limits_{i = 1}^{n} Q_{i} - c \cdot \mathop \sum \limits_{i = 1}^{n} \left( {Q_{i} - D_{i} } \right)^{ + } $$

$$ i = 1,2, \ldots ,n $$

Get the optimal equilibriums by solving the above expected profit functions

$$ \overline{z}_{i} \left( w \right) = F_{i}^{ - 1} \left( {\frac{p - w}{p}} \right) $$

$$ w^{SB} = \frac{{p\left\{ {c\mathop \sum \nolimits_{i = 1}^{n} f_{i} \left( {z_{i} } \right)^{ - 1} + p\left[ {n\left( {a - bp} \right) + \mathop \sum \nolimits_{i = 1}^{n} \overline{z}_{i} \left( {w^{SB} } \right)} \right]} \right\}}}{{\left( {p + c} \right)\mathop \sum \nolimits_{i = 1}^{n} f_{i} \left( {z_{i} } \right)^{ - 1} }} $$