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HomeHealth EconomicsEvolutionary recreation evaluation of constructing a sustainable clever aged care service platform

Evolutionary recreation evaluation of constructing a sustainable clever aged care service platform


Strategic stability of the digital expertise firm

The anticipated revenues ((E_{11}), (E_{12})) for the digital expertise firm when taking part and never taking part in worth co-creation, in addition to the typical anticipated income ((overline{{E_{1} }})) are as follows:

$$left{ {start{array}{*{20}l} start{gathered} E_{11} = yzleft( {G_{d} + R_{d} + lambda IQ – C_{d} – Delta C_{d} } proper) + yleft( {1 – z} proper)left( {G_{d} + lambda IQ – C_{d} – Delta C_{d} } proper) + left( {1 – y} proper)zleft( {G_{d} + lambda zeta IQ – C_{d} – beta Delta C_{d} } proper) + left( {1 – y} proper)left( {1 – z} proper)left( {G_{d} + lambda alpha zeta IQ – C_{d} – beta Delta C_{d} } proper) finish{gathered} hfill start{gathered} E_{12} = yzleft( {G_{d} – C_{d} – L_{d} } proper) + yleft( {1 – z} proper)left( {G_{d} – C_{d} – L_{d} } proper) + left( {1 – y} proper)zleft( {G_{d} – C_{d} – L_{d} } proper) + left( {1 – y} proper)left( {1 – z} proper)left( {G_{d} – C_{d} – L_{d} } proper) finish{gathered} hfill {overline{{E_{1} }} = xE_{11} + left( {1 – x} proper)E_{12} } hfill finish{array} } proper.$$

(1)

The replicator dynamic equation is expressed as:

$$Fleft( x proper) = frac{dx}{{dt}} = xleft( {E_{11} – overline{{E_{1} }} } proper) = x{mkern 1mu} left( {1 – x} proper)left( start{gathered} L_{d} – Delta C_{d} {mkern 1mu} beta + left( {beta – 1} proper)Delta C_{d} {mkern 1mu} y + R_{d} {mkern 1mu} y{mkern 1mu} z hfill + IQlambda {mkern 1mu} left( {{mkern 1mu} alpha {mkern 1mu} zeta + yleft( {1 – {mkern 1mu} alpha zeta } proper) + {mkern 1mu} {mkern 1mu} left( {1 – {mkern 1mu} alpha } proper)left( {1 – y} proper)zeta z} proper) hfill finish{gathered} proper)$$

(2)

We get hold of the first-order spinoff of the replicator dynamic equation with respect to (x).

$$frac{dFleft( x proper)}{{dx}} = left( {2{mkern 1mu} x – 1} proper){mkern 1mu} Gleft( z proper)$$

(3)

the place (Gleft( z proper) = – L_{d} + Delta C_{d} {mkern 1mu} beta + left( {beta – 1} proper)Delta C_{d} {mkern 1mu} y – R_{d} {mkern 1mu} y{mkern 1mu} z – IQlambda {mkern 1mu} left( {{mkern 1mu} alpha {mkern 1mu} zeta + yleft( {1 – {mkern 1mu} alpha zeta } proper) + {mkern 1mu} {mkern 1mu} left( {1 – {mkern 1mu} alpha } proper)left( {1 – y} proper)zeta z} proper)).

Based on the soundness theorem of differential equations, the strategic equilibrium factors of the digital expertise firm should fulfill (Fleft( x proper) = 0) and ({{dFleft( x proper)} mathord{left/ {vphantom {{dFleft( x proper)} {dx}}} proper. kern-0pt} {dx}} < 0). It may be deduced that ({{partial Gleft( z proper)} mathord{left/ {vphantom {{partial Gleft( z proper)} {partial z}}} proper. kern-0pt} {partial z}} = {mkern 1mu} lambda {mkern 1mu} zeta I{mkern 1mu} Qleft( {1 – {mkern 1mu} y} proper){mkern 1mu} {mkern 1mu} left( {alpha – 1} proper) – R_{d} {mkern 1mu} y < 0), implying that (Gleft( z proper)) is a reducing perform of (z). We will derive (Gleft( z proper) = 0) when (z = z_{1}^{*} = frac{{Delta C_{d} {mkern 1mu} left( {left( {1 – y} proper)beta + y} proper) – L_{d} – IQ{mkern 1mu} lambda left( {alpha zeta left( {1 – y} proper) + y} proper)}}{{R_{d} {mkern 1mu} y + lambda zeta IQleft( {y – 1} proper)left( {alpha – 1} proper)}}), at which level ({{dFleft( x proper)} mathord{left/ {vphantom {{dFleft( x proper)} {dx}}} proper. kern-0pt} {dx}} equiv 0), rendering it indeterminate for the digital expertise firm to determine a secure technique. When (z < z_{1}^{*}), we derive (Gleft( z proper) > 0) and (left. {frac{dFleft( x proper)}{{dx}}} proper|_{x = 0} < 0), during which case (x = 0) turns into the evolutionary secure technique (ESS) for the digital expertise firm. Conversely, when (z > z_{1}^{*}), we get hold of (Gleft( z proper) < 0) and (left. {frac{dFleft( x proper)}{{dx}}} proper|_{x = 1} < 0), which means that (x = 1) is ESS. The part diagram for the technique evolution of the digital expertise firm is proven in Fig. 2.

Fig. 2
figure 2

The part diagram of the technique evolution for the digital expertise firm.

Proposition 1:

Through the evolutionary course of, the chance of the digital expertise firm selecting to take part in worth co-creation will increase because the participation chances of the opposite two events rise.

Proposition 1 signifies that the extra actively the social group operates and the stronger the service supplier’s dedication to delivering high-quality companies, the extra inclined the digital expertise firm is to interact in worth co-creation. Conversely, if the social group neglects to supply digital coaching for the aged, fails to promptly reply to their suggestions, doesn’t cooperate within the integration of digital sources, and workout routines lax oversight over the service supplier—permitting low-quality supplier to entry IESP—belief in IESP among the many aged diminishes. This erosion of belief results in diminished consumer engagement and consequently lowers IESP’s income. In such a situation, the digital expertise firm has little incentive to take part in worth co-creation actions, reminiscent of iterative IESP updates. Subsequently, for the sustainable growth of IESP, it’s crucial that the federal government strengthens oversight of the social group’s operational habits and establishes a high quality normal and analysis system for the aged care service supplier.

Proposition 2:

The chance of the digital expertise firm opting to interact in worth co-creation technique is positively correlated with the advantages derived from IESP ((IQ)), reputational good points ((R_{d})), and reputational loss incurred from non-participation ((L_{d})). Conversely, it’s negatively correlated with the extra prices related to worth co-creation ((Delta C_{d})).

Proposition 2 signifies that growing the income generated by IESP and lowering the extra operational prices for the digital expertise firm will encourage its participation in worth co-creation. To attain this, the federal government ought to improve the promotion of IESP by collaborating with grassroots organizations, reminiscent of neighborhood committees and neighborhood councils, to hold out offline campaigns that bridge the digital divide among the many aged, thereby growing the variety of energetic customers on IESP and reaching economies of scale. Moreover, the federal government can incentivize the digital expertise firm to interact in worth co-creation by reducing the price of information acquisition. This may be completed by opening public information, establishing data-sharing platforms, and creating and selling information requirements. Furthermore, each reputational good points and losses can inspire the digital expertise firm to take part in worth co-creation. Thus, the federal government ought to prioritize subsequent collaborations with corporations that actively have interaction in worth co-creation. Conversely, for corporations that fail to cooperate in upkeep after undertaking acceptance, the federal government ought to scale back their alternatives for future collaboration.

Strategic stability of the social group

The anticipated revenues ((E_{21}), (E_{22})) for the social group when taking part and never taking part in worth co-creation, in addition to the typical anticipated income ((overline{{E_{2} }})) are as follows:

$$left{ {start{array}{*{20}l} start{gathered} E_{21} = xzleft( {left( {1 – lambda } proper)IQ + R_{s} – C_{s} – Delta C_{s} } proper) + left( {1 – x} proper)zleft( {Iwidetilde{Q} – C_{s} – Delta C_{s} } proper) + xleft( {1 – z} proper)left( {left( {1 – lambda } proper)IQ – C_{s} – Delta C_{s} } proper) + left( {1 – x} proper)left( {1 – z} proper)left( {Iwidetilde{Q} – C_{s} – Delta C_{s} } proper) finish{gathered} hfill start{gathered} E_{22} = xzleft( {left( {1 – lambda } proper)zeta IQ – C_{s} – L_{s} } proper) + left( {1 – x} proper)zleft( {zeta Iwidetilde{Q} – C_{s} – L_{s} } proper) + xleft( {1 – z} proper)left( {left( {1 – lambda } proper)alpha zeta IQ – C_{s} – L_{s} } proper) + left( {1 – x} proper)left( {1 – z} proper)left( {alpha zeta Iwidetilde{Q} – C_{s} – L_{s} } proper) finish{gathered} hfill {overline{{E_{2} }} = yE_{21} + left( {1 – y} proper)E_{22} } hfill finish{array} } proper.$$

(4)

The replicator dynamic equation is expressed as:

$$start{gathered} Fleft( y proper) = frac{dy}{{dt}} = yleft( {E_{21} – overline{{E_{2} }} } proper) = y{mkern 1mu} left( {1 – y} proper){mkern 1mu} left( start{gathered} L_{s} – Delta {textual content{C}}_{s} + Iwidetilde{Q}left( {1 – alpha {mkern 1mu} zeta } proper) + left( {IQleft( {1 – lambda } proper) – Iwidetilde{Q}} proper)left( {1 – alpha zeta } proper)x – Iwidetilde{Q}left( {1 – alpha } proper)zeta z hfill {mkern 1mu} + left( {Iwidetilde{Q}left( {1 – alpha } proper) + {mkern 1mu} left( {lambda – 1} proper){mkern 1mu} left( {1 – alpha } proper)IQ} proper)zeta x{mkern 1mu} z + R_{s} x{mkern 1mu} z hfill finish{gathered} proper) finish{gathered}$$

(5)

The primary-order spinoff of the replicator dynamic equation with respect to (y) is derived as:

$$frac{dFleft( y proper)}{{dy}} = left( {2{mkern 1mu} y – 1} proper){mkern 1mu} Hleft( z proper)$$

(6)

the place (start{gathered} Hleft( z proper) = – L_{s} + Delta {textual content{C}}_{s} – Iwidetilde{Q}left( {1 – alpha {mkern 1mu} zeta } proper) – left( {IQleft( {1 – lambda } proper) – Iwidetilde{Q}} proper)left( {1 – alpha zeta } proper)x + Iwidetilde{Q}left( {1 – alpha } proper)zeta {mkern 1mu} z {mkern 1mu} – left( {Iwidetilde{Q}left( {1 – alpha } proper) + {mkern 1mu} left( {lambda – 1} proper){mkern 1mu} left( {1 – alpha } proper)IQ} proper)zeta x{mkern 1mu} z – R_{s} x{mkern 1mu} z finish{gathered}).

By way of the same fixing course of, we get hold of (Hleft( z proper) = 0) when (z = z_{2}^{*} = frac{{left( {IQleft( {1 – lambda } proper) – Iwidetilde{Q}} proper)left( {1 – alpha zeta } proper)x + Iwidetilde{Q}left( {1 – alpha zeta } proper) + L_{s} – Delta {textual content{C}}_{s} }}{{Iwidetilde{Q}zeta left( {1 – alpha } proper)left( {1 – x} proper) – left( {{mkern 1mu} left( {lambda – 1} proper){mkern 1mu} left( {1 – alpha } proper)IQzeta + R_{s} } proper)x}}), at which level ({{dFleft( y proper)} mathord{left/ {vphantom {{dFleft( y proper)} {dy}}} proper. kern-0pt} {dy}} equiv 0), which means that the social group can not decide a secure technique, as illustrated in Fig. 3(a). There exist two attainable secure states, (y = 0) or (y = 1), when (z ne z_{2}^{*}).

Fig. 3
figure 3

The part diagram of the technique evolution for the social group.

When (frac{partial Hleft( z proper)}{{partial z}} > 0):

  • If (z < z_{2}^{*}), it follows that (left. {frac{dFleft( y proper)}{{dy}}} proper|_{y = 1} < 0), and (y = 1) is ESS, as illustrated by arrow (2) in Fig. 3(b).

  • If (z > z_{2}^{*}), it follows that (left. {frac{dFleft( y proper)}{{dy}}} proper|_{y = 0} < 0), and (y = 0) is ESS, as illustrated by arrow (1) in Fig. 3(c).

When (frac{partial Hleft( z proper)}{{partial z}} < 0):

  • If (z < z_{2}^{*}), it follows that (left. {frac{dFleft( y proper)}{{dy}}} proper|_{y = 0} < 0), and (y = 0) is ESS, as illustrated by arrow (1) in Fig. 3(b).

  • If (z > z_{2}^{*}), it follows that (left. {frac{dFleft( y proper)}{{dy}}} proper|_{y = 1} < 0), and (y = 1) is ESS, as illustrated by arrow (2) in Fig. 3(c).

By way of the above calculations, we will get hold of the Proposition 3.

Proposition 3:

If (R_{s} > IQleft( {1 – lambda } proper){mkern 1mu} left( {1 – alpha } proper)zeta) and (x{mkern 1mu} > x_{1}^{*} = frac{{Iwidetilde{Q}{mkern 1mu} zeta left( {1 – alpha } proper)}}{{Iwidetilde{Q}left( {1 – alpha {mkern 1mu} } proper)zeta + R_{s} – IQ{mkern 1mu} left( {1 – lambda } proper)left( {1 – alpha {mkern 1mu} } proper)zeta {mkern 1mu} }}) are each met, (frac{partial Hleft( z proper)}{{partial z}} < 0). In any other case, (frac{partial Hleft( z proper)}{{partial z}} > 0).

Proposition 3 signifies that if the reputational good points acquired by the social group outweigh its losses incurred from low-quality service supplier coming into IESP, then the chance of the social group taking part in worth co-creation will increase because the chance of the service supplier’s participation will increase, supplied that the digital expertise firm’s participation chance exceeds a sure threshold. In any other case, the chance of the group taking part decreases because the chance of the service supplier’s participation rises.

Thus, the reputational good points of the social group function a pivotal consider figuring out the total engagement of all stakeholders in worth co-creation. That is rooted in the truth that the reputational good points are solely achievable when all three events actively take part in worth co-creation, resulting in the profitable institution of a high-satisfaction IESP. Such energetic involvement permits the social group to realize superior operational efficiency metrics, leading to larger authorities rankings and extra favorable insurance policies. When reputational good points are substantial, even when the service supplier is prone to supply high-quality companies, the social group stays motivated to implement strict laws, guaranteeing that no low-quality supplier good points entry to IESP. Conversely, when reputational good points are inadequate, the social group could go for relaxed oversight when the service supplier tends to ship high-quality companies, thus lowering operational prices and exhibiting “free-rider” habits. Subsequently, the federal government should rigorously assess the rankings of social organizations and supply extra favorable insurance policies to these with larger rankings.

Strategic stability of the aged care service supplier

The anticipated revenues ((E_{31}), (E_{32})) for the service supplier when taking part and never taking part in worth co-creation, in addition to the typical anticipated income ((overline{{E_{3} }})) are as follows:

$$left{ {start{array}{*{20}l} start{gathered} E_{31} = xyleft( {R_{pH} – C_{pH} + G_{p} + Delta R_{pH} – C_{p} } proper) + left( {1 – x} proper)yleft( {R_{pH} – C_{pH} + G_{p} + Delta R_{p} – C_{p} } proper) + xleft( {1 – y} proper)left( {R_{pH} – C_{pH} + Delta R_{pH} – C_{p} } proper) + left( {1 – x} proper)left( {1 – y} proper)left( {R_{pH} – C_{pH} + Delta R_{p} – C_{p} } proper) finish{gathered} hfill start{gathered} E_{32} = xyleft( {R_{pL} – C_{pL} } proper) + left( {1 – x} proper)yleft( {R_{pL} – C_{pL} } proper) + xleft( {1 – y} proper)left( {R_{pL} – C_{pL} + Delta R_{pL} – C_{p} } proper) + left( {1 – x} proper)left( {1 – y} proper)left( {R_{pL} – C_{pL} + Delta R_{p} – C_{p} } proper) finish{gathered} hfill {overline{{E_{3} }} = zE_{31} + left( {1 – z} proper)E_{32} } hfill finish{array} } proper.$$

(7)

The replicator dynamic equation is expressed as:

$$Fleft( z proper) = frac{dz}{{dt}} = zleft( {E_{31} – overline{{E_{3} }} } proper) = z{mkern 1mu} left( {1 – z} proper){mkern 1mu} left( start{gathered} C_{pL} – C_{pH} + R_{pH} – R_{pL} + left( {Delta R_{pH} {mkern 1mu} – Delta R_{pL} } proper){mkern 1mu} x hfill + left( {Delta R_{p} {mkern 1mu} + G_{p} – C_{p} } proper){mkern 1mu} y + left( {Delta R_{pL} {mkern 1mu} – Delta R_{p} } proper)x{mkern 1mu} y hfill finish{gathered} proper)$$

(8)

The primary-order spinoff of the replicator dynamic equation with respect to (z) is derived as:

$$frac{dFleft( z proper)}{{dz}} = left( {2{mkern 1mu} z – 1} proper){mkern 1mu} Jleft( x proper)$$

(9)

the place (start{gathered} Jleft( x proper) = – C_{pL} + C_{pH} – R_{pH} + R_{pL} – left( {Delta R_{pH} {mkern 1mu} – Delta R_{pL} } proper){mkern 1mu} x – left( {Delta R_{p} {mkern 1mu} + G_{p} – C_{p} } proper){mkern 1mu} y – left( {Delta R_{pL} {mkern 1mu} – Delta R_{p} } proper)x{mkern 1mu} y finish{gathered}).

By way of the same fixing course of, we get hold of (Jleft( x proper) = 0) when (x = x^{*} = frac{{ – C_{pL} + C_{pH} – R_{pH} + R_{pL} – left( {Delta R_{p} {mkern 1mu} + G_{p} – C_{p} } proper){mkern 1mu} y}}{{Delta R_{pH} {mkern 1mu} – Delta R_{pL} + {mkern 1mu} left( {Delta R_{pL} {mkern 1mu} – Delta R_{p} } proper)y}}{mkern 1mu}), at which level ({{dFleft( z proper)} mathord{left/ {vphantom {{dFleft( z proper)} {dz}}} proper. kern-0pt} {dz}} equiv 0), indicating that the service supplier is unable to find out a secure technique. It may be deduced that (frac{partial Jleft( x proper)}{{partial x}} = Delta R_{pL} – Delta R_{pH} + left( {Delta R_{p} {mkern 1mu} – Delta R_{pL} } proper){mkern 1mu} y < 0). When (x < x^{*}), we derive (Jleft( x proper) > 0) and (left. {frac{dFleft( z proper)}{{dz}}} proper|_{z = 0} < 0), during which case (z = 0) turns into ESS for the service supplier. Conversely, when (x > x^{*}), we get hold of (left. {frac{dFleft( z proper)}{{dz}}} proper|_{z = 1} < 0), which means that (z = 1) is ESS. The part diagram for the technique evolution of the service supplier is proven in Fig. 4.

Fig. 4
figure 4

The part diagram of the technique evolution for the service supplier.

Proposition 4:

Through the evolutionary course of, the chance of the aged care service supplier opting to interact in worth co-creation will increase because the participation chances of the opposite two events rise.

Proposition 4 signifies that the stronger the willingness of the digital expertise firm and social group to interact in worth co-creation, the extra inclined the aged care service supplier is to supply high-quality companies. It is because when the digital expertise firm participates in worth co-creation, it enhances the visibility of high-quality service supplier by analyzing consumer suggestions information, thereby widening the hole in market share and income between high-quality and low-quality service suppliers. Moreover, when the social group imposes strict oversight on service supplier, low-quality service suppliers are precluded from accessing IESP, depriving them of on-line buyer acquisition alternatives. Subsequently, growing coverage incentives for the digital expertise firm and social group that take part in worth co-creation can successfully encourage the aged care service supplier to enhance service high quality.

Proposition 5:

The chance of the aged care service supplier opting to interact in worth co-creation methods is positively correlated with the extra income generated by enhancing service high quality ((R_{pH} – R_{pL}), (Delta R_{pH} {mkern 1mu} – Delta R_{pL})) and authorities subsidies ((G_{p})), and negatively correlated with the prices of digital transformation for becoming a member of the platform ((C_{p})) and the extra prices incurred in enhancing service high quality ((C_{pH} – C_{pL})).

Proposition 5 signifies that the service supplier is incentivized to supply high-quality companies solely when enhancing service high quality results in larger earnings. Subsequently, enhancing the cost capability of the aged is a basic method to enhancing the service high quality. Moreover, the federal government can widen the earnings hole between high-quality and low-quality service suppliers by standardizing score standards and growing incentives and monetary subsidies for high-quality suppliers. For instance, insurance policies just like the “reward as an alternative of subsidy” initiative launched by Shanghai could be efficient.

Stability evaluation of the system’s equilibrium methods

Contemplating that the technique set on this binary-choice recreation consists of participation or non-participation, we focus solely on the evaluation of pure methods. This method is chosen to facilitate sensible utility and understanding, which is supported by many analysis findings54,55. From (Fleft( x proper) = 0), (Fleft( y proper) = 0) and (Fleft( z proper) = 0), we will derive eight attainable technique combos, specifically (left( {0,0,0} proper)), (left( {1,0,0} proper)), (left( {0,1,0} proper)), (left( {1,1,0} proper)), (left( {0,0,1} proper)), (left( {0,1,1} proper)), (left( {1,0,1} proper)) and (left( {1,1,1} proper)). To find out the system’s equilibrium methods, we first assemble the Jacobian matrix, following the method outlined by Friedman56.

$$J = left[ {begin{array}{*{20}c} {{{partial Fleft( x right)} mathord{left/ {vphantom {{partial Fleft( x right)} {partial x}}} right. kern-0pt} {partial x}}} & {{{partial Fleft( x right)} mathord{left/ {vphantom {{partial Fleft( x right)} {partial y}}} right. kern-0pt} {partial y}}} & {{{partial Fleft( x right)} mathord{left/ {vphantom {{partial Fleft( x right)} {partial z}}} right. kern-0pt} {partial z}}} {{{partial Fleft( y right)} mathord{left/ {vphantom {{partial Fleft( y right)} {partial x}}} right. kern-0pt} {partial x}}} & {{{partial Fleft( y right)} mathord{left/ {vphantom {{partial Fleft( y right)} {partial y}}} right. kern-0pt} {partial y}}} & {{{partial Fleft( y right)} mathord{left/ {vphantom {{partial Fleft( y right)} {partial z}}} right. kern-0pt} {partial z}}} {{{partial Fleft( z right)} mathord{left/ {vphantom {{partial Fleft( z right)} {partial x}}} right. kern-0pt} {partial x}}} & {{{partial Fleft( z right)} mathord{left/ {vphantom {{partial Fleft( z right)} {partial y}}} right. kern-0pt} {partial y}}} & {{{partial Fleft( z right)} mathord{left/ {vphantom {{partial Fleft( z right)} {partial z}}} right. kern-0pt} {partial z}}} end{array} } right] = left[ {begin{array}{*{20}c} {J_{11} } & {J_{12} } & {J_{13} } {J_{21} } & {J_{22} } & {J_{23} } {J_{31} } & {J_{32} } & {J_{33} } end{array} } right]$$

the place (J_{11} = left( {2{mkern 1mu} x – 1} proper){mkern 1mu} left( { – L_{d} + Delta C_{d} {mkern 1mu} beta + left( {beta – 1} proper)Delta C_{d} {mkern 1mu} y – R_{d} {mkern 1mu} y{mkern 1mu} z – IQlambda {mkern 1mu} left( {{mkern 1mu} alpha {mkern 1mu} zeta + yleft( {1 – {mkern 1mu} alpha zeta } proper) + {mkern 1mu} {mkern 1mu} left( {1 – {mkern 1mu} alpha } proper)left( {1 – y} proper)zeta z} proper)} proper)),(J_{12} = x{mkern 1mu} left( {1 – x} proper){mkern 1mu} left( {left( {beta – 1} proper)Delta C_{d} {mkern 1mu} + IQlambda left( {1 – {mkern 1mu} alpha {mkern 1mu} zeta + {mkern 1mu} left( {alpha {mkern 1mu} – 1} proper)z{mkern 1mu} zeta } proper) + R_{d} {mkern 1mu} {mkern 1mu} z} proper)), (J_{13} = x{mkern 1mu} left( {1 – x} proper){mkern 1mu} left( {IQzeta lambda {mkern 1mu} left( {1 – {mkern 1mu} alpha } proper)left( {{mkern 1mu} 1 – {mkern 1mu} y{mkern 1mu} } proper) + R_{d} {mkern 1mu} y} proper)), (J_{21} = y{mkern 1mu} left( {1 – y} proper){mkern 1mu} left( {left( {IQleft( {1 – lambda } proper) – Iwidetilde{Q}} proper)left( {1 – alpha zeta } proper) + left( {Iwidetilde{Q}left( {1 – alpha } proper) + {mkern 1mu} left( {lambda – 1} proper){mkern 1mu} left( {1 – alpha } proper)IQ} proper){mkern 1mu} zzeta + R_{s} z} proper)), (J_{22} = left( {2{mkern 1mu} y – 1} proper){mkern 1mu} left( start{gathered} – L_{s} + Delta {textual content{C}}_{s} – Iwidetilde{Q}left( {1 – alpha {mkern 1mu} zeta } proper) – left( {IQleft( {1 – lambda } proper) – Iwidetilde{Q}} proper)left( {1 – alpha zeta } proper)x + Iwidetilde{Q}left( {1 – alpha } proper)zeta {mkern 1mu} z {mkern 1mu} – left( {Iwidetilde{Q}left( {1 – alpha } proper) + {mkern 1mu} left( {lambda – 1} proper){mkern 1mu} left( {1 – alpha } proper)IQ} proper)zeta x{mkern 1mu} z – R_{s} x{mkern 1mu} z finish{gathered} proper)), (J_{23} = y{mkern 1mu} left( {1 – y} proper){mkern 1mu} left( { – Iwidetilde{Q}zeta left( {1 – alpha } proper) + left( {Iwidetilde{Q}left( {1 – alpha } proper) + {mkern 1mu} left( {lambda – 1} proper){mkern 1mu} left( {1 – alpha } proper)IQ} proper)xzeta {mkern 1mu} + R_{s} x} proper)), (J_{31} = z{mkern 1mu} left( {1 – z} proper){mkern 1mu} left( {Delta R_{pH} {mkern 1mu} – Delta R_{pL} + left( {Delta R_{pL} {mkern 1mu} – Delta R_{p} } proper){mkern 1mu} y} proper)), (J_{32} = z{mkern 1mu} left( {1 – z} proper){mkern 1mu} left( {Delta R_{p} {mkern 1mu} + G_{p} – C_{p} + left( {Delta R_{pL} {mkern 1mu} – Delta R_{p} } proper)x{mkern 1mu} } proper)), (J_{33} = left( {2z – 1} proper)left( { – C_{pL} + C_{pH} – R_{pH} + R_{pL} – left( {Delta R_{pH} {mkern 1mu} – Delta R_{pL} } proper){mkern 1mu} x – left( {Delta R_{p} {mkern 1mu} + G_{p} – C_{p} } proper){mkern 1mu} y – left( {Delta R_{pL} {mkern 1mu} – Delta R_{p} } proper)x{mkern 1mu} y} proper)).

Based on Lyapunov’s system stability idea, an equilibrium level could be decided as an ESS if all of the eigenvalues of the Jacobian matrix have detrimental actual components. The soundness evaluation of the technique combos is offered in Desk 3.

Desk 3 Stability evaluation of technique combos.

Proposition 6:

If (D_{2} < 0), it may be inferred that (D_{1} > 0), (D_{3} > 0), (D_{4} > 0),(D_{6} > 0), (D_{5} < 0), (D_{7} < 0) and (D_{8} < 0).

Corollary 6 signifies that when (D_{2} < 0) is happy, the digital expertise firm’s secure equilibrium technique in pure technique combos will all the time be to interact in worth co-creation, whatever the decisions made by the opposite two events. It is because (D_{2} < 0) represents the minimal further profit that the digital expertise firm can get hold of by taking part in worth co-creation (the extra profit gained when the digital firm chooses to take part whereas the opposite two events don’t), which nonetheless exceeds the loss incurred if it doesn’t take part. Notably, lowering the operational prices incurred by the digital firm’s participation, growing the gross sales quantity and fee on IESP, in addition to the revenue distribution ratio for the digital firm, and enhancing the reputational harm suffered by the digital firm when it chooses to not take part, can all contribute to the success of (D_{2} < 0).

Proposition 7:

If (S_{6} < 0), it may be inferred that (S_{1} > 0), (S_{2} > 0), (S_{4} > 0), (S_{7} > 0), (S_{3} < 0), (S_{5} < 0) and (S_{8} < 0).

Proposition 7 means that when (S_{6} < 0) situation is met, whatever the decisions made by the opposite two events, the secure equilibrium technique for the social group in a pure technique mixture is to take part in worth co-creation. Not like the digital expertise firm, the minimal probability for the social group to interact in worth co-creation doesn’t come up when the opposite two events are absent, however relatively when the service supplier participates whereas the digital expertise firm doesn’t. That is evident in the truth that (S_{3} < 0) is extra relaxed than (S_{6} < 0). When the service supplier actively participates in worth co-creation by providing high-quality companies, the social group’s lack of stringent supervision doesn’t negatively influence IESP’s popularity or scale back its income, making the motivation for the social group to take part in worth co-creation decrease than when the service supplier doesn’t take part. When the situation (S_{6} < 0) is met, guaranteeing that the social group’s advantages from taking part in worth co-creation exceed these of not taking part, the group will invariably select to interact in worth co-creation. As IESP gross sales enhance by digital cognitive coaching, the additional prices of taking part in worth co-creation lower, and the popularity loss for not taking part grows, making it simpler to fulfill situation (S_{6} < 0).

Proposition 8:

If (P_{4} < 0), it may be inferred that (P_{1} > 0), (P_{2} > 0) and (P_{7} < 0). If (P_{6} < 0), it may be inferred that (P_{3} > 0), (P_{5} > 0) and (P_{8} < 0). When (Delta R_{p} + G_{p} – C_{p} > 0), (P_{4} < 0) implies (P_{6} < 0).

Proposition 8 demonstrates that when the digital expertise firm engages in worth co-creation, the constraints required to realize the service supplier’s participation in worth co-creation are extra relaxed than when the digital firm doesn’t take part. This suggests that the involvement of the digital firm can encourage the service supplier to undertake a participatory technique. Particularly, when the sum of the minimal on-line earnings earned by the high-quality service supplier and authorities subsidies exceeds its digital transformation prices (i.e., (Delta R_{p} + G_{p} – C_{p} > 0)), (P_{4} < 0) can deduce (P_{6} < 0), suggesting that the participation of the social group can inspire the service supplier to decide on a participatory technique. It is because, underneath such situations, the service supplier has the motivation to hitch IESP, and the rigorous supervision of the social group makes delivering high-quality companies a prerequisite for the service supplier to entry IESP. Conversely, when the digital transformation prices for the service supplier are larger, the supplier lacks the motivation to hitch IESP. Subsequently, along with growing the revenue margin between high-quality and low-quality offline companies to fulfill the situation (P_{4} < 0), it is usually essential to reinforce the service supplier’s on-line income, enhance authorities subsidies for high-quality service suppliers, and scale back the digital transformation prices for the service supplier.

Proposition 9:

When Situation 8 is happy, the system reaches a perfect state, the place the evolutionarily secure technique for all three events is to interact in worth co-creation. Moreover, Situation 2 and Situation 8 could coexist, as could Situations 3, 4, and eight. To make sure that the system evolves towards the best state, i.e., to make sure that solely Situation 8 is happy, it’s essential to maximise (R_{d}), (R_{s}), (G_{p}) and (Delta R_{pH}), whereas minimizing (C_{p}).

We will discover that, in comparison with the strictest constraints outlined in Proposition 6 to eight that compel a single social gathering to interact in worth co-creation, the constraints in Situation 8 are considerably relaxed in reaching the participation of all events in worth co-creation. This suggests that when all three events select to interact in co-creation, a win-win state of affairs could be realized. To attain this best state, it’s essential to reinforce the reputational advantages for the digital expertise firm and the social group, enhance subsidies for high-quality service suppliers, widen the earnings hole between high-quality and low-quality service suppliers, and scale back the digital transformation prices for the service suppliers.

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