## 3 Evolutionary Game Theoretical Model
### Model Calculation and Stability Analysis
(2021) [50] and [PERSON] et al. (2022) [51]. According to this method, we can obtain the calculation formulas of \(detJ\) and \(trJ\), as shown in Formulas (11) and (12). If the conditions \(detJ>0\) and \(trJ<0\) are met, then the potential equilibrium points can be considered as the stable strategy of the evolutionary game, i.e., the Evolutionary Stable Strategy (ESS). \[detJ=\frac{\partial F(x)}{\partial x}\times\frac{\partial F(y)}{\partial y}- \frac{\partial F(y)}{\partial x}\times\frac{\partial F(x)}{\partial y} \tag{11}\]  
\[trJ=\frac{\partial F(x)}{\partial x}+\frac{\partial F(y)}{\partial y} \tag{12}\]  
After substituting the five potential equilibrium points into the Jacobian matrix, Table 3 was obtained. We used the cost-benefit analysis method to discuss the stability of the game strategies of the two sides in different situations. From the perspective of the users, \(aM+b(E+I+R)\) or \(\Delta\) represents the medical service support, emotional support, information support, and rewards given by the OHCs after users disclose their personal information, i.e., the total benefits of users. \(cL_{u}\) and \(dL_{u}\) are the expected loss of user privacy leakage under the \"positive protection\" strategy and \"negative protection\" strategy implemented by the OHCs, respectively, i.e., the total costs of users under the \"positive protection\" strategy and \"negative protection\" strategy,respectively. Therefore, there are three situations in the relationship between the total benefits and the total costs of users: \(\Delta<cL_{u}\), \(cL_{u}<\Delta<dL_{u}\), and \(\Delta>dL_{u}\). From the perspective of OHCs, \(B\) is the increase in the influence and recognition of the communities and other benefits received by the OHCs after users disclose their personal information, i.e., the total benefits of the OHCs. \(C_{2}+(c-d)L_{c}\) is the sum of all types of extra costs paid by the OHCs after implementing the \"positive protection\" strategy and the reduced expected loss of user privacy leakage, as compared with the \"negative protection\" strategy, i.e., the total costs of the OHCs. It is worth noting that \(C_{2}+(c-d)L_{c}\) may be greater than 0 or less than 0, which needs to be discussed in terms of classification. Therefore, the relationship between the total benefits and the total costs of OHCs also exist in three situations: \(B<C_{2}+(c-d)L_{c}\) and \(C_{2}+(c-d)L_{c}>0\), \(B>C_{2}+(c-d)L_{c}\) and \(C_{2}+(c-d)L_{c}>0\), and \(B>C_{2}+(c-d)L_{c}\) and \(C_{2}+(c-d)L_{c}>0\). On the basis of the above discussion, after analyzing the costs and benefits of both players, the following nine scenarios can be obtained, as shown in Table 4.  
**Proposition 1**. : _When the conditions are those in Scenarios 1 to 3, \(D_{1}\) (0, 0) is an ESS, and users and OHCs will choose (non-disclosure, negative protection)._  
In these scenarios, since the sum of the benefits obtained by users for disclosing their private personal information is less than the sum of the costs under the \"positive protection\" strategy, i.e., \(\Delta<cL_{u}\), users tend not to disclose private personal information \((x\to 0)\). On the other hand, for the OHCs, there is no benefit to their \"positive protection\" strategy as users gradually choose not to disclose their private information. For cost reasons, they will tend to choose to implement the \"negative protection\" strategy \((y\to 0)\). Therefore, the evolutionary game process will eventually converge to \(D_{1}\) (0, 0), which can be seen in Figure 3(a). **Proposition 2**.