Monopoly or competition: strategic analysis of a retailing technology service provision

Abstract

To improve the consumer shopping experience and to remain competitive, an increasing number of retailers continue to increase their investment in technology services and actively seek help from technology service providers to transform and upgrade their retail business. Technology service providers deliver new technological solutions for retailers by providing professional retailing technology services (e.g., building omnichannel retailing modes). Considering this context, we seek to understand whether retailers expect to use technology services and service providers’ expectations of providing retailing services. In this paper, we build a two-tier supply chain with one upstream technology service provider and one downstream retailer (or two competitive retailers) to analyse the strategic choice of the retailer(s) and the service provider. We find that the retailer(s) can purchase a professional retailing technology from the technology service provider and that the service provider is willing to provide technology to two competing retailers, which can increase market demand and yield higher profits. Furthermore, we find the impact of the technological contribution level, retail competition intensity, the power structure, and the technology marketing effect on decision outcomes is significant. Therefore, retailers should consider the influence of these factors when service providers deliver a retailing technology service.

Appendix

Proof of Proposition 1

According to the equilibrium solutions of Lemmas 1 and 2, we use derivation rule to obtain the results of proposition 1. We omit the straightforward algebraic steps here.$\square$

Proof of Proposition 2

• (1) PBC mode

According to Lemma 2, we have

$$\begin{aligned} \frac{{\partial w^{PBC} }}{\partial \eta } & = \frac{{\alpha \gamma \beta^{2} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }} > 0,\;\frac{{\partial \theta^{PBC} }}{\partial \eta } = \frac{2\alpha \gamma \beta }{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }} > 0, \\ \frac{{\partial Q_{i}^{PBC} }}{\partial \eta } & = \frac{{2\alpha \gamma^{2} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }} > 0,\;\frac{{\partial Q_{j ,j \ne i}^{PBC} }}{\partial \eta } > 0,\;\frac{{\partial \varPi_{S} (PBC )}}{\partial \eta } = \frac{{2\alpha^{2} \gamma^{2} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }} > 0. \\ \end{aligned}$$

Further, we obtain $\frac{{\partial p_{i}^{PBC} }}{\partial \eta } = \frac{{\partial p_{j ,j \ne i}^{PBC} }}{\partial \eta } = \frac{{2\alpha \gamma (\beta^{2} - \gamma )}}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }} = \frac{{2\alpha \gamma^{2} (G - 1)}}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }}$, and $G = \frac{{\beta^{2} }}{\gamma }$. Thus, when $0 < G < 1$, we have $\frac{{\partial p_{i}^{PBC} }}{\partial \eta } = \frac{{\partial p_{j ,j \ne i}^{PBC} }}{\partial \eta } < 0$; when $1 < G < \frac{2(2 + \eta )}{1 + \eta }$, we have $\frac{{\partial p_{i}^{PBC} }}{\partial \eta } = \frac{{\partial p_{j ,j \ne i}^{PBC} }}{\partial \eta } > 0$.

• (2) PNC mode

Under the PNC mode, we have $\frac{{\partial w^{PNC} }}{\partial \eta } = \frac{{2\alpha \gamma \beta^{2} }}{{\left[ {4\gamma - A\beta^{2} } \right]^{2} }}\frac{\partial A}{\partial \eta }$, and $\frac{\partial A}{\partial \eta } = \frac{{32 + 144\eta + 232\eta^{2} + 160\eta^{3} + 44\eta^{4} }}{{\left[ {4(1 + \eta )(2 + 4\eta + \eta^{2} )} \right]^{2} }} > 0$, we let $\frac{\partial A}{\partial \eta } = A_{1}$, obtain $\frac{{\partial w^{PNC} }}{\partial \eta } = \frac{{2A_{1} \alpha \gamma \beta^{2} }}{{\left[ {4\gamma - A\beta^{2} } \right]^{2} }} > 0$, and $\frac{{\partial \varPi_{S} (PNC )}}{\partial \eta } = \frac{{4A_{1} \alpha^{2} \gamma^{2} }}{{\left[ {4\gamma - A\beta^{2} } \right]^{2} }} > 0$, $\frac{{\partial \theta^{PNC} }}{\partial \eta } = \frac{{4A_{1} \alpha \gamma \beta }}{{\left[ {4\gamma - A\beta^{2} } \right]^{2} }} > 0$, $\frac{{\partial Q_{i}^{PNC} }}{\partial \eta } > 0$, $\frac{{\partial Q_{j ,j \ne i}^{PNC} }}{\partial \eta } > 0$.

Further, we have $\frac{{\partial p_{i}^{PNC} }}{\partial \eta } = \frac{{A_{2} \alpha \gamma^{2} \left( {G - \frac{{8 + 16\eta + 12\eta^{2} }}{{A_{2} }}} \right)}}{{\left[ {(2 + 4\eta + \eta^{2} )(4\gamma - A\beta^{2} )} \right]^{2} }}$, and $A_{2} = (6 + 11\eta + 2\eta^{2} )(2 + 4\eta + \eta^{2} )A_{1} + (2 + 4\eta + 3\eta^{2} )A$. Thus, when $0 < G < \frac{{8 + 16\eta + 12\eta^{2} }}{{A_{2} }}$, we obtain $\frac{{\partial p_{i}^{PNC} }}{\partial \eta } < 0$; when $\frac{{8 + 16\eta + 12\eta^{2} }}{{A_{2} }} < G < \frac{4}{A}$, we obtain $\frac{{\partial p_{i}^{PNC} }}{\partial \eta } > 0$.

Similarly, we have

$$\frac{{\partial p_{j .j \ne i}^{PNC} }}{\partial \eta } = \frac{{A_{3} \alpha \gamma^{2} \left( {G - \frac{{32 + 160\eta + 256\eta^{2} + 160\eta^{3} + 40\eta^{4} }}{{A_{3} }}} \right)}}{{\left[ {2(1 + \eta )(2 + 4\eta + \eta^{2} )(4\gamma - A\beta^{2} )} \right]^{2} }}$$

(22)

$A_{3} = (12 + 34\eta + 25\eta^{2} + 4\eta^{3} )(4 + 12\eta + 10\eta^{2} + 2\eta^{3} )A_{1} + (8 + 40\eta + 64\eta^{2} + 40\eta^{3} + 10\eta^{4} )A$.

Thus, when $0 < G < \frac{{32 + 160\eta + 256\eta^{2} + 160\eta^{3} + 40\eta^{4} }}{{A_{3} }}$, we obtain $\frac{{\partial p_{j .j \ne i}^{PNC} }}{\partial \eta } < 0$; when $\frac{{32 + 160\eta + 256\eta^{2} + 160\eta^{3} + 40\eta^{4} }}{{A_{3} }} < G < \frac{4}{A}$, we obtain $\frac{{\partial p_{j .j \ne i}^{PNC} }}{\partial \eta } > 0$. In summary, when $\hbox{max} G < G < \frac{2(2 + \eta )}{1 + \eta }$,we have $\frac{{\partial p_{i}^{PBC} }}{\partial \eta } = \frac{{\partial p_{j ,j \ne i}^{PBC} }}{\partial \eta } > 0$, $\frac{{\partial p_{i}^{PNC} }}{\partial \eta } > 0$, $\frac{{\partial p_{j ,j \ne i}^{PNC} }}{\partial \eta } > 0$; when $0 < G < \hbox{min} G_{1}$, we have $\frac{{\partial p_{i}^{PBC} }}{\partial \eta } = \frac{{\partial p_{j ,j \ne i}^{PBC} }}{\partial \eta } < 0$, $\frac{{\partial p_{i}^{PNC} }}{\partial \eta } < 0$, $\frac{{\partial p_{j ,j \ne i}^{PNC} }}{\partial \eta } < 0$, $\hbox{max} G_{1} = \hbox{max} \left\{ {1 ,\frac{{8 + 16\eta + 12\eta^{2} }}{{A_{2} }} ,\frac{{32 + 160\eta + 256\eta^{2} + 160\eta^{3} + 40\eta^{4} }}{{A_{3} }}} \right\}$, $\hbox{min} G_{1} = \hbox{min} \left\{ {1 ,\frac{{8 + 16\eta + 12\eta^{2} }}{{A_{2} }} ,\frac{{32 + 160\eta + 256\eta^{2} + 160\eta^{3} + 40\eta^{4} }}{{A_{3} }}} \right\}$, $A_{1} = \frac{{32 + 144\eta + 232\eta^{2} + 160\eta^{3} + 44\eta^{4} }}{{\left[ {4(1 + \eta )(2 + 4\eta + \eta^{2} )} \right]^{2} }}$, $A_{2} = (6 + 11\eta + 2\eta^{2} )(2 + 4\eta + \eta^{2} )A_{1} + (2 + 4\eta + 3\eta^{2} )A$, $A_{3} = (12 + 34\eta + 25\eta^{2} + 4\eta^{3} )(4 + 12\eta + 10\eta^{2} + 2\eta^{3} )A_{1} + (8 + 40\eta + 64\eta^{2} + 40\eta^{3} + 10\eta^{4} )A$. $\square$

Proof of Proposition 3

• (1) PBC mode

According to Lemma 2, we have $\frac{{\partial \varPi_{S} (PBC )}}{\partial \eta } = \frac{{2\alpha^{2} \gamma^{2} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} }} > 0$, $\frac{{\partial \varPi_{{R_{i} }} (PBC)}}{\partial \eta } = \frac{{\partial \varPi_{{R_{j,j \ne i} }} (PBC)}}{\partial \eta } = \frac{{\alpha^{2} \gamma^{3} (1 + \eta )\left( {G - \frac{2\eta }{1 + \eta }} \right)}}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{3} }}$. Thus, when $0 < G < \frac{2\eta }{1 + \eta }$, we obtain $\frac{{\partial \varPi_{{R_{i} }} (PBC)}}{\partial \eta } = \frac{{\partial \varPi_{{R_{j,j \ne i} }} (PBC)}}{\partial \eta } < 0$; when $\frac{2\eta }{1 + \eta } < G < \frac{2(2 + \eta )}{1 + \eta }$, we obtain $\frac{{\partial \varPi_{{R_{i} }} (PBC)}}{\partial \eta } = \frac{{\partial \varPi_{{R_{j,j \ne i} }} (PBC)}}{\partial \eta } > 0$.

• (2) PNC Mode

Under the PNC mode, we have $\frac{\partial A}{\partial \eta } = \frac{{32 + 144\eta + 232\eta^{2} + 160\eta^{3} + 44\eta^{4} }}{{\left[ {4(1 + \eta )(2 + 4\eta + \eta^{2} )} \right]^{2} }} > 0$, $\frac{\partial A}{\partial \eta } = A_{1}$, thus $\frac{{\partial \varPi_{S} (PNC )}}{\partial \eta } = \frac{{4A_{1} \alpha^{2} \gamma^{2} }}{{\left[ {4\gamma - A\beta^{2} } \right]^{2} }} > 0$. Further, we have

$$\frac{{\partial \varPi_{{R_{i} }} (PNC)}}{\partial \eta } = \frac{{\alpha^{2} \gamma^{3} (4\gamma - A\beta^{2} )(AA_{4} + A_{5} )\left( {G - \frac{{4A_{8} }}{{AA_{8} + A_{9} }}} \right)}}{{\left[ {2(1 + \eta )\left( {2 + 4\eta + \eta^{2} } \right)^{2} \left( {4\gamma - A\beta^{2} } \right)^{2} } \right]^{2} }}$$

(23)

$$\begin{aligned} A_{4} & = \left[ \begin{aligned} (8 + 40\eta + 70\eta^{2} + 48\eta^{3} + 9\eta^{4} )(40 + 144\eta + 168\eta^{2} + 72\eta^{3} + 10\eta^{4} ) - \hfill \\ (8 + 40\eta + 72\eta^{2} + 56\eta^{3} + 18\eta^{4} + 2\eta^{5} )(40 + 140\eta + 144\eta^{2} + 36\eta^{3} ) \hfill \\ \end{aligned} \right] > 0, \\ A_{5} & = 2A_{1} (8 + 40\eta + 72\eta^{2} + 56\eta^{3} + 18\eta^{4} + 2\eta^{5} )(8 + 40\eta + 70\eta^{2} + 48\eta^{3} + 9\eta^{4} ) > 0. \\ \end{aligned}$$

Thus, when $0 < G < \frac{{4A_{4} }}{{AA_{4} + A_{5} }}$, we obtain $\frac{{\partial \varPi_{{R_{i} }} (PNC)}}{\partial \eta } < 0$; when $\frac{{4A_{4} }}{{AA_{4} + A_{5} }} < G < \frac{4}{A}$, we obtain $\frac{{\partial \varPi_{{R_{i} }} (PNC)}}{\partial \eta } > 0$.

Similarly, we have

$$\frac{{\partial \varPi_{{R_{j,j \ne i} }} (PNC)}}{\partial \eta } = \frac{{\alpha^{2} \gamma^{3} (4\gamma - A\beta^{2} )\left[ {A(A_{7} + A_{8} - A_{6} ) + A_{9} } \right]\left[ {G - \frac{{4(A_{7} + A_{8} - A_{6} )}}{{A(A_{7} + A_{8} - A_{6} ) + A_{9} }}} \right]^{{}} }}{{\left[ {4(1 + \eta )^{2} \left( {2 + 4\eta + \eta^{2} } \right)^{2} \left( {4\gamma - A\beta^{2} } \right)^{2} } \right]^{2} }}$$

(24)

$$\begin{aligned} A_{6} & = (192 + 1072\eta + 2272\eta^{2} + 2232\eta^{3} + 990\eta^{4} + 150\eta^{5} )(8 + 40\eta + 72\eta^{2} + 56\eta^{3} + 18\eta^{4} + 2\eta^{5} ), \\ A_{7} & = 2(16 + 96\eta + 220\eta^{2} + 232\eta^{3} + 105\eta^{4} + 18\eta^{5} )(8 + 40\eta + 72\eta^{2} + 56\eta^{3} + 18\eta^{4} + 2\eta^{5} ), \\ A_{8} & = 2(1 + \eta )(16 + 96\eta + 220\eta^{2} + 232\eta^{3} + 105\eta^{4} + 18\eta^{5} )(40 + 144\eta + 168\eta^{2} + 72\eta^{3} + 10\eta^{4} ), \\ A_{9} & = 2A_{1} (1 + \eta )(16 + 96\eta + 220\eta^{2} + 232\eta^{3} + 105\eta^{4} + 18\eta^{5} )(16 + 80\eta + 144\eta^{2} + 112\eta^{3} + 36\eta^{4} + 4\eta^{5} ). \\ \end{aligned}$$

Thus, when $0 < G < \frac{{4(A_{7} + A_{8} - A_{6} )}}{{A(A_{7} + A_{8} - A_{6} ) + A_{9} }}$, we obtain $\frac{{\partial \varPi_{{R_{j,j \ne i} }} (PNC)}}{\partial \eta } < 0$; when $\frac{{4(A_{7} + A_{8} - A_{6} )}}{{A(A_{7} + A_{8} - A_{6} ) + A_{9} }} < G < \frac{4}{A}$, we obtain $\frac{{\partial \varPi_{{R_{j,j \ne i} }} (PNC)}}{\partial \eta } > 0$. In summary, when $0 < G < \hbox{min} G_{2}$, we have $\frac{{\partial \varPi_{{R_{i} }}^{{}} (PBC)}}{\partial \eta } = \frac{{\partial \varPi_{{R_{j,j \ne i} }}^{{}} (PBC)}}{\partial \eta } < 0$, $\frac{{\partial \varPi_{{R_{i} }}^{{}} (PNC)}}{\partial \eta } < 0$, $\frac{{\partial \varPi_{{R_{j,j \ne i} }}^{{}} (PNC)}}{\partial \eta } < 0$; when $\hbox{max} G_{2} < G < \frac{2(2 + \eta )}{1 + \eta }$, we have $\frac{{\partial \varPi_{{R_{i} }}^{{}} (PBC)}}{\partial \eta } = \frac{{\partial \varPi_{{R_{j,j \ne i} }}^{{}} (PBC)}}{\partial \eta } > 0$, $\frac{{\partial \varPi_{{R_{i} }}^{{}} (PNC)}}{\partial \eta } > 0$, $\frac{{\partial \varPi_{{R_{j,j \ne i} }}^{{}} (PNC)}}{\partial \eta } > 0$, $\hbox{min} G_{2} = \hbox{min} \left\{ {\frac{2\eta }{1 + \eta },\frac{{4A_{4} }}{{AA_{4} + A_{5} }},\frac{{4(A_{7} + A_{8} - A_{6} )}}{{A(A_{7} + A_{8} - A_{6} ) + A_{9} }}} \right\}$, $\hbox{max} G_{2} = \hbox{max} \left\{ {\frac{2\eta }{1 + \eta },\frac{{4A_{4} }}{{AA_{4} + A_{5} }},\frac{{4(A_{7} + A_{8} - A_{6} )}}{{A(A_{7} + A_{8} - A_{6} ) + A_{9} }}} \right\}$.$\square$

Proof of Proposition 4

• (1) Comparison of $w^{C}$ and $w^{M}$

According to Lemmas 1 and 2, under the PBC mode, we have

$$w^{PBC} - w^{M} = \frac{\alpha \gamma (2 + \eta )}{{2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )}} - \frac{2\alpha \gamma }{{4\gamma - \beta^{2} }} = \frac{{\alpha \gamma \eta \beta^{2} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} (4\gamma - \beta^{2} )}} > 0$$

(25)

Under the PNC mode, we have

$$w^{PNC} - w^{M} = \frac{2\alpha \gamma }{{4\gamma - A\beta^{2} }} - \frac{2\alpha \gamma }{{4\gamma - \beta^{2} }} = \frac{{2\alpha \gamma \beta^{2} (A - 1)}}{{(4\gamma - A\beta^{2} )(4\gamma - \beta^{2} )}}$$

(26)

Since $A = \frac{{8 + 28\eta + 29\eta^{2} + 8\eta^{3} }}{{4(1 + \eta )\left( {2 + 4\eta + \eta^{2} } \right)^{{}} }}$, we have $A - 1 > 0$, thus $w^{PNC} - w^{M} > 0$. Further, we obtain

$$w^{PBC} - w^{PNC} = \frac{{\alpha \gamma \left[ {4\gamma + \beta^{2} [2(1 + \eta ) - A(2 + \eta )]} \right]^{{}} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} (4\gamma - \beta^{2} )}}$$

(27)

Since $2(1 + \eta ) - A(2 + \eta ) > 0$, we have $w^{PBC} - w^{PNC} > 0$, that is $w^{PBC} > w^{PNC}$.

• (2) Comparison of $\theta^{C}$ and $\theta^{M}$

Under the PBC mode, we have

$$\theta^{PBC} - \theta^{M} = \frac{\alpha \beta (1 + \eta )}{{2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )}} - \frac{\alpha \beta }{{4\gamma - \beta^{2} }} = \frac{2\alpha \gamma \eta \beta }{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} (4\gamma - \beta^{2} )}} > 0$$

(28)

Under the PNC mode, we have

$$\theta^{PNC} - \theta^{M} = \frac{A\alpha \beta }{{4\gamma - A\beta^{2} }} - \frac{\alpha \beta }{{4\gamma - \beta^{2} }} = \frac{4\alpha \gamma \beta (A - 1)}{{(4\gamma - A\beta^{2} )(4\gamma - \beta^{2} )}} > 0$$

(29)

Further, we obtain

$$\theta^{PBC} - \theta^{PNC} = \frac{\alpha \gamma \beta [4(1 + \eta ) - 2A(2 + \eta )]}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} (4\gamma - \beta^{2} )}}$$

(30)

Since $4(1 + \eta ) - 2A(2 + \eta ) > 0$, we have $\theta^{PBC} - \theta^{PNC} > 0$, that is $\theta^{PBC} > \theta^{PNC}$.

• (3) Comparison of $Q^{C}$ and $Q^{M}$

According to Lemmas 1 and 2, under the PBC mode, we have

$$Q^{PBC} - Q^{M} = \frac{\alpha \gamma (1 + \eta )}{{2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )}} - \frac{\alpha \gamma }{{4\gamma - \beta^{2} }} = \frac{{2\alpha \gamma^{2} \eta }}{{\left( {4\gamma - \beta^{2} } \right)\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} }} > 0$$

(31)

By the same calculation, we obtain $Q^{PNC}$ is greater than $Q^{M}$ under the PNC mode.

• (4) Comparison of $p_{i}^{C}$ and $P^{M}$, $p_{j ,j \ne i}^{C}$ and $P^{M}$

Under the PBC mode, we have

$$p_{i}^{PBC} - p_{{}}^{M} = p_{j ,j \ne i}^{PBC} - p_{{}}^{M} = \frac{{2\alpha \gamma^{2} \eta \left( {\frac{{\beta^{2} }}{\gamma } - 1} \right)}}{{\left( {4\gamma - \beta^{2} } \right)\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} }} = \frac{{2\alpha \gamma^{2} \eta \left( {G - 1} \right)}}{{\left( {4\gamma - \beta^{2} } \right)\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} }}$$

(32)

Thus, when $0 < G < 1$, we obtain $p_{i}^{PBC} = p_{j ,j \ne i}^{PBC} < P^{M}$; when $1 < G < \frac{2(2 + \eta )}{1 + \eta }$, we obtain $p_{i}^{PBC} = p_{j ,j \ne i}^{PBC} > P^{M}$.

Under the PNC mode, we have

$$p_{i}^{PNC} - p_{{}}^{M} = \frac{{A_{10} \alpha \gamma^{2} \left( {\frac{{\beta^{2} }}{\gamma } - \frac{{4(\eta + \eta^{2} )}}{{A_{10} }}} \right)}}{{\left( {4\gamma - \beta^{2} } \right)\left( {4\gamma - A\beta^{2} } \right)\left( {2 + 4\eta + \eta^{2} } \right)}},\;p_{j ,j \ne i}^{PNC} - p_{{}}^{M} = \frac{{A_{11} \alpha \gamma^{2} \left( {\frac{{\beta^{2} }}{\gamma } - \frac{{4(2\eta + 5\eta^{2} + 2\eta^{3} )}}{{A_{11} }}} \right)}}{{2(1 + \eta )\left( {2 + 4\eta + \eta^{2} } \right)^{{}} \left( {4\gamma - \beta^{2} } \right)\left( {4\gamma - A\beta^{2} } \right)}}$$

(33)

$A_{10} = (A - 1)(6 + 11\eta + 2\eta^{2} ) + A(\eta + \eta^{2} )$. Thus, when $0 < G < \frac{{4(\eta + \eta^{2} )}}{{A_{10} }}$, we obtain $p_{i}^{PNC} < P^{M}$; when $\frac{{4(\eta + \eta^{2} )}}{{A_{10} }} < G < \frac{4}{A}$, we obtain $p_{i}^{PNC} > P^{M}$.

$A_{11} = (A - 1)(12 + 34\eta + 25\eta^{2} + 4\eta^{3} ) + A(2\eta + 5\eta^{2} + 2\eta^{3} )$. Thus, when $0 < G < \frac{{4(2\eta + 5\eta^{2} + 2\eta^{3} )}}{{A_{11} }}$, we have $p_{j ,j \ne i}^{PNC} < P^{M}$; when $\frac{{4(2\eta + 5\eta^{2} + 2\eta^{3} )}}{{A_{11} }} < G < \frac{4}{A}$, we have $p_{j ,j \ne i}^{PNC} > P^{M}$.

• (5) Comparison of $p_{i}^{PBC}$ and $p_{j ,j \ne i}^{PBC}$, $p_{i}^{PNC}$ and $p_{j ,j \ne i}^{PNC}$

  $$p_{i}^{PBC} - p_{i}^{PNC} = \frac{{A_{12} \alpha \gamma^{2} \left( {\frac{{\beta^{2} }}{\gamma } - \frac{{2\eta^{2} }}{{A_{6} }}} \right)}}{{\left[ {2\gamma ( 2+ \eta )- \beta^{2} ( 1+ \eta )} \right]\left( {2 + 4\eta + \eta^{2} } \right)^{{}} \left( {4\gamma - A\beta^{2} } \right)}}$$

  (34)

$A_{12} = (6 + 17\eta + 13\eta^{2} + 2\eta^{3} ) - A(6 + 14\eta + 7\eta^{2} + \eta^{3} ) > 0$. Thus, when $0 < G < \frac{{2\eta^{2} }}{{A_{12} }}$, we have $p_{i}^{PBC} < p_{i}^{PNC}$; when $\frac{{2\eta^{2} }}{{A_{12} }} < G < \frac{4}{A}$, we have $p_{i}^{PBC} > p_{i}^{PNC}$.

$$p_{j ,j \ne i}^{PBC} - p_{j ,j \ne i}^{PNC} = \frac{{A_{13} \alpha \gamma^{2} \left( {\frac{{\beta^{2} }}{\gamma } - \frac{{2\eta^{3} }}{{A_{7} }}} \right)}}{{2\left[ {2\gamma ( 2+ \eta )- \beta^{2} ( 1+ \eta )} \right](1 + \eta )\left( {2 + 4\eta + \eta^{2} } \right)^{{}} \left( {4\gamma - A\beta^{2} } \right)}}$$

(35)

$A_{13} = (12 + 46\eta + 59\eta^{2} + 29\eta^{3} + 4\eta^{4} ) - A(12 + 40\eta + 42\eta^{2} + 16\eta^{3} + 2\eta^{4} ) > 0$. Thus, when $0 < G < \frac{{2\eta^{3} }}{{A_{13} }}$, we have $p_{j ,j \ne i}^{PBC} < p_{j ,j \ne i}^{PNC}$; when $\frac{{2\eta^{3} }}{{A_{13} }} < G < \frac{4}{A}$, we have $p_{j ,j \ne i}^{PBC} > p_{j ,j \ne i}^{PNC}$.

In summary, when $0 < G < \hbox{min} G_{3}$, we obtain $p_{i}^{PBC} = p_{j ,j \ne i}^{PBC} < p_{j ,j \ne i}^{PNC} < p_{i}^{PNC} < p_{{}}^{M}$; when $\hbox{max} G_{3} < G < \frac{2(2 + \eta )}{1 + \eta }$, we obtain $p_{{}}^{M} < p_{j ,j \ne i}^{PNC} < p_{i}^{PNC} < p_{i}^{PBC} = p_{j ,j \ne i}^{PBC}$.

$$\begin{aligned} & \min G_{3} = \min \left\{ {1,\frac{{4{\text{(}}\eta + 4\eta ^{2} {\text{)}}}}{{A_{{10}} }}{\text{,}}\frac{{4{\text{(2}}\eta + 5\eta ^{2} + 2\eta ^{2} {\text{)}}}}{{A_{{11}} }}{\text{,}}\frac{{2\eta ^{2} }}{{A_{{12}} }}{\text{,}}\frac{{2\eta ^{3} }}{{A_{{13}} }}} \right\},\;\max G_{3} = \max \left\{ {1,\frac{{4{\text{(}}\eta + 4\eta ^{2} {\text{)}}}}{{A_{{10}} }}{\text{,}}\frac{{4{\text{(2}}\eta + 5\eta ^{2} + 2\eta ^{2} {\text{)}}}}{{A_{{11}} }}{\text{,}}\frac{{2\eta ^{2} }}{{A_{{12}} }}{\text{,}}\frac{{2\eta ^{3} }}{{A_{{13}} }}} \right\}, \\ & A_{{10}} = (A - 1)(6 + 11\eta + 2\eta ^{2} ) + A(\eta + \eta ^{2} ),\;A_{{11}} = (A - 1)(12 + 34\eta + 25\eta ^{2} + 4\eta ^{3} ) + A(2\eta + 5\eta ^{2} + 2\eta ^{3} ), \\ & A_{{12}} = (6 + 17\eta + 13\eta ^{2} + 2\eta ^{3} ) - A(6 + 14\eta + 7\eta ^{2} + \eta ^{3} ), \\ & A_{{13}} = (12 + 46\eta + 59\eta ^{2} + 29\eta ^{3} + 4\eta ^{4} ) - A(12 + 40\eta + 42\eta ^{2} + 16\eta ^{3} + 2\eta ^{4} ). \\ \end{aligned}$$

$$\square$$

Proof of Proposition 5

• (1) Comparison of $\varPi_{S} (C)$ and $\varPi_{S} (M)$

Under the PBC mode, we have

$$\varPi_{S} (PBC) - \varPi_{S} (M) = \frac{{\alpha^{2} \gamma (1 + \eta )}}{{2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )}} - \frac{{\alpha^{2} \gamma }}{{2(4\gamma - \beta^{2} )}} = \frac{{\alpha^{2} \gamma \left[ {4\gamma \eta + 2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} }}{{2\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} (4\gamma - \beta^{2} )}} > 0$$

(36)

Under the PNC mode, we have

$$\varPi_{S} (PNC) - \varPi_{S} (M) = \frac{{A\alpha^{2} \gamma }}{{4\gamma - A\beta^{2} }} - \frac{{\alpha^{2} \gamma }}{{2(4\gamma - \beta^{2} )}} = \frac{{\alpha^{2} \gamma \left[ {4\gamma (A - 1) + A(4\gamma - \beta^{2} )} \right]^{{}} }}{{2(4\gamma - A\beta^{2} )(4\gamma - \beta^{2} )}} > 0$$

(37)

Further, we obtain

$$\varPi_{S} (PBC) - \varPi_{S} (PNC) = \frac{{\alpha^{2} \gamma [4(1 + \eta ) - 2A(2 + \eta )]}}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{{}} (4\gamma - \beta^{2} )}} > 0$$

(38)

Thus, $\varPi_{S} (PBC) > \varPi_{S} (PNC)$.

• (2) Comparison of $\varPi_{R} (M)$, $\varPi_{{R_{i} }} (C)$ and $\varPi_{{R_{j,j \ne i} }} (C)$

Under the PBC mode, we have

$$\varPi_{R} (M) - \varPi_{{R_{i} }} (PBC) = \varPi_{R} (M) - \varPi_{{R_{j,j \ne i} }} (PBC) = \frac{{\alpha^{2} \gamma^{4} \left[ {4\eta^{2} + G^{2} (\eta + \eta^{2} ) - G(\eta + \eta^{2} )} \right]^{{}} }}{{\left[ {2\gamma (2 + \eta ) - \beta^{2} (1 + \eta )} \right]^{2} \left( {4\gamma - \beta^{2} } \right)^{2} }}$$

(39)

Let $F(G) = 4\eta^{2} + G^{2} (\eta + \eta^{2} ) - G(\eta + \eta^{2} ) = 0$, we obtain $G_{{}}^{*} = 2 \pm \frac{2}{{\sqrt {1 + \eta } }}$, and $2 + \frac{2}{{\sqrt {1 + \eta } }} > \frac{2(2 + \eta )}{1 + \eta }$.

Thus, when $0 < G < 2 - \frac{2}{{\sqrt {1 + \eta } }}$, we have $\varPi_{R} (M) > \varPi_{{R_{i} }} (PBC) = \varPi_{{R_{j,j \ne i} }} (PBC)$; when $2 - \frac{2}{{\sqrt {1 + \eta } }} < G < \frac{2(2 + \eta )}{1 + \eta }$, we have $\varPi_{R} (M) < \varPi_{{R_{i} }} (PBC) = \varPi_{{R_{j,j \ne i} }} (PBC)$.

Under the PNC mode, we have

$$\varPi_{R} (M) - \varPi_{{R_{i} }} (PNC) = \frac{{\alpha^{2} \gamma^{4} \left[ {aG^{2} + bG + c} \right]^{{}} }}{{2(1 + \eta )\left( {2 + 4\eta + \eta^{2} } \right)^{2} \left( {4\gamma - A\beta^{2} } \right)^{2} \left( {4\gamma - \beta^{2} } \right)^{2} }}$$

(40)

Let $F(G) = aG^{2} + bG + c = 0$, we obtain $G_{{}}^{*} = \frac{{ - b \pm \sqrt {b^{2} - 4ac} }}{2a}$, and $\frac{{ - b + \sqrt {b^{2} - 4ac} }}{2a} > \frac{4}{A}$; $a = 2A^{2} (1 + \eta )(2 + 4\eta + \eta^{2} )^{2} - B(2 + 3\eta )$, $b = - 8[2A(1 + \eta )(2 + 4\eta + \eta^{2} )^{2} - B(2 + 3\eta )]$, $c = 16[2(1 + \eta )(2 + 4\eta + \eta^{2} )^{2} - B(2 + 3\eta )]$. Thus, when $0 < G < \frac{{ - b - \sqrt {b^{2} - 4ac} }}{2a}$, we have $\varPi_{R} (M) > \varPi_{{R_{i} }} (PNC)$; when $\frac{{ - b - \sqrt {b^{2} - 4ac} }}{2a} < G < \frac{4}{A}$, we have $\varPi_{R} (M) < \varPi_{{R_{i} }} (PNC)$. In the same way, when $0 < G < \frac{{ - b_{1} - \sqrt {b_{1}^{2} - 4a_{1} c_{1} } }}{{2a_{1} }}$, we have $\varPi_{R} (M) > \varPi_{{R_{j,j \ne i} }} (PNC)$; when $\frac{{ - b_{1} - \sqrt {b_{1}^{2} - 4a_{1} c_{1} } }}{{2a_{1} }} < G < \frac{4}{A}$, we have $\varPi_{R} (M) < \varPi_{{R_{j,j \ne i} }} (PNC)$. Wherein, $a_{1}$, $b_{1}$ and $c_{1}$ can be obtained by $\varPi_{R} (M) - \varPi_{{R_{j,j \ne i} }} (PNC) = 0$. Therefore, when $0 < G < \frac{{ - b_{1} - \sqrt {b_{1}^{2} - 4a_{1} c_{1} } }}{{2a_{1} }}$, we have $\varPi_{{R_{j,j \ne i} }} (PNC) < \varPi_{{R_{i} }} (PBC) = \varPi_{{R_{j,j \ne i} }} (PBC) < \varPi_{{R_{i} }} (PNC)$, when $\frac{{ - b_{1} - \sqrt {b_{1}^{2} - 4a_{1} c_{1} } }}{{2a_{1} }} < G < \frac{4}{A}$, we have $\varPi_{{R_{j,j \ne i} }} (PNC) < \varPi_{{R_{i} }} (PNC) < \varPi_{{R_{i} }} (PBC) = \varPi_{{R_{j,j \ne i} }} (PBC)$.

In summary, when $0 < G < \hbox{min} G_{4}$, $\varPi_{{R_{j,j \ne i} }} (PNC) < \varPi_{{R_{i} }} (PBC) = \varPi_{{R_{j,j \ne i} }} (PBC) < \varPi_{{R_{i} }} (PNC) < \varPi_{R} (M)$; when $\hbox{max} G_{4} < G < \frac{2(2 + \eta )}{1 + \eta }$, we have $\varPi_{R} (M) < \varPi_{{R_{j ,j \ne i} }} (PNC) < \varPi_{{R_{i} }} (PNC) < \varPi_{{R_{i} }} (PBC) = \varPi_{{R_{j ,j \ne i} }} (PBC)$.

$$\begin{aligned} & \hbox{min} G_{4} = \hbox{min} \left\{ {1 ,2 - \frac{2}{{\sqrt {1 + \eta } }} ,\frac{{ - b_{1} - \sqrt {b_{1}^{2} - 4a_{1} c_{1} } }}{{2a_{1} }}} \right\},\;\hbox{max} G_{3} = \hbox{max} \left\{ {1 ,2 - \frac{2}{{\sqrt {1 + \eta } }} ,\frac{{ - b_{1} - \sqrt {b_{1}^{2} - 4a_{1} c_{1} } }}{{2a_{1} }}} \right\}, \\ & a = 2A^{2} (1 + \eta )(2 + 4\eta + \eta^{2} )^{2} - B(2 + 3\eta ),\;b = - 8[2A(1 + \eta )(2 + 4\eta + \eta^{2} )^{2} - B(2 + 3\eta )], \\ & c = 16[2(1 + \eta )(2 + 4\eta + \eta^{2} )^{2} - B(2 + 3\eta )]. \\ \end{aligned}$$

$\square$

Proof of Proposition 6

In the case of effective implementation of the EPBC mode, when $\kappa_{i} > \kappa_{j,j \ne i}$, we have

$$\begin{aligned} & p_{i}^{EPBC} - p_{j,j \ne i}^{EPBC} = \frac{{(\kappa_{i} - \kappa_{j,j \ne i} )\theta^{EPBC} }}{(2 + 3\eta )} > 0,\;Q_{i}^{EPBC} - Q_{j,j \ne i}^{EPBC} = \frac{{(1 + \eta )(\kappa_{i} - \kappa_{j,j \ne i} )\theta^{EPBC} }}{(2 + 3\eta )} > 0 \\ & \varPi_{{R_{i} }} (EPBC) - \varPi_{{R_{j,j \ne i} }} (EPBC) = (p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} - (p_{j,j \ne i}^{EPBC} - w^{EPBC} )Q_{j,j \ne i}^{EPBC} > 0. \\ \end{aligned}$$

When $\kappa_{i} < \kappa_{j,j \ne i}$, we have

$$\begin{aligned} & p_{i}^{EPBC} - p_{j,j \ne i}^{EPBC} = \frac{{(\kappa_{i} - \kappa_{j,j \ne i} )\theta^{EPBC} }}{(2 + 3\eta )} < 0,\;Q_{i}^{EPBC} - Q_{j,j \ne i}^{EPBC} = \frac{{(1 + \eta )(\kappa_{i} - \kappa_{j,j \ne i} )\theta^{EPBC} }}{(2 + 3\eta )} < 0, \\ & \varPi_{{R_{i} }} (EPBC) - \varPi_{{R_{j,j \ne i} }} (EPBC) = (p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} - (p_{j,j \ne i}^{EPBC} - w^{EPBC} )Q_{j,j \ne i}^{EPBC} < 0. \\ \end{aligned}$$

Finally, combined with the constraint of effective implementation of the EPBC mode, we obtain $0 < \kappa_{i} + \kappa_{j ,j \ne i} < \sqrt {8\gamma (2 + \eta )/(1 + \eta )} - 2\beta$.

Let $\kappa = \kappa_{i} + \kappa_{j ,j \ne i} = \sqrt {8\gamma (2 + \eta )/(1 + \eta )} - 2\beta$, we have $\frac{d\kappa }{d\beta } = - 2 < 0$, That is, $\kappa$ decreases with $\beta$. Therefore, the total technology marketing effect ($\kappa = \kappa_{i} + \kappa_{j ,j \ne i}$) and the technology spillover effect $\beta$ form a complementary relationship; when $\kappa_{i}$ is constant, we can verify that the technology marketing effect $\kappa_{j ,j \ne i}$ and the technology spillover effect $\beta$ form a complementary relationship; when $\kappa_{j ,j \ne i}$ is unchanged, we can also verify that the technology marketing effect $\kappa_{i}$ and the technology spillover effect $\beta$ form a complementary relationship. $\square$

Proof of Proposition 7

• (1) After comparing PBC mode and EPBC mode, we have

  $$\varPi_{S} (EPBC) - \varPi_{S} (PBC) = w^{EPBC} (Q_{i}^{EPBC} + Q_{j ,j \ne i}^{EPBC} )- \gamma (\theta^{EPBC} )^{2} - \frac{{\alpha^{2} (1 + \eta )}}{2(2 + \eta ) - G(1 + \eta )} > 0$$

  (41)

Therefore, the optimal strategic mode of technology service provider is EPBC mode.

• (2) After comparing PBC mode and EPBC mode, we have

  $$\varPi_{{R_{i} }} (EPBC) - \varPi_{{R_{i} }} (PBC) = (p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} - F - \frac{{\alpha^{2} (1 + \eta )}}{{\left[ {2(2 + \eta ) - G(1 + \eta )} \right]^{2} }}$$

  (42)

Let $\varPi_{{R_{i} }} (EPBC) - \varPi_{{R_{i} }} (PBC) = 0$, we obtain

$$F = (p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} - \frac{{\alpha^{2} (1 + \eta )}}{{\left[ {2(2 + \eta ) - G(1 + \eta )} \right]^{2} }}$$

(43)

When $\kappa_{i} > \kappa_{j,j \ne i}$, two thresholds of $F$ are

$$F_{1} = (p_{j}^{EPBC} - w^{EPBC} )Q_{j}^{EPBC} \left| {_{{\kappa_{i}^{ *} + \kappa_{j,j \ne i}^{ *} }} } \right. - \frac{{\alpha^{2} (1 + \eta )}}{{\left[ {2(2 + \eta ) - G(1 + \eta )} \right]^{2} }}$$

(44)

$$F_{2} = (p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} \left| {_{{\kappa_{i}^{ *} + \kappa_{j,j \ne i}^{ *} }} } \right. - \frac{{\alpha^{2} (1 + \eta )}}{{\left[ {2(2 + \eta ) - G(1 + \eta )} \right]^{2} }}$$

(45)

Wherein, $\kappa_{i}^{ *} + \kappa_{j,j \ne i}^{ *} = \arg \hbox{max} \{ 8\gamma (2 + \eta ) - (1 + \eta )(2\beta + \kappa_{i} + \kappa_{j ,j \ne i} )^{2} \}$, thus $F_{1} < F_{2}$.

If $F > F_{2}$, we have $\varPi_{{R_{i} }} (EPBC) - \varPi_{{R_{i} }} (PBC) < 0$, $\varPi_{{R_{j} }} (EPBC) - \varPi_{{R_{j} }} (PBC) < 0$, that is, the optimal strategic mode of retailer $R_{i}$ and $R_{j,j \ne i}$ is PBC; if $F_{1} < F < F_{2}$, we have $\varPi_{{R_{i} }} (EPBC)\left| {_{{\kappa_{i}^{ *} + \kappa_{j,j \ne i}^{ *} }} } \right. - \varPi_{{R_{i} }} (PBC) > 0$, $\varPi_{{R_{j} }} (EPBC) - \varPi_{{R_{j} }} (PBC) < 0$, that is, the optimal strategic mode of retailer $R_{i}$ is EPBC mode, while the optimal strategic mode of retailer $R_{j,j \ne i}$ is PBC mode; if $F < F_{1}$, we have $\varPi_{{R_{i} }} (EPBC) - \varPi_{{R_{i} }} (PBC) > 0$, $\varPi_{{R_{j} }} (EPBC) - \varPi_{{R_{j} }} (PBC) > 0$, that is, the optimal strategic mode of retailer $R_{i}$ and $R_{j,j \ne i}$ is EPBC mode. When $\kappa_{i} < \kappa_{j,j \ne i}$, the same conclusion can be obtained according to the above analysis. Therefore, there exists two thresholds $F^{*} > 0$ and $F^{**} ( \ge F^{*} )$ such that: if $F < F^{*}$, EPBC mode is the optimal strategic choice of two retailers; if $F^{*} < F < F^{**}$, the strategic mode choice between retailer $R_{i}$ and $R_{j,j \ne i}$ aren’t consistent (that is, retailer $R_{j,j \ne i}$ with smaller technology marketing effect will choose PBC mode, while retailer $R_{i}$ with greater technology marketing effect will choose EPBC mode); if $F > F^{**}$, PBC mode is the optimal strategic choice of two retailers.

$$F^{*} = \hbox{min} \left\{ {(p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} ,(p_{j}^{EPBC} - w^{EPBC} )Q_{j}^{EPBC} } \right\}\left| {_{{\kappa_{i}^{*} + \kappa_{j,j \ne i}^{*} }} } \right. - \frac{{\alpha^{2} (1 + \eta )}}{{\left[ {2(2 + \eta ) - G(1 + \eta )} \right]^{2} }}$$

(46)

$$F^{ * *} = \hbox{max} \left\{ {(p_{i}^{EPBC} - w^{EPBC} )Q_{i}^{EPBC} ,(p_{j}^{EPBC} - w^{EPBC} )Q_{j}^{EPBC} } \right\}\left| {_{{\kappa_{i}^{ *} + \kappa_{j,j \ne i}^{ *} }} } \right. - \frac{{\alpha^{2} (1 + \eta )}}{{\left[ {2(2 + \eta ) - G(1 + \eta )} \right]^{2} }}$$

(47)

$$\kappa_{i}^{*} + \kappa_{j,j \ne i}^{*} = \arg \hbox{max} \{ 8\gamma (2 + \eta ) - (1 + \eta )(2\beta + \kappa_{i} + \kappa_{j,j \ne i} )^{2} \}$$

(48)

$\square$