## 4 Analysis
### Total Benefit of Demand Sharing
There exists a threshold  
\[\overline{\mu_{2}}=\Lambda-\mu_{1}+\sqrt{\frac{c_{1}\mu_{1}}{V_{1}}}+\frac{c_{1} }{2V_{2}}+\sqrt{\frac{c_{1}^{2}}{4V_{2}^{2}}+\frac{c_{1}\Lambda_{1}+c_{2} \Lambda_{2}-c_{1}\mu_{1}+c_{1}\sqrt{c_{1}\mu_{1}/V_{1}}}{V_{2}}}. \tag{6}\]  
If \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\), then the HD would utilize its own optimal pricing strategy and share its remaining patients \(\Lambda_{12}^{0}\) with the HS; additionally, the HS would serves all the patients from Region-2, that is, \(\lambda_{2}^{*}=\Lambda_{2}\) and accept a part of the demand switching from the HD; which is given as:  
\[\lambda_{12}^{*}=\mu_{2}-\Lambda_{2}-\sqrt{\frac{c_{1}\mu_{2}-c_{1}\Lambda_{2} +c_{2}\Lambda_{2}}{V_{2}}}. \tag{7}\]  
The corresponding total optimal revenue is as follows:  
\[\pi_{c}^{*}=V_{1}\mu_{1}+V_{2}\mu_{2}+2c_{1}-2\sqrt{c_{1}\mu_{1}V_{1}}-2\sqrt{ V_{2}(c_{1}\mu_{2}-c_{1}\Lambda_{2}+c_{2}\Lambda_{2})}.\]  
Otherwise, that is, \(\overline{\mu_{2}}<\mu_{2}\), under the medical alliance, the HD will share at least \(\Lambda_{12}^{0}\) with the HS and may deviate from its own optimal strategy; the HS will accept all patients shared by HD. Hence, all patients from both Region-1 and Region-2 will be served by the two hospitals. Proposition 2 shows the condition, that is, \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\), that even with demand sharing and medical alliance, the HD will still hold its optimal strategy and the two hospitals will be unable to serve all the patients in both the regions. On the other hand, if \(\overline{\mu_{2}}<\mu_{2}\), then the HS can serve more patients than HD's remaining demand under its optimal strategy. In this case, the HD will not implement its own optimal strategy due to share more patients with the HS. Compared to the independent operations, HD will earn less revenue by serving its own region. However, this revenue loss will be lesser than the revenue gain as a result of HS's higher utilization. In this case, the two hospitals can serve all patients from the two regions. The following lemma characterizes the market shares of the two hospitals under the centralized operation when the HS capacity is sufficiently large. Suppose that \(\overline{\mu_{2}}<\mu_{2}\), if there exists an equilibrium for the medical alliance, then the corresponding equilibrium effective arrival rate of HD \(\lambda_{1}^{*}\) should be the solution to the equation  
\[(V_{1}-V_{2})(\mu_{1}-\lambda_{1}^{*})^{2}(\mu_{2}-\Lambda+ \lambda_{1}^{*})^{2}+[c_{1}\mu_{2}-c_{1}\Lambda_{2}+c_{2}\Lambda_{2}](\mu_{1}- \lambda_{1}^{*})^{2}\] \[-c_{1}\mu_{1}(\mu_{2}-\Lambda+\lambda_{1}^{*})^{2}=0, \tag{8}\]  
where \(\Lambda=\Lambda_{1}+\Lambda_{2}\). If the two hospitals' service quality is equal, that is, \(V_{1}=V_{2}\), then there will be a unique equilibrium, and the demand shared by HD will be given by  
\[\lambda_{12}^{*}=\Lambda_{1}-\mu_{1}-(\mu_{1}+\mu_{2}-\Lambda)\frac{c_{1}\mu_ {1}-\sqrt{c_{1}\mu_{1}(c_{2}\Lambda_{2}+c_{1}\mu_{2}-c_{1}\Lambda_{2})}}{c_{2 }\Lambda_{2}+c_{1}\mu_{2}-c_{1}\Lambda_{2}-c_{1}\mu_{1}}. \tag{9}\]  
Equation (8) is the necessary condition for equilibrium, which is the first order condition of (4). When the service quality of the two hospitals become equal, they become different in waiting time and pricing for the patients in Region-1. Lemma 1 gives the optimal demand sharing strategy for the medical alliance when \(V_{1}=V_{2}\). Here, we focus on the case of \(V_{1}=V_{2}\), and discuss the case of different service quality later. Let us use \(\
abla\pi_{c}\) to denote the total revenue gain of the medical alliance. It is defined as the difference between the total revenue of the medical alliance and the sum of revenues of two independent hospitals, that is, \(\
abla\pi_{c}=\pi_{c}-\pi_{1}^{0}-\pi_{2}^{0}\). We conduct a numerical study to illustrate the impact of the service rate of the HS, \(\mu_{2}\), on the total revenue gain, when \(V_{1}=V_{2}\), as shown in Figure 2. We observe that the revenue gain ratio curve can be divided into three segments.