## Appendix A Proof for Proposition 1, Proposition 2 and Lemma 1  
According to the Theorem 1 of [29], facing two classes of patients, which are characterized by \(c_{1}>c_{2}\), the optimal pricing and admission control policy for HS is that the lower delay cost enters first, followed by, possibly, part of the other class. It means that HS should first accept patients only from Market-2 and continue doing this until the class is captured; subsequently, if there is sufficient service capacity, then HS should accept patients from Market-1 until it captures the whole sharing demand from Market-1. HS uses different prices to control two streams of demand. However, if HS accepts two types of patients, all patients are faced with the same waiting time in the queue. The optimal problem is to maximize the total revenues shown in Equation (4) under the equilibrium constraints of the patient behavior, which is defined in Equation (1). Before further analysis, let us discuss HS's optimal problem, which is given as follows,  
\[\max_{p_{2},p_{12}}\pi_{2}(p_{2},p_{12})=p_{12}\lambda_{12}+p_{2}\lambda_{2}. \tag{1}\]  
According to Theorem 1 of [29], given the potential sharing demand rate \(\Lambda_{12}\), the optimal solution of the problem defined in Equation (1) satisfies the following:  
1. Suppose that \(\mu_{2}^{0}<\mu_{2}\leq\underline{\mu_{2}}\), then the optimal pricing strategy is as follows: \[p_{2}^{*}=V_{2}-\sqrt{\frac{c_{2}}{\mu_{2}-\Lambda_{2}}},\;p_{12}^{*}>V_{2}- \sqrt{\frac{c_{1}}{\mu_{2}-\Lambda_{2}}}.\] Additionally, the corresponding actual demand rates are as follows: \[\lambda_{2}^{*}=\Lambda_{2},\;\lambda_{12}^{*}=0.\]
2. Suppose that \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}^{\prime}}=\Lambda_{2}+ \Lambda_{12}+\frac{c_{1}}{2V_{2}}+\sqrt{\frac{c_{1}^{2}}{4V_{2}^{*}}+\frac{c_ {2}\Lambda_{2}+c_{1}\Lambda_{2}}{V_{2}}}\), then the optimal prices satisfy the following: \[p_{2}^{*}=V_{2}-c_{2}\sqrt{\frac{V_{2}}{c_{1}\mu_{2}-c_{1}\Lambda_{2}+c_{2} \Lambda_{2}}},\;p_{12}^{*}=V_{2}-c_{1}\sqrt{\frac{V_{2}}{c_{1}\mu_{2}-c_{1} \Lambda_{2}+c_{2}\Lambda_{2}}}.\]Additionally, the corresponding actual demand rates are as follows:  
\[\lambda_{2}^{*}=\Lambda_{2},\,\lambda_{12}^{*}=\mu_{2}-\Lambda_{2}-\sqrt{\frac{c_{ 1}\mu_{2}-c_{1}\Lambda_{2}+c_{2}\Lambda_{2}}{V_{2}}}.\]
3. Suppose that \(\overline{\mu_{2}^{\prime}}<\mu_{2}\), then the optimal prices satisfy the following: \[p_{2}^{*}=V_{2}-\frac{c_{2}}{\mu_{2}-\Lambda_{2}-\Lambda_{12}},\,p_{12}^{*}=V_ {2}-\frac{c_{1}}{\mu_{2}-\Lambda_{2}-\Lambda_{12}}.\] Additionally, the corresponding actual demand rates are as follows: \[\lambda_{2}^{*}=\Lambda_{2},\,\lambda_{12}^{*}=\Lambda_{12}.\] Subsequently, let us analyze HD's optimal problem, which is given as follows, \[\max_{p_{1}}\pi_{1}(p_{1})=p_{1}\lambda_{1}.\] (11) We can obtain the optimal strategy of HD based on the monopolistic pricing results of [27], which is given as follows, 1. Suppose that \(\Lambda_{1}\geq\mu_{1}-\sqrt{\frac{c_{1}\mu_{1}}{V_{1}}}\), then the optimal pricing strategy is \(p_{1}^{*}=V_{1}-\sqrt{\frac{c_{1}V_{1}}{\mu_{1}}}\). Additionally, the corresponding actual demand rate is \(\lambda_{1}^{*}=\mu_{1}-\sqrt{\frac{c_{1}\mu_{1}}{V_{1}}}\). 2. Suppose that \(\Lambda_{1}<\mu_{1}-\sqrt{\frac{c_{1}\mu_{1}}{V_{1}}}\), then the optimal price is \(p_{1}^{*}=V_{1}-\frac{c_{1}}{\mu_{1}-\Lambda_{1}}\). Additionally, the corresponding actual demand rate is \(\lambda_{1}^{*}=\Lambda_{1}\). Here, we analyze the centralized optimal problem shown in Equation (4). Since the threshold \(\underline{\mu_{2}}\) does not depend on the HD's decision, and the optimal strategy of the HS is to capture the demand of Market-2 only when \(\mu_{2}^{0}<\mu_{2}\leq\underline{\mu_{2}}\), the two hospitals are independent. Two hospitals independently operate as a monopolist. Hence, in this case, centralized operation will not result in a revenue gain. However, if \(\mu_{2}>\underline{\mu_{2}}\), then HS can raise its revenue by continuing to accept a part of the sharing demand. Hence, we prove the Proposition 1.