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It is common knowledge that speculative behavior among elderly care service providers has become an important issue that cannot be ignored in “Internet + Community Elderly Care.” The omnichannel feedback mechanism of the elderly and the government's regulatory role can inhibit the speculative behavior of elderly care service providers. The omnichannel feedback mechanism includes the online evaluation conducted by the online evaluation system after the elderly purchase services through online channels and the offline evaluation conducted by the social network after the elderly purchase services through offline channels. Moreover, the return visits of the elderly care service information platforms and the complaint behaviors of the elderly are also part of the content of the Omni-feedback mechanism. The government can restrict the behavior of elderly care service providers and elderly care service information platforms through the policy adjustment of regulations, subsidies, and penalties. These policy adjustments can guide the healthy development of the “Internet + Community Elderly Care” industry. The relationships between the various subjects in “Internet + Community Elderly Care” are shown in Figure 1.

This study used the evolutionary game as a research method because the regulation of “Internet + Community Elderly Care” services often involves multiple stakeholders with limited rationality, and the strategic choice of each stakeholder is influenced by many factors such as income, cost, and mutual influence. Stakeholders interact with each other through multiple rounds of strategy selection to achieve a stable state of strategy selection, which is the same as the characteristics of evolutionary game theory (EGT) (38). At present, EGT has been successfully applied to various economic and social issues, such as environmental governance (39–42), supply chain collaboration (43, 44), green building development (45, 46), online car-hailing platform regulations (47–49) and elderly service provision (50, 51). Therefore, EGT is more suitable for us to study the regulation of “Internet + Community Elderly Care” services.

The following hypotheses are proposed to construct the four-party evolutionary game model.

Hypothesis 1. Government departments, elderly care service providers, elderly care service information platforms, and elderly people are selected as the subjects of the four-party game. The probability of elderly care service providers offering high-quality (H.Q.) or low-quality service (L.Q.) is (x, 1 − x). The probability of the elderly care service information platforms performing positive return visits (P.V.) or negative return visits (N.V.) is (y, 1 − y). The probability of government departments enforcing positive regulations (P.R.) or negative regulations (N.R.) is (z, 1 − z). The probability that elderly people choose online evaluation (ON) or offline evaluation (OF) toward the services they have purchased is (w, 1 − w). Here, x, y, z, w ∈ [0, 1].

Hypothesis 2. The elderly care service providers can earn (1 − α)Rs for services offered through the elderly care service information platforms channel, where Rs is the total revenue of both the elderly care service providers and information platforms, α is the percentage of revenue to the elderly care service information platforms. The cost of providing high-quality service to elderly people is Csh, and the cost of providing low-quality service is Csl(Csh > Csl). When the elderly care service providers provide low-quality services, they will be subjected to administrative penalties from the government and are required to compensate the elderly who purchased the service for Io.

Hypothesis 3. The elderly care service information platforms make return visits to customers who purchase and use elderly care services at the cost of Cei. When the elderly care service providers publish their service information through the elderly care service information platforms, elderly care service information platforms will collect αRs as commission income. The operating cost of the elderly care service information platforms is Ceo. When the elderly care service information platforms return positively, they will receive operating subsidies S from government departments. When the elderly care service information platforms return negatively and the elderly care service providers are confirmed to be providing low-quality services, they will be fined Fe by the government.

Hypothesis 4. The cost of positive regulations by the government is Cg. When the government imposes regulations negatively, the elderly may file a complaint, which may result in the government being fined by its superior government with penalties Fg. When offering high-quality services, elderly care service providers will produce social welfare Rg. When providing low-quality services, elderly care service providers may hurt the interests and even the health of the elderly. It would disrupt the development of the “Internet + Community Elderly Care” industry and cause social loss Dg.

Hypothesis 5. When providing online evaluations, the elderly will make some effort, such as learning the software for making appointments, evaluating, and spending some time, which costs Cw. On the other hand, offline evaluations incur Cm for the use of a smart terminal calling device such as a push-to-talk or phone. The elderly care service providers provide high-quality services that will bring physical and mental pleasure to the elderly, generating gainful utility Ro. On the contrary, the low-quality services provided by elderly care service providers will cause financial, physical, and mental damage Do to elderly people. When their rights are violated, the elderly may have a certain probability of complaining to the relevant government to safeguard their rights, with a rate of β, β ∈ [0, 1], and a cost of Co.

Hypothesis 6. The size of the offline social network for elderly people is γ ∈ [0, 1], which indicates the proportion of the potential customer affected by offline evaluations compared to the potential customer affected by online evaluations. In the omnichannel feedback mechanism, the elderly care service providers who provide high-quality services will gain the trust of elderly people and their families. They may gain more new clients and then have a reputational gain Is. The elderly care service providers who offer low-quality services will suffer reputational losses Ds due to damage to their reputation and loss of market share. Similarly, positive return visits by the elderly care service information platforms will promote reputational gains Ie, while negative ones will cause reputational losses De.

It follows from the game payment matrix that the expected revenue of the elderly care service providers who choose to provide high-quality services (low-quality services) is Ux(U_{1 − x}), and the average expected revenue of the elderly care service providers is Ūx: (1)Ux=(1-α)Rs-Csh+wIs+(1-w)Isγ (2)U1-x=(1-α)Rs-Csl-wDs-(1-w)Dsγ-zFs     -yIo-(1-y)zIo-(1-y)(1-z)(βFs+βIo) (3)Ūx=xUx+(1-x)U1-x By using the Malthusian equation, the replicator dynamic equation and the first-order derivative of the elderly care service providers can be written as follows: (4)F(x)=dx/dt=x(Ux-Ūx)=x(1-x)G(y,z,w) (5)F′(x)=(1-2x)G(y,z,w) (6)G(y,z,w)=w(1-γ)(Is+Ds)+zFs+Csl-Csh+Isγ+Dsγ              +yIo+(1-y)zIo+(1-y)(1-z)(βFs+βIo) The dynamic replicator equation shows that some factors influence the service provision of elderly care service providers, such as the strategic choice of other players and all other factors that are closely related to the cost-benefit of the elderly care service providers. Using the stability theorem of the differential equation, the probability of decision-making of the elderly care service providers in a stable state must satisfy F(x) = 0 and F′(x) < 0.

Proposition 1. When w > w₀, elderly care service providers provide a high-quality service. When w < w₀, elderly care service providers provide low-quality service. When w = w₀, its stabilization strategy cannot be determined. The threshold w₀ is defined as follows: wo={Csh-Csl-Isγ-Dsγ-zFs-yIo-(1-y)[(1-z)(βFs           +βIo)+zoI]}/(1-y)(Is+Ds).

Proof of Proposition 1. Because ∂G(y, z, w)/∂w > 0 G(y, z, w) is an increasing function of w. When w < w₀, we have that G(y, z, w) < 0, F(x)|_{x = 0} = 0, F′(x)|x=0<0, and then x = 0 has stability. When w > w₀, G(y, z, w) > 0, F(x)|_{x = 1} = 0, F′(x)|x=1<0, and then x = 1 has stability. When w = w₀, G(y, z, w) = 0, F(x) = 0 and F′(x) = 0, which cannot determine a stable strategy. The proof of Proposition 1 is complete.

Proposition 1 suggests that an increase in the probability of online evaluations by elderly people would lead to a shift in the stabilization strategy of elderly care service providers from providing low-quality services to supplying high-quality services. However, an increase in the proportion of offline evaluations by elderly people will make the elderly care service providers take risks and eventually choose to provide low-quality services. This implies that the online evaluation behavior of elderly people can be an effective deterrent to the supply of low-quality services by elderly service providers. The elderly should be motivated to take the initiative and actively learn how to operate the software for elderly services.

From Proposition 1, the evolutionary trajectory of elderly care service providers' strategies is obtained, as shown in Figure 2.

From Figure 2, the volume of part Vx0 is the probability that the elderly care service providers will choose to provide low-quality services. The volume of part Vx1 is the probability that it will supply high-quality services. Let a = Csh − Csl − Isγ − Dsγ − yIo − β(1 − y)(Fs + Io) and b = (1 − y)(βFs + βIo − Io)−Fs. Then w₀ = (a + bz)/(1 − γ)(Is + Ds), it is easy to obtain the following: (7)Vx0=∫01∫01(a+bz)/(1-γ)(Is+Ds)dzdx              =(2a+b)/2(1-γ)(Is+Ds) (8)Vx1=1-Vx0=1-(2a+b)/2(1-γ)(Is+Ds)

Corollary 1.1. The elderly care service providers will be more inclined to provide low-quality services when there is an increase in cost shrinkage arising from the provision of low-quality services. On the contrary, when reputational gains or losses increase, or when penalties for illegal acts of the elderly care service platforms increase, the elderly care service providers will be more inclined to provide high-quality services.

Proof of Corollary 1.1. The first order derivative from Vx1 for Csh − Csl, Is, Ds, Fs, respectively, gives the following: ∂Vx1/∂(Csh-Csl)=-1/(1-γ)(Is+Ds)<0, ∂Vx1/∂Is=[γ(1-γ)(Is+Ds)+(1-γ)(2a+b)]           /2(1-γ)2(Is+Ds)2>0, ∂Vx1/∂Ds=[γ(1-γ)(Is+Ds)+(1-γ)(2a+b)]           /2(1-γ)2(Is+Ds)2>0, ∂Vx1/∂Fs=(2β-2βy-βz+βyz+z)           /2(1-γ)(Is+Ds)>0. The proof of Corollary 1.1 is complete.

Corollary 1.2. Government should impose penalties Fs greater than a certain threshold F′s to ensure that elderly care service providers provide high-quality services. The following scenarios may occur. For example, the gains of low-quality services in illegal increase; the reputational gains and losses decrease; the likelihood of online evaluation decline; the size of offline social networks decreases; compensations decrease; and the number of complaints from elderly people decreases. Under these scenarios, the government needs to increase fines. Here, the threshold F′s is defined as follows: F′s={Csh-Csl-w(1-γ)(Is+Ds)-Isγ-Dsγ-yIo           -(1-y)[(1-z)βIo+zIo]}/[z+(1-y)(1-z)β].

Proof of Corollary 1.2. It follows from F′(x)|x=1<0 and Proposition 1 that Fs > F′s. By calculating the first-order partial derivative of Csh − Csl, Is, Ds, w, γ, Io, and β for F′s, respectively, it is easy to conclude that ∂F′s/∂(Csh − Csl) > 0, ∂F′s/∂Is < 0, ∂F′s/∂Ds < 0, ∂F′s/∂w < 0, ∂F′s/∂γ < 0, ∂F′s/∂Io < 0 and ∂F′s/∂β < 0. Therefore, F′s is positively correlated with Csh − Csl and negatively correlated with Is, Ds, w, γ, Io and β, respectively. The proof of Corollary 1.2 is complete.

The expected revenue for the positive return visit (negative return visit) strategy chosen by the elderly care service information platforms is Uy(U_{1 − y}), and its average expected revenue is Ūy: (9)Uy=αRs-Ceo-Cei+wIe+(1-w)Ieγ+S (10)U1-y=αRs-Ceo-wDe-(1-w)Deγ           -(1-x)[zFe+(1-z)βFe] (11)Ūy=yUy+(1-y)U1-y The replicator dynamic equation and first-order derivative of the elderly care service information platforms can be calculated as follows: (12)F(y)=dy/dt=y(Uy-Ūy)=y(1-y)H(x,z,w) (13)F′(y)=(1-2y)H(x,z,w) (14)H(x,z,w)=w(1-γ)(Ie+De)+Ieγ+Deγ            +S-Cei+(1-x)[zFe+(1-z)βFe] From Equations 12–14, it can be seen that the strategy of the elderly care service information platforms mainly depends on the choices of the three remaining parties, the strength of the government penalties or subsidies, and the size of reputational gains and losses. By using the stability theorem of a differential equation, the decision probability of the elderly care service information platforms in a stable state must satisfy F(y) = 0 and F′(y) < 0.

Proposition 2. When w > w₁, the elderly care service information platforms choose positive return visits. When w < w₁, the elderly care service information platforms will choose negative return visits. When w = w₁, its stabilization strategy cannot be determined. The threshold w1 is defined as follows: w1= { Cei-S-Ieγ-Deγ          -(1-x)[zFe+(1-z)βFe] } /(1-γ)(Ie+De).

Proof of Proposition 2. Since ∂H(x, z, w)/∂w > 0, H(x, z, w) is an increasing function with regard to w. When w < w₁, we have that H(x, z, w) < 0, F(y)|_{y = 0} = 0, F′(y)|y = 0<0, and then y = 0 has stability. When w > w₁, H(x, z, w) > 0, F(y)|_{y = 1} = 0, F′(y)|y = 1<0, and then y = 1 has stability. When w = w₁, we can obtain that H(x, z, w) = 0, and then F(y) = 0 and F′(y) = 0. Now, no stable strategy can be identified. The proof of Proposition 2 is complete.

It follows from Proposition 2 that an increase in the probability of online evaluation by elderly people will make the elderly care service information platforms more inclined to choose positive return visits. In contrast, an increase in the proportion of elderly people who evaluate offline will reduce both the possible reputational gains and losses faced by elderly care service information platforms. With the lower self-regulatory business gains and higher speculative business gains, the strategy of elderly care service information platforms will shift from positive return visits to negative return visits.

From Proposition 2, the evolutionary trajectory of elderly care service information platforms' strategy is obtained, as shown in Figure 3.

Figure 3 shows the volumes of parts Vy0 and Vy1 are the probabilities of the elderly care service information platforms choosing negative return visits and positive return visits, respectively. Let c = Cei − S − Ieγ − Deγ and d = zFe + (1 − z)βFe. Then, w₁ = (c − d + dx)/(1 − γ)(Ie + De), and we have the following equation: (15)Vy0=∫01∫01(c-d+dx)/(1-γ)(Ie+De)dxdy              =(2c-d)/2(1-γ)(Ie+De) (16)Vy1=1-Vy0=1-(2c-d)/2(1-γ)(Ie+De)

Corollary 2.1. When the cost of positive return visits by elderly care service information platforms increases, those platforms will be more inclined to a negative return visit. When reputational gains or losses increase, the complaint rate of elderly people increases, the government imposes penalties, and the elderly care service information platforms will be more inclined to make positive return visits.

Proof of Corollary 2.1. The first order derivative from Vy1 for Cei, β, Ie, De and Fe, respectively, gives ∂Vy1/∂Cei < 0, ∂Vy1/∂β > 0, ∂Vy1/∂Ie > 0, ∂Vy1/∂De > 0, ∂Vy1/∂Fe > 0. The proof of Corollary 2.1 is complete.

Corollary 2.2. The government subsidies S to the elderly care service information platforms should be greater than a certain threshold S′ to effectively use these platforms. When some issues emerge, the government should increase policy subsidies for elderly care service information platforms. These scenarios include an increase in the cost of return visits, a decrease in the probability of online evaluations, a decrease in reputational gains and losses, a decrease in the size of offline social networks, an increase in the probability of providing high-quality services, a decrease in the intensity of government supervision and punishment, and a decrease in the complaint rate among the elderly. Here, the threshold S′ is defined as follows: S′=Cei-Ieγ-Deγ-w(1-γ)(Ie+De)           -(1-x)[zFe+(1-z)βFe].

Proof of Corollary 2.2. It follows from F′(y)|y=1<0 and Proposition 2 that, S > S′. It can be easily observed from the first order partial derivative of Cei, w, Ie, De, γ, x, z, Fe, and β for S′ that ∂S′/∂Cei > 0, ∂S′/∂w < 0, ∂S′/∂Ie < 0, ∂S′/∂De < 0, ∂S′/∂γ < 0, ∂S′/∂x > 0, ∂S′/∂z < 0, ∂S′/∂Fe < 0, and ∂S′/∂β < 0, respectively. Thus, S′ is positively correlated with Cei and x, and negatively correlated with w, Ie, De, γ, z, Fe, and β. The proof of Corollary 2.2 is complete.

The expected revenue for the positive regulation (negative regulation) strategy implemented by the government is Uz(U_{1 − z}), and its average expected revenue is Ūz: (17)Uz=xRg-yS-Cg+(1-x)[Fs-Dg+(1-y)Fe] (18)U1-z=xRg-yS+(1-x)[(1-y)β(Fs+Fe-Fg)-Dg] (19)Ūz=zUz+(1-z)U1-z The replicator dynamic equation and first-order derivative of government can be formulated as follows: (20)F(z)=dz/dt=z(Uz-Ūz)=z(1-z)P(x,y) (21)F′(z)=(1-2z)P(x,y) (22)P(x,y)=(1-x)[Fs+(1-y)Fe-(1-y)β(Fs+Fe-Fg)]                       -Cg It can be seen from Equations 20–22 that the government's strategic choice is influenced by that of the elderly care service providers and the elderly care service information platforms, as well as revenue and expenditure from various government departments. By using the stability theorem of the differential equation, the decision probability of government in a stable state must satisfy F(z) = 0 and F′(z) < 0.

Proposition 3. When x > x₀, the government enforces negative regulations. When x < x₀, the government enforces positive regulations. When x = x₀, the stabilization strategy of the government cannot be determined. Here, the threshold is defined as follows:

x0=[Fs+(1-y)(Fe-βFs-βFe+βFg)-Cg]                /[Fs+(1-y)(Fe-βFs-βFe+βFg)].

Proof of Proposition 3. Because ∂P(x, y)/∂x < 0, P(x, y) is a decreasing function about x. When x > x₀, we have that P(x, y) < 0, F(z)|_{z = 0} = 0, F′(z)|z = 0<0, and then z = 0 has stability. When x < x₀, it follows that P(x, y) > 0, F(z)|_{z = 1} = 0, F′(z)|z = 1<0, and then z = 1 has stability. When x = x₀, P(x, y) = 0, F(z) = 0 and F′(z) = 0, which cannot determine a stable strategy. The proof of Proposition 3 is complete.

Proposition 3 suggests that if the elderly care service providers provide high-quality services, the cost of positive regulations by government departments is not equal to the benefit. Thus, the government will switch from positive to negative regulations. When elderly care service providers offer low-quality services, the government faces significant social losses and the risk of accountability from higher levels and would prefer a positive regulatory strategy.

From Proposition 3, the evolutionary trajectory of the government's strategy is obtained, as shown in Figure 4.

From Figure 4, we can see that the volumes of parts Vz0 and Vz1 are the probabilities of the government implementing negative and positive regulations. If we assume m = Fe − βFs − βFe + βFg, then x₀ = 1 − Cg/[Fs + (1 − y)m], and we have the following equation: (23)Vz1=∫01∫011-Cg/[Fs+(1-y)m]dydz              =1-Cgln  | 1+m/Fs | /m (24)Vz0=1-Vz1=Cgln  | 1+m/Fs | /m

Corollary 3.1. The cost of positive regulations by the government negatively affects the probability of positive regulations when Fg > Fs. Here, Fg > Fs represents that the administrative penalties imposed by the superior government are greater than the revenue from penalties imposed by the government on the elderly care service providers.

Proof of Corollary 3.1. When Fg > Fs, we can see that m > 0, such that Vz1 = 1 − Cg ln (1 + m/Fs)/m. Using the first-order derivative, we can obtain that ∂Vz1/∂Cg = −ln (1 + m/Fs)/m < 0. When Fg < Fs, the probability of positive regulations by the government is influenced by a combination of factors, such as the rate of elderly people's complaints, the level of penalties imposed on the elderly service information platforms, and the level of policy subsidies for the elderly service information platforms. In this case, it should be discussed according to the specific situation. The proof of Corollary 3.1 is complete.

Corollary 3.2. If the administrative penalties Fg imposed by the superior government on the junior government are greater than a certain threshold F′g, it will choose a positive regulatory strategy. Under some scenarios, the superior government should increase the administrative penalties imposed on the junior government to urge him to regulate positively. These scenarios include an increase in the cost of positive regulations, a decrease in the revenue from penalties, a decrease in the complaint rate among elderly people, an increase in the probability of offering high-quality services, and an increase in positive return visit strategies. Here, the threshold F′g is defined as follows:

F′g={Cg-(1-x)[Fs+(1-y)(Fe-βFs-βFe)]}                /(1-x)(1-y)β.

Proof of Corollary 3.2. By using Proposition 3 F′(z)|z = 1<0, we get Fg > F′g. It follows from the first order partial derivative of Cg, Fe, Fs, β, x, and y for F′g that ∂F′g/∂Cg > 0, ∂F′g/∂Fe < 0, ∂F′g/∂Fs < 0, ∂F′g/∂β < 0, ∂F′g/∂x > 0, and ∂F′g/∂y > 0, respectively. Thus, F′g is positively correlated with Cg, x and y, and negatively correlated with Fe, Fs and β. The proof of Corollary 3.2 is complete.

The expected revenue for the online evaluation (offline evaluation) strategy chosen by the elderly is Uw(U_{1 − w}), and its average expected revenue is Ūw: (25)Uw=xRo-Cw+(1-x)[(1-y)zIo+β(1-y)(1-z)              (Io-Co)+yIo-Do] (26)U1-w=xRo-Cm+(1-x)[(1-y)zIo+β              (1-y)(1-z)(Io-Co)+yIo-Do] (27)Ūw=wUw+(1-w)U1-w The replicator dynamic equation and first-order derivative for elderly people can be formulated as follows: (28)F(w)=dw/dt=w(Uw-Ūw)=w(1-w)(Cm-Cw) (29)F′(w)=(1-2w)(Cm-Cw) Equations 28 and 29 imply that the strategy of elderly people mainly depends on the cost of online and offline evaluations. Using the stability theorem of the differential equation, the decision probability of the elderly in a steady state must satisfy F(w) = 0 and F′(w) < 0.

Proposition 4. When the cost of the online evaluation is higher than that of offline evaluations, i.e., Cm − Cw < 0, elderly people will choose offline evaluations. When the cost of the online evaluation is lower than that of offline evaluation, i.e., Cm − Cw > 0, elderly people will choose online evaluation. When the cost of the online evaluation is equal to the cost of offline evaluations, i.e., Cm − Cw = 0, the stabilization strategy for elderly people cannot be determined.

Proof of Proposition 4. If Cm − Cw < 0, then F(w)|_{w = 0} = 0 and F′(w)|w=0<0. Furthermore, w = 0 has stability. If Cm − Cw > 0, F(w)|_{w = 1} = 0 and F′(w)|w=1<0, then w = 1 has stability. If Cm − Cw = 0, then F(w) = 0, F′(w) = 0, and no stable strategy for elderly people can be identified. Proposition 4 is complete.

Proposition 4 suggests that the strategy choice of elderly people mainly depends on the cost of offline and online evaluations. Elderly people will always choose the evaluation method with the lowest cost as their optimal strategic choice. Therefore, the government should urge the relevant units to provide the elderly with operation training for software for elderly services. This training would reduce the perceived cost of online evaluations for elderly people and thus encourage them to conduct them.

When the cost of online evaluations for elderly people is higher than that of offline evaluations, there are eight equilibrium points in the replicated dynamic system. The stability analysis of the equilibrium points is shown in Table 2.

It can be seen from Table 2 that there are two instability points when elderly people are evaluated offline. The remaining six equilibrium points can potentially become more stable when certain conditions are met. As the elderly care service providers in the four pure strategy equilibrium points E₁, E₃, E₄, and E₆ all choose to offer low-quality services, these four equilibrium points are then undesirable strategy combinations. Some measures should be taken to improve the reputational gains and losses, the penalties imposed by the government, the complaint rate of elderly people, and the amount of compensation given to elderly people to prevent these points from becoming ESS. Thus, the losses incurred by the elderly care service providers fully cover the cost reduction incurred when they provide low-quality services. For both points E₂ and E₅, the elderly care service providers choose to offer high-quality services. Thus, these two equilibrium points are desirable strategy combinations. When the expected revenue from supplying high-quality services is greater than that from supplying low-quality services, both E₂ and E₅ may become stable points in the system. Specifically, the system will eventually stabilize at E₂ when the sum of the reputational gains and losses and operating subsidies earned by the elderly services information platforms is less than the cost of positive return visits. Conversely, the system will stabilize at E₅.

For points E₂ and E₅, it can be seen that when the elderly care service information platforms choose a negative return visit, the absence of their supervision function will result in insufficient constraints on the elderly care service providers. However, this insufficient constraint may be made up by the penalties imposed by the government on the elderly care service providers when the elderly actively defend their rights. This is the reason a course on legal awareness is offered to elderly people.

When the cost of online evaluations for elderly people is lower than that of offline evaluations, i.e., Cm − Cw > 0, there might be eight equilibrium points in the replicated dynamic system. The stability analysis of the equilibrium strategy combination is shown in Table 3. As can be seen from Table 3, there are also two unstable points E₁₅ and E₁₆, four undesirable equilibrium points E₉, E₁₁, E₁₂, and E₁₄ that may become ESS and two desirable equilibrium points that can become ESS. The analysis process is similar to that of the offline evaluation of elderly people.

By comparing the conditions of the corresponding stability points in the online and offline evaluations, it can be concluded that the online evaluation channel for the elderly can increase the reputation losses and gains of elderly care service providers and elderly care service information platforms. Specifically, compared with offline evaluations, online evaluations make it more difficult for the system to create undesirable stability points and more likely to create desirable stability points. This is due to the fact that, in the case of the online evaluation, an undesirable stability point requires a higher cost reduction for elderly care service providers to reach stability. However, a desirable stability point can be achieved with lower compensation, fewer complaints from elderly people, and fewer penalties imposed by government departments.

Therefore, community-based elderly service centers should be relied upon to train the elderly in the community to operate the software for elderly care services. The positive effects of online evaluations should also be publicized to reduce the perceived cost of online evaluations for elderly people. It is important for the healthy development of “Internet + Community Elderly Care”.

System dynamics (52) (S.D.) is often applied to study complex systems and analyze evolutionary trends (53–55). Some authors recently used system dynamics to simulate and analyze evolutionary game problems (56–58). In this study, SD was incorporated into the study of the omnichannel feedback mechanism of elderly people to clearly depict the complex relationships and mechanisms of interaction among the subjects. Moreover, VENSIM 5.6b software was used for simulation analysis of the evolutionary game SD model, which involved government departments, elderly care service providers, elderly care service information platforms, and elderly people's participation. The proposed four-party evolutionary game SD model is shown in Figure 5. It consists of four sub-models: government departments sub-SD Model, elderly care service providers sub-SD Model, elderly care service information platforms sub-SD model and elderly people sub-SD Model. The four state variables, four rate variables, and a set of other intermediate and external variables in the model were determined by the model's construction. Furthermore, the functional relationships in the model were determined by the expectation function and the replicator dynamic equation.

To make the system stable to the ideal stability point, the following three conditions should be satisfied, i.e., Cw − Cm < 0, Cei − Ie − De < 0, and Csh − Csl − Is − Ds − Io < 0. Combined with the actual situation, the corresponding parameters are set as follows: Rs = 100, Csh = 50, Csl = 25, Is = 20, Ds = 22, S = 10, Ceo = 15, Cei = 5, Ie = 10, De = 12, α = 10%, β = 1%, γ = 5%, Fs = 20, Fe = 5, Fg = 200, Rg = 40, Cg = 10, Dg = 40, Cm = 10, Cw = 8, Io = 10, Ro = 20, Do = 30, and Co = 5. In addition, we assumed that INITIAL TIME = 0, FINAL TIME = 3, and TIME STEP = 0.0125.