4.1. Stability Analysis of Unilateral Evolutionary Strategies
- (1)
Stability analysis of elderly evolutionary strategies
Assuming that the elderly chooses “participation” and “non-participation” to supervise the smart aging platform, the expected benefits are respectively
and
, the average benefit is
, then:
The replication dynamic equation for the elderly evolutionary strategy is:
When
,
, when all values of
taken are steady state and both strategies of the elderly are the ESS. When
, let
to get the equilibrium points
and
, then we have
,
,
is the stable point, and participation is the ESS. When
, let
to get the equilibrium points
and
, then there are
,
, then
is the stable point, and non-participation is the ESS. The elderly dynamic trend phase diagram is shown in
Figure 2.
As can be seen from
Figure 2, if the loss of privacy is greater than the gain in service,
becomes larger and the shaded cross-section moves in the positive direction of the
y-axis, the elderly tend not to participate in the strategy. If the loss of privacy is smaller than the gain of service, the
space becomes larger and the shaded cross-section shifts to the negative direction of
y-axis, the elderly tend to participate in the strategy. In addition, as
decreases, the
space becomes larger, and the elderly still tend to not participate.
- (2)
Stability analysis of the evolutionary strategy of the platform service providers
Let the expected benefits of high-quality protection and low-quality protection of public privacy in the platform service providers be
and
, respectively, and the average benefit
, then:
The replication dynamic equation for the evolutionary strategy of the platform service providers is:
When
,
, when all values of
taken are steady state and both strategies of the platform service providers are ESS. When
, let
to get the equilibrium points
and
, then we have
,
, then
is the stable point, and high-quality protection is the ESS. When
, let
to get the equilibrium points
and
, then there are
,
, then
is the stable point, and low-quality protection is the ESS. The dynamic trend phase diagram of the platform service providers is shown in
Figure 3.
As can be seen from
Figure 3, if other parameters remain unchanged and the input cost of the platform service provider decreases, the shaded cross-section shifts to the lower left, the
space increases, and the platform service provider tends to the high-quality protection strategy. Similarly, the higher the level of government incentives and penalties, the more the platform service providers choose the high-quality protection strategy. When the loss of privacy leakage and future revenue become larger, the platform service providers tend to choose the high-quality protection strategy. When the
difference becomes smaller, the
space becomes larger, and the platform service provider chooses the low-quality protection strategy.
- (3)
Analysis of the stability of the government’s evolutionary strategy
Assuming that the privacy protection level of the government’s high input supervision and low input supervision platforms is
and
, respectively, and the average income is
, the same can be obtained:
The replication dynamic equation for the evolutionary strategy of the government is:
When
,
, when all values of
taken are steady state and both strategies of the government are the ESS. When
, let
to get the equilibrium points
and
, then we have
,
; then
is the stable point, and high input supervision is the ESS. When
, let
to get the equilibrium points
and
, then
,
, and
is the stable point, and low input supervision is the ESS. The government dynamic trend phase diagram is shown in
Figure 4.
As can be seen from
Figure 4, the space of
and
in the initial strategy space is related to the cost of government regulation, and s6 becomes larger when the cost of regulation increases, which tends towards low input regulation. When the government chooses the strategy of high input regulation, it is found that there is a higher the probability that the platform service provider chooses low-quality protection; the higher the fine, the more government tends to choose high input regulation. The government tends to prefer high input regulation.
4.2. Stability Analysis of the Evolutionary Strategy of the Tripartite System
A stability study of the evolutionary stabilization strategy of the tripartite system will be performed for further evaluation of the ideal evolutionary stabilization strategy of the tripartite system and the main elements impacting the behavioral patterns of the tripartite subjects.
- (1)
Determination of the point of progressive stability
A tripartite evolutionary dynamical system consisting of replicated dynamic equations for every single party:
From Equations (13)–(15), the dynamical system is solved in conjunction to obtain the system equilibrium points , , , , , , , , and . It follows from the probabilities in the model assumptions that , so .
In the dynamic replication system of multi-subject evolutionary game, strategy X is asymptotically stable in the dynamic replication system of the multi-group evolutionary game when and only when X is a strict Nash equilibrium [
39], so only the asymptotic stability of the equilibrium points of the system involving pure strategies need to be analyzed, and the equilibrium points of the system satisfying the condition include eight points from
to
. According to the Lyapunov system stability discriminant, when all the eigenvalues of the Jacobi matrix are less than zero, the equilibrium point is asymptotically stable [
40]; when at least one eigenvalue in the Jacobi matrix is greater than zero, the equilibrium point is not asymptotically stable. Denote the Jacobi matrix as
:
The eigenvalues corresponding to each of the eight system equilibrium points may be found in
Table 3 by substituting each of the partial derivative function equations in the matrix of Equation (16).
Combining the parameter sizes in the model assumptions reveals that , , , and . Thus, , , , , and are not asymptotically stable points.
The usage of an aging app by elderly families can solve difficulties such as mobility issues, medical services for the elderly, and emergency support, all of which are important for the healthy development of aged services. The platform service providers’ high-quality protection of the privacy of elderly families and their families may enhance their willingness to participate in the supervision of the smart aging platforms. The government may adopt a free-riding mentality to save money. With the improvement of platform service providers’ awareness of privacy protection, they actively adopt the strategy of high-quality protection. To lower the expense of regulation, the government can pick low input regulation. We can select
as the optimal asymptotic stabilization point, i.e., (participation, high-quality protection, low input regulation) as the best ESS, because only the three eigenvalues are negative. The three-party game space is:
This is a more perfect equilibrium state for the smart aging platform’s stable development, and it is the optimal strategic goal we are pursuing. As a result, serves as a reference target for examining the asymptotic stability of and .
- (2)
Stability analysis of
When , , and , is the asymptotic stability point and the ESS is non-participation, low-quality protection, high input regulation. From , it is clear that if the service benefits received by the elderly are less than the loss caused by privacy leakage, the elderly will choose the non-participation strategy. Therefore, platform service providers should strengthen the level of privacy protection for the elderly, and at the same time expand the service income for the elderly, so as to encourage the elderly to participate in the supervision of the smart aging platform. According to , the sum of government subsidies and fines for platform service providers is less than the difference between the platform’s input cost when there is high-quality protection and low-quality protection, and the platform service providers tend to low-quality protection of user privacy in the case of high input regulation by the government. The government can raise the subsidy for high-quality protection while lowering the penalty for low-quality protection, or platform service providers can expand the input cost of low-quality protection and reduce the input cost of high-quality protection to promote high-quality protection of the privacy of elderly home users. The analysis of shows that if the sum of the cost of regulation and the subsidy given to the senior care app platform is less than the fine received from the platform, the government will choose the high input regulation strategy. However, to save money, the government frequently employs the strategy of low input regulation. If the amount of subsidy is increased, it will encourage the platform service providers to take the initiative to provide high-quality protection of the privacy of elderly families and their family users, thereby enhancing the government’s credibility. An increase in the number of subsidies can benefit the elderly, the platform service providers, and the government.
- (3)
Stability analysis of
When , and , is the asymptotic stability point and ESS is non-participation, low-quality protection, low input regulation. If is similar to the stability analysis of , the platform service providers should increase the level of privacy protection for public users while also expanding the service benefits obtained by the elderly, to encourage the elderly to participate in supervising the smart aging platform. If , i.e., the investment cost of low-quality protection is less than that of high-quality protection, the platform service providers will choose low-quality protection. To enable the platform service providers to choose the strategy of high-quality protection, the input cost should be expanded or reduced , so that the difference between the input cost of the service providers when choosing the two strategies is not significant as far as possible. When we look at , we can see that if the government actions high input projects, we obtain high input results. If the cost of regulation plus the subsidies granted to the platform service providers exceeds the penalty earned from the platform, the government will opt for low input regulation, which is exactly what the government expects.