## 4 Analysis
### Revenue Allocation in the Medical Alliance  
Although demand sharing can significantly increase the overall revenue of both hospitals, in the case of \(\mu_{2}>\mu_{2}\), the collaboration can occur only when each hospital can earn more revenue individually when compared to the independent operation without demand sharing. Therefore, how to allocate the revenue gain between two hospitals in such a way that both hospitals have the incentive to collaborate is an important issue. This section shows that an optimal allocation can be performed by determining an appropriate commission fee. We now use a two-player bargaining game to characterize the collaboration between HD and HS. Bargaining games have been extensively utilized to model the negotiation processes on prices between sellers and buyers, firm mergers, and acquisitions of small firms. This research area has been surveyed by [33]. Given that the negotiation over the commission fee for each switching patients involves HD and HS, we solve the bargaining game between the HD and the HS by utilizing the cooperative-game solution (also called the generalized Nash bargaining solution (NBS)) [34]. Our analysis framework can be applied to other types of bargaining solutions, such as the Kalai-Smorodinsky bargaining solution [35; 36]. The cooperative-game solution can be obtained by solving the following problem:\[ \begin{array}{ll}\max_{p_{1},p_{2},p_{12}^{0},s}&[\pi_{1}^{*}(p_{1},s)-\pi_{1}^{0 }]^{a}[\pi_{2}^{*}(p_{2},p_{12}^{0})-\pi_{2}^{0})]^{\beta}\\ s.t.&\pi_{1}^{*}(p_{1},s)-\pi_{1}^{0}>0;\\ &\pi_{2}^{*}(p_{2},p_{12}^{0})-\pi_{2}^{0}>0.\end{array}  \tag{10}\]  
The disagreement payoffs for the HD and the HS are their revenues earned when they operate independently. Let \(\pi_{1}^{*}(s)\) and \(\pi_{2}^{*}(p_{12}^{0},s)\) denote the revenues of HD and HS, respectively, when they cooperate. The parameters \(\alpha,\beta>0\), (\(\alpha+\beta=1\)) represent the bargaining power of the HD and the HS, respectively. The two constraints require that both hospitals can make more revenue with demand sharing than without it (or independent operations). The next lemma gives a method for determining the commission fee that facilitates demand sharing. **Lemma 2**. : _In the equilibrium, the two hospitals set their optimal prices as a centralized hospital. The optimal commission fee \(s^{*}\) can be derived by solving_  
\[ \begin{array}{ll}\max_{\chi}&[\pi_{1}^{*}(p_{1}^{*},s)-\pi_{1}^{0}]^{a}[\pi_ {2}^{*}(p_{2}^{*},p_{12}^{*}-s)-\pi_{2}^{0}]^{\beta}\\ s.t.&\pi_{1}^{*}(p_{1}^{*},s)-\pi_{1}^{0}>0;\\ &\pi_{2}^{*}(p_{2}^{*},p_{12}^{*}-s)-\pi_{2}^{0}>0,\end{array}  \tag{11}\]  
_where \(p_{1}^{*},p_{2}^{*},p_{12}^{*}\) are given in the proofs of the propositions and lemma in Section 4.1._  
Subsequently, the revenue \(\pi_{1}^{*},\pi_{2}^{*}\) and the optimal commission fee \(s^{*}\) can be derived according to Proposition 2 and Lemma 1. For a certain range of the HS's capacity, the optimal commission fee can be expressed explicitly as stated in the following proposition. **Proposition 3**. : _For \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\), in equilibrium, the negotiated commission fee \(s^{*}\) for each switching patient is independent of HD's service capacity \(\mu_{1}\) and potential demand \(\Lambda_{1}\), which is given as_  
\[s^{*}=\frac{\alpha V_{2}}{\alpha+\beta}[1-\sqrt{\frac{c_{1}}{V_{2}(\mu_{2}- \Lambda_{2})}+\frac{c_{2}\Lambda_{2}}{V_{2}(\mu_{2}-\Lambda_{2})^{2}}}]. \tag{12}\]  
_Additionally, the equilibrium commission fee \(s^{*}\) decreases in \(c_{1},c_{2},\Lambda_{2}\), but increases in \(V_{2},\mu_{2}\)._  
Under the condition of \(\underline{\mu_{2}}<\mu_{2}\leq\overline{\mu_{2}}\), the HD shares a part of patients with the HS, without experiencing any impact on its optimal pricing strategy for Region-1. Since the HS now accepts some extra patients from the HD, the expected waiting time becomes longer than the expected waiting time without demand sharing. This means that the HS must reduce its price for his original patients in Region-2.