## 4 Analysis
### Revenue Allocation in the Medical Alliance
Hence, to compensate for this revenue loss of HS, the commission fee is less than a proportion of the co-payment of the switching patients, that is, \(s^{*}<\alpha p_{12}^{*}/(\alpha+\beta)\). In addition, we find that this policy of revenue allocation mainly depends on HS's region characteristics, its service rate, and switching patients' delay sensitivity. Although the result of additional idle HS capacity leading to higher total revenue gain and commission fee is intuitive, quantifying the HS capacity range and the optimal commission fee offers medical personnel valuable information. We present a numerical example in Figure 3 to illustrate the impact of the parameters of HS, \(c_{2},\Lambda_{2},\mu_{2}\), on the revenue gain due to demand sharing. Figure 3 shows the following: (1) the revenue gain can be significant (e.g., more than 20%) depending on the parameters (i.e., \(\Lambda_{2}\) and \(\mu_{2}\)); (2) the revenue gain is monotonous with respect to \(c_{2},\Lambda_{2},\mu_{2}\), it is more sensitive to the demand level at the HS market. Unlike the case of \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\), wherein the HD can share its demand without affecting its optimal strategy, the case of \(\overline{\mu_{2}}<\mu_{2}\) is more complicated. We can still get the equilibrium commission fee for this case with \(V_{1}=V_{2}\). The non-equal service value case will be discussed later. **Lemma 3**. : _For \(V_{1}=V_{2}\) and \(\overline{\mu_{2}}<\mu_{2}\), in equilibrium, the negotiated commission fee \(s^{*}\) for each switching patient is_  
\[s=\frac{\alpha}{\alpha+\beta}[V_{2}+\frac{(\sqrt{V_{1}\mu_{1}}- \sqrt{c_{1}})^{2}-V_{2}\Lambda_{2}}{\lambda_{12}^{*}}+\frac{c_{2}\Lambda_{2}}{( \mu_{2}-\Lambda_{2})\lambda_{12}^{*}}\] \[-\frac{c_{1}}{\mu_{2}-\Lambda_{2}-\lambda_{12}^{*}}+\frac{c_{1} \Lambda_{1}-c_{2}\Lambda_{2}-c_{1}\lambda_{12}^{*}}{(\mu_{2}-\Lambda_{2}- \lambda_{12}^{*})\lambda_{12}^{*}}],\]  
_where \(\lambda_{12}^{*}\) is given in Lemma 1._  
Unfortunately, although the closed-form of the equilibrium commission fee is available for the \(\overline{\mu_{2}}<\mu_{2}\) case with the condition \(V_{1}=V_{2}\), we cannot get the properties of Proposition 3 analytically. Thus, we conduct a numerical study to investigate the impact of the two regions' parameters (i.e., \(c_{1},c_{2},\Lambda_{1},\Lambda_{2},\mu_{1},\mu_{2}\)) on the equilibrium commission fee and revenue gain. The total revenue gain is equal to \((V_{1}-c_{1}/\left(\mu_{1}-\Lambda_{1}+\lambda_{12}^{*}\right))(\Lambda_{1}- \lambda_{12}^{*})-(\sqrt{V_{1}\mu_{1}}-\sqrt{c_{1}})^{2}+s^{*}\lambda_{12}^{*}\), which is actually the revenue from commission \(s^{*}\lambda_{12}^{*}\) minus the revenue loss in Region-1 \(\tau_{1}^{0}-(V_{1}-c_{1}/(\mu_{1}-\Lambda_{1}+\lambda_{12}^{*}))(\Lambda_{1}- \lambda_{12}^{*})\). The revenue loss is attributed to the fact that the HD shares more demand than its own optimal pricing strategy. The impacts of all parameters, except for \(c_{1}\), are similar to the case \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\). The total revenue gain, i.e., \((V_{1}-c_{1}/(\mu_{1}-\Lambda_{1}+\lambda_{12}^{*}))(\Lambda_{1}-\lambda_{12 }^{*})-(\sqrt{V_{1}\mu_{1}}-\sqrt{c_{1}})^{2}+s^{*}\lambda_{12}^{*}\), is not necessarily monotonous with respect to the unit waiting cost \(c_{1}\). Figure 4 shows that the revenue from commission fee, that is \(s^{*}\lambda_{12}^{*}\), is decreasing in the delay sensitivity of patients in Region-1 \(c_{1}\). This is mainly because, with more delay sensitive patients, there is an increase in the contribution of the idle capacity of HS toward this alliance, which reduces HD's revenue allocating from the medical alliance. However, in Region-1, the revenue loss, i.e., \(\pi_{1}^{0}-(V_{1}-c_{1}/(\mu_{1}-\Lambda_{1}+\lambda_{12}^{*}))(\Lambda_{1}- \lambda_{12}^{*})\), is also decreasing and concave in \(c_{1}\).