An evolutionary game-based simulation study of a multi-agent governance system for smart senior care services in China

By analyzing the competing interests of the government, smart senior care technology service providers, and older adults, this study assesses the conundrum of the tripartite game system of smart senior care services. To promote and ensure the development and high-quality of smart senior care services, the government—as the head of the system—must financially subsidize both older adults and smart senior care technology service providers, as well as regulate the behavior of providers. These actions count as governmental strategies, and they not only emphasize the government’s leadership role but also provide the government with intangible advantages, such as appreciation from the public.

The advent of smart senior care technology service providers, which focus on addressing the various needs of older adults and serve as representatives of market forces, has significantly decreased the burden of the aging population on the Chinese government. The main interest of these providers is increasing own economic revenue and brand influence.

Older adults are the main users of the aforementioned smart senior care services, using them to satisfy their various needs, raise quality of life, and enjoy own spirituality, and they worry about topics such as costs and service quality. As illustrated in Figs. 1 and 2, this study created a tree decision and logic diagram, respectively, of the tripartite evolutionary game agents to allow for a more intuitive description of agent interactions. Below we present the study hypotheses.

Hypothesis 1: the combination of the tripartite game agents can be represented as \(Q\), \(Q = \left\{ {g, \, c, \, e \, } \right\}\). In this model, the set of \(g\) symbolizes the government, \(c\) symbolizes smart senior care technology service providers, and \(e\) symbolizes older adults. All three are finite rational decision-makers because there is knowledge asymmetry between them.

Hypothesis 2: \(Pg \,\) = {strict regulation, relaxed regulation} describes the combination of strategies of the government. Specifically, “strict regulation” refers to the government’s strict adherence to the issued directives and guidelines aimed at identifying and exposing service providers who break the law, fall short of industry standards, or jeopardize the rights and interests of older adults. “Relaxed regulation” refers to the government’s passive attitude toward smart senior care technology service providers’ betrayal of older adults’ confidence and legal violations.

Moreover,\(Pc\) = {trustworthy, breach of trust} describes the combination of strategies of smart senior care technology service providers. “Trustworthy” refers to the providers’ actions toward guaranteeing service quality and preserving older adults’ legal rights and interests. A “breach of trust” occurs when a provider offers subpar services to older adults to increase own profits.

Furthermore, \(Pe\) = {use, no use} describes the combination of strategies of older adults. In particular, “use” refers to older adults and their families making a conscious decision to use smart senior care services to enhance their quality of life. “No use” refers to older adults and their families adopting a reserved attitude toward services, not accepting them, or accepting them in an non-engaging manner.

Hypothesis 3: the probability of the government choosing “strict regulation” is \(x\)(0 ≤ \(x\) ≤ 1), and that of choosing “relaxed regulation” is 1-\(x\). The probability of a provider choosing “trustworthy” is \(y\)(0 ≤ \(y\) ≤ 1), and that of choosing “breach of trust” is 1-\(y\). Moreover, the probability of older adults choosing “use” is \(z\)(0 ≤ \(z\) ≤ 1), and that of choosing “no use” is 1-\(z\).

The relevant model parameters and meaning of parameters for the tripartite game agents are listed in Table 1.

The payment benefit matrix for the tripartite game agents—the government, the provider of smart senior care technology services, and older adults—is displayed in Table 2 based on the foregoing hypothesis.

The expected benefit \(G_1\), \(G_2\) for “strict regulation” and “relaxed regulation” by the government, and the average expected benefit \(\overline {G_1}\) by the government are:

It can therefore be concluded from Eqs. (1), (2) and (3) that the government's replication dynamics equation Eq. (4) is:

The expected benefit \(C_1\), \(C_2\) for the “trustworthy” and “breach of trust” strategies by the smart senior care technology service providers, and the average expected benefit \(\overline {C_1}\) are:

The replication dynamic equation for the smart senior care technology service provider from Eqs. (5), (6) and (7) can therefore be derived Eq. (8) as:

The expected benefit \(E_1\), \(E_2\) for the “use” and “no use” by the elderly, and the average expected benefit \(\overline {E_1}\) by older adults are:

The replication dynamics equation for older adults from Eqs. (9), (10) and (11) can therefore be derived Eq. (12) as:

When Eqs. (4), (8) and (12) satisfy the condition of \(F\left( x \right) = 0\), \([F\left( y \right) = 0\) and \(F\left( z \right) = 0\), eight local equilibrium points for the pure strategy (\(x\), \(y\), \(z\)) and local equilibrium points for the mixed strategy (\(x^{\ast}\), \(y^{\ast}\),\(z^{\ast}\)) are obtained. Although there are many possibilities for a mixed-strategy equilibrium point in asymmetric evolutionary games, each agent often chooses a deterministic approach based on the information that one already possesses. This implies that the equilibrium point in an asymmetric evolutionary game must be a rigorous Nash equilibrium, which is purely a strategy equilibrium [47, 48]. Therefore, the system produces eight local equilibrium points, as follows: (0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 1, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 0), and (1, 1, 1).

The Lyapunov First Method [49] is used to determine equilibrium point stability. In this method, if all the eigenvalues of the Jacobi matrices are negative, the local equilibrium point is an evolutionary stable equilibrium point. If there are eigenvalues of zero and all other eigenvalues are negative, the stability of the local equilibrium point cannot be determined by the eigenvalues. If there are positive eigenvalues, the local equilibrium point is unstable. The determination of each equilibrium point is presented in Table 3.

Thus, it can be determined that 1 (0, 0, 0), 2 (0, 0, 1), 3 (0, 1, 1), 4 (0, 1, 0), 6 (1, 0, 1), and 7 (1, 1, 0) are not evolutionarily stable equilibrium points. The stability strategy for the two remaining points, 5 (1, 0, 0) and 8 (1, 1, 1), was analyzed as described hereinafter.

Scenario 1: the evolutionary stable equilibrium point 5 (1, 0, 0) is reached when \({M_{g1}} - {S_{g1}} - {L_{c1}} - {M_{g2}} - {S_{g2}}\) < 0, \({N_c} + {L_{c1}} - {M_{c1}} - c + {M_{c2}}\) < 0, and \({N_e} + {L_{c2}} + {B_e} - {M_e} + {H_{e2}}\) < 0. The “strict regulation” strategy was developed because when the government regulates with strictness, the sum of the increase in appreciation by older adults for the government, in government credibility and performance, and in fines for breach of trust of providers far outweighs the cost of strict regulation. Moreover, although the government provides financial subsidies for providers to carry out the smart senior care technology services, the “breach of trust” strategy becomes effective when the sum of the total profit and additional benefits gained by the breach of trust is much greater than the cost of operations and the fines submitted by providers to the government. The “no use” strategy becomes effective for older adults when the costs of using the services are higher than the incentives, subsidies, and advantages afforded by the government’s strict regulation.

Scenario 2: when \({N_c} + {M_{g1}} + {N_e} - {S_{g1}} - {M_{g2}}\) < 0,\({M_{c1}} + c - {S_c} - {L_{c2}} - {N_c} - {L_{c1}} - {M_{c2}}\) < 0, and \({M_e} - {H_{e1}} - {N_e} - {B_e}\) < 0, the evolutionary stable equilibrium point 8 (1, 1, 1) is reached. As aforementioned, the government receives appreciation from older adults and an improvement in its credibility and performance when it regulates with strictness; thus, the “strict regulation” approach is adopted when the government’s gains outweigh the benefits that it delivers to providers and the costs associated with regulation. When the recognition and brand benefits that a provider receives for being trustworthy surpass the compensation and fines that must be paid to older adults and the government for a breach of trust, the “trustworthy” strategy is used. For older adults, both the government and technology service providers affect their strategic decisions: the government offers financial subsidies to older adults who choose to use smart senior care services, and when the government strictly regulates the service quality of providers, it rewards older adults who actively use smart senior care services. Meanwhile, the providers offer services that bring convenience, physical health, and spiritual satisfaction to older adults. Thus, when all these benefits of the services to older adults outweigh the purchase cost, the “use” strategy emerges.

This study's parameter setting is partially based on actual senior care organization situations from Anhui Province, China, as well as local governmental regulations. We choose Anhui Jing'an Nursing Court Demonstration Base as an example in this instance [50, 51]. It has 301 beds and charges between CNY 2,679 to CNY 8,779 a month, with an average of CNY 5,000. We estimate that the range of bed occupancy is between 250 and 300 based on these facts. Furthermore, it costs 1.35 million annually to maintain trust for a smart senior care technology service provider to deliver high-quality service, and it costs CNY 750,000 annually to breach trust. The insurance liability range for the compensation limit in the event that the smart senior care technology service provider breaches trust is CNY 10000 to 200,000 [52]. Meanwhile, smart senior care demonstration bases that fulfill provincial requirements will be granted a minimum one-time award of CNY 200,000 [53]. Another part of the parameter setting is derived from relevant references. For instance, China's provincial-level civil affairs agencies offer older adults with good assessment abilities an annual operating subsidy of CNY 2,400 to 4,800 [54], and those who do not meet the standard will be fined between CNY 30,000 and 100,000 [55]. We also took into account the social benefits and costs of whole-process and nodal regulate by a Chinese municipal government, in addition to the additional benefits that resulted from the speculative actions of developers [56]. Ultimately, we determined the suitably defined parameters for this study by considering the advice of pertinent subject-matter experts as well as the previously provided data. The specific parameters are as follows: \({M_{c1}}\) = 135, \({M_{c2}}\) = 75, \({N_c}\) = 70, \({L_{c1}}\) = 10, \(c\) = 100, \({L_{c2}}\) = 50, \({S_c}\) = 80, \({M_e}\) = 165,\({H_{e1}}\) = 120, \({H_{e2}}\) = 60, \({B_e}\) = 30, \({N_e}\) = 70, \({M_{g1}}\) = 150, \({M_{g2}}\) = 100, \({S_{g1}}\) = 200, \({S_{g2}}\) = 75.

As shown in Fig. 3a, to verify the evolutionary trend of \(x\) over time, the initial willingness value of the government (\(x\) = 0.5) is fixed at a constant and the values of \(y\) and \(z\) increase from 0.1 to 0.9. The government’s rate of convergence is impacted by the changes in the initial values for older adults and providers, but it eventually converges on 1 and adopts the “strict regulation” strategy. That is, the government’s rate of convergence will be slower and the evolutionary cycle will be longer when the initial values of older adults and smart senior care technology service providers are higher.

As shown in Fig. 3b, to verify the evolutionary trend of \(y\) over time, the initial willingness value of the providers (\(y\) = 0.5) is fixed at a constant and the values of \(x\) and \(z\) increase from 0.1 to 0.9. When the government and older adults’ initial willingness values are between 0.1 and 0.5, the providers’ initial willingness value declines, then rises, and eventually stabilizes at 1; furthermore, as the initial values of \(x\) and \(z\) increase, the rate of development of \(y\) accelerates and the evolutionary period shortens. Additionally, when the initial willingness values of the government and older adults were greater than 0.5, \(y\) eventually stabilizes at 1. Therefore, although the main evolutionary line tendency of providers clearly changes, it has little impact on the outcomes.

Figure 3c depicts the evolution trend of \(z\) over time when the initial willingness value of older adults (\(z\) = 0.5) is fixed and those of \(x\) and \(y\) increase from 0.1 to 0.9. Over time, the general trend of older adults’ strategy decision increases steadily with little variation in the rate of convergence and evolutionary period.

First, from the perspective of the government, the overall evolutionary trends shown in Fig. 4a and b are similar; there were no discernible changes in the system’s evolutionary rate as the values rise and eventually form a cycle. This indicates that the government’s regulatory decisions are not highly impacted by the outcomes of strict regulation, namely changes in government monitoring costs and the advantages of this strategy (e.g., credibility and performance). This contradicts the results of other studies that contend that the probability of governmental regulation is inversely correlated with its expenses [57, 58] and positively correlated with reputational profits [59]. The inference is that the government, as the authority, continues to play a dominant role in the governance system through regulation regardless of interference from outside parameters, and that its role and functional attributes make it less susceptible to such interference. Therefore, the government can ultimately decide to regulate strictly regardless of changes in the strategies of the other agents, as depicted in Fig. 4a.

It is also important that the government provides only modest financial assistance because more assistance will simply increase financial burden and negatively impact social welfare. In this instance, the system is circular, as is demonstrated in Fig. 5a with \({N_c}\) as 90 and in Fig. 6c with \({N_e}\) as 90, making the agent’s strategy impossible to be determined. This conclusion is in line with other research' findings that excessive subsidies will result in a significant cost burden [60, 61]. As a result, the government ought to implement "dynamic subsidies" and "dynamic incentives" policies, as well as arbitrarily determine the proportion of incentives to subsidies.

In order to ensure the quality and safety of the services, the government should simultaneously continuously establish a sound regulatory system to standardize the management framework for senior care service providers and introduce social forces—such as the public, third-party regulators, etc.—to conduct assessment and testing. Additionally, the government should set up a support system that includes financial, technical, and training support to help smart senior care technology service providers improve their capabilities and provide the technical guidance services that older adults require. Finally, pertinent policies and regulations should be developed in order to set clear standards and direction for the development of smart senior care.

Second, from the perspective of smart senior care technology service providers, providing them with extra benefits make them vulnerable to engaging in a “breach of trust” strategy [62]. This study implied that providers are a market force that chases profit maximization between by navigating public welfare and profit, making providers vulnerable to a “breach of trust” strategy. Therefore, in addition to offering high-quality services, smart senior care technology service providers should also intensify their marketing initiatives to encourage older adults to make purchases in order to strike a balance between costs and benefits and lessen providers' reliance on additional benefits. Meanwhile, an increase in the fines for a breach of trust, in the compensation for older adults [63], and in the offer of incentives [64] would relieve the financial pressure on providers and lead them to use a “trustworthy” strategy. It is important to note that the results of our study, which imposed fines for breach of trust, are different from those of our predecessors. Tsebelisg, Liu, and Zhang contend that punishing companies does not sufficiently constrain their violating behaviours to produce satisfactory outcomes [43, 65, 66]. Therefore, the government should make efforts and find methods to strengthen market regulations, market order regulations, service quality, improve the well-being of older adults using the services, and encourage providers to fulfill their social responsibilities.

Additionally, in order to provide more comprehensive and effective senior services, smart senior care technology service providers should keep innovating technologically and creating products and services suited to the needs of older adults. They should also cooperate and share with each other and increase publicity to encourage older adults to actively participate and provide feedback, as well as to continuously improve and optimize their offerings.

Finally, from the standpoint of older adults, it is possible to significantly increase their demand for and willingness to utilize smart senior care services by reducing service costs [67] and offering them financial subsidies and incentives [64]. Contrary to Tian's results, he argues that subsidies to producers are more beneficial than subsidies to users [68]. This may be because smart senior care services are relatively expensive for older adults or even beyond their financial capabilities, as these services involve large investments in technological development, equipment acquisition, maintenance, and expert personnel skills support. To reduce the burden on older adults, the government should provide them with more incentives and subsidies. This will increase their digital literacy and enable them to integrate better into modern society.

Older adults should actively engage in the usage and learning of smart senior care services, and comprehend and master pertinent technologies and knowledge. Simultaneously, they ought to offer input and recommendations to the government and service providers in order to help them enhance and optimize smart senior care services and encourage collaborative development. Additionally, they need defend their own legal rights and interests, strengthen the capacity of smart senior care services to identify, and stop fraud and information leaks.

The numerical settings for this study were sourced from Anhui Province, China, and the entire game system involves only the government, providers, and older adults. Thus, owing to the constrained data sources and agent parameter assumptions, the results of the simulation are not fully applicable at the national level. Future researchers should attempt to use other available methodologies and more agents to evaluate model results while considering the aforementioned limitations when utilizing simulations for assessment and prediction.