## Appendix B Proof for Lemma 2, Proposition 3 and Lemma 3  
Proof.: To determine the NBS, we solve the optimization problem defined in Equation (10). We first determine the optimal commission fee, \(s^{*}\), for the given pricing \(p_{1}^{*},p_{2}^{*},p_{12}^{*}\). Subsequently, we solve for the optimal pricing. For the given \(p_{1}^{*},p_{2}^{*},p_{12}^{*}\), it can be shown that Equation (10) is strictly concave in \(s\), and hence the optimal commission fee \(s^{*}\) is unique. To solve for \(s^{*}\), we first write the KKT conditions. Let \(\
u_{1}\) and \(\
u_{2}\) be the Lagrangian multipliers. Subsequently, the KKT conditions are as follows:  
\[\lambda_{12}[p_{2}\lambda_{2}+p_{12}\lambda_{12}-p_{1}\lambda_{1} -2s\lambda_{12}+(\sqrt{\
u\mu_{1}}-\sqrt{c_{1}})^{2}\] \[-\Lambda_{2}(V-\frac{c_{2}}{\mu_{2}-\Lambda_{2}})+\
u_{1}-\
u_{2}] =0;\] \[\
u_{1}[\pi_{1}^{*}(p_{1},s)-(\sqrt{\
u\mu_{1}}-\sqrt{c_{1}})^{2}] =0;\] \[\
u_{2}[\pi_{2}^{*}(p_{2},p_{12}^{0})-\Lambda_{2}(V-\frac{c_{2}}{ \mu_{2}-\Lambda_{2}})] =0;\] \[\
u_{1},\
u_{2} \geq 0.\]  
From the KKT condition, we obtain  
\[s^{*}=\frac{p_{2}\lambda_{2}+p_{12}\lambda_{12}-p_{1}\lambda_{1}+(\sqrt{\
u \mu_{1}}-\sqrt{c_{1}})^{2}-\Lambda_{2}(V-\frac{c_{2}}{\mu_{2}-\Lambda_{2}})}{ 2\lambda_{12}}.\] (A3)  
To obtain the optimal pricing, we rewrite the Equation (10) by utilizing Equation (A3):  
\[\max_{p_{1},p_{2},p_{12}}[\frac{p_{1}\lambda_{1}+p_{2}\lambda_{2}+p_{12} \lambda_{12}-(\sqrt{\
u\mu_{1}}-\sqrt{c_{1}})^{2}-\Lambda_{2}(V-\frac{c_{2}}{ \mu_{2}-\Lambda_{2}})}{2}]^{2},\] (A4)  
which is equivalent to the optimization problem defined in Equation (4). Hence, we have Lemma 2.  
Suppose that \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\), according to Proposition 2, the optimal problem defined in Equation (10) can be rewritten as follows:  
\[\max_{s}{(s\lambda_{12}^{*})^{\alpha}}[p_{2}^{*}\Lambda_{2}+(p_{12}^{*}-s) \lambda_{12}^{*}-\Lambda_{2}(V_{2}-\frac{c_{2}}{\mu_{2}-\Lambda_{2}})]^{\beta}\] (A5)Given the first order condition, we have the following:  
\[s =\frac{\alpha}{\alpha+\beta}p_{12}^{*}+\frac{\alpha[p_{2}^{*}\Lambda_{ 2}-V_{2}\Lambda_{2}+c_{2}\Lambda_{2}/(\mu_{2}-\Lambda_{2})]}{(\alpha+\beta) \lambda_{12}^{*}}\] \[=\frac{\alpha}{\alpha+\beta}[V_{2}-c_{1}w^{*}-\frac{c_{2}\Lambda_ {2}w^{*}}{\mu_{2}-\Lambda_{2}}].\]  
where the optimal waiting time is given as follows:  
\[w^{*}=\frac{1}{\mu_{2}-\Lambda_{2}-\lambda_{12}^{*}}=\sqrt{\frac{V_{2}}{c_{1}( \mu_{2}-\Lambda_{2})+c_{2}\Lambda_{2}}}\]  
Hence, in this case, the equilibrium commission fee is  
\[s^{*}=\frac{\alpha V_{2}}{\alpha+\beta}[1-\sqrt{\frac{c_{1}}{V_{2}(\mu_{2}- \Lambda_{2})}+\frac{c_{2}\Lambda_{2}}{V_{2}(\mu_{2}-\Lambda_{2})^{2}}}].\]  
With the above commission given and optimal prices given in Proposition 2, the corresponding gains of optimal revenues from demand sharing can be calculated.  
Suppose that \(\overline{\mu_{2}}<\mu_{2}\), the optimal problem defined in Equation (10) can be rewritten as follows:  
\[\max_{s}{[p_{1}^{*}\lambda_{1}^{*}+s\lambda_{12}^{*}-(\sqrt{V_{2}\mu_{1}}- \sqrt{c_{1}})^{2}]^{\alpha}[p_{2}^{*}\Lambda_{2}+(p_{12}^{*}-s)\lambda_{12}^{* }-\Lambda_{2}(V_{2}-\frac{c_{2}}{\mu_{2}-\Lambda_{2}})]^{\beta}}\]  
Given the first order condition, we have the following:  
\[s=\frac{\alpha}{\alpha+\beta}p_{12}^{*}+\frac{\alpha[p_{2}^{*} \Lambda_{2}-p_{1}^{*}\lambda_{1}^{*}+(\sqrt{V_{1}\mu_{1}}-\sqrt{c_{1}})^{2}- V_{2}\Lambda_{2}+c_{2}\Lambda_{2}/(\mu_{2}-\Lambda_{2})]}{(\alpha+\beta)\lambda_{12}^{* }},\]  
which is equivalent to  
\[s=\frac{\alpha}{\alpha+\beta}[V_{2}+\frac{(\sqrt{V_{1}\mu_{1}}- \sqrt{c_{1}})^{2}-V_{2}\Lambda_{2}+c_{2}\Lambda_{2}/(\mu_{2}-\Lambda_{2})}{ \lambda_{12}^{*}}\] \[-\frac{c_{1}}{\mu_{2}-\Lambda_{2}-\lambda_{12}^{*}}+\frac{c_{1} \Lambda_{1}-c_{2}\Lambda_{2}-c_{1}\lambda_{12}^{*}}{(\mu_{2}-\Lambda_{2}- \lambda_{12}^{*})\lambda_{12}^{*}}].\]