Information services and omnichannel retailing strategy choices of e-commerce platforms with supplier competition

Abstract

The integration of offline channels and online channels has become a direction for the transformation and development of many e-commerce platforms, so omnichannel retailing has become an important strategy. Thus, we build a theoretical model between a platform and two suppliers to study the platform’s optimal information service and omnichannel strategy choice (under a reselling agreement vs. under a hybrid retailing agreement). The results show that when the product competition intensity is low, an omnichannel strategy under a reselling agreement is conducive to increasing the offline information service level and profits of the platform. When the product competition intensity is high, the omnichannel strategy under a hybrid retailing agreement is the optimal choice, and the platform can set a smaller referral fee to transfer part of the profit to suppliers to encourage suppliers to enter omnichannel retailing. Furthermore, the platform can improve the information service level using new retailing technology, which can create a higher product showrooming effect. Finally, we introduce the case of China’s household appliances to verify the effectiveness of our research and suggest that the platform establishes experience stores that can weaken the competitive relationship between suppliers and the platform to achieve a win–win strategy.

Appendices

Appendix 1

1.1 Proof of Proposition 1

According to (8), we have

$$ \frac{{\partial \hat{\Pi }_{A}^{\emptyset } }}{\partial \gamma } = \frac{{\partial \hat{\Pi }_{B}^{\emptyset } }}{\partial \gamma } = \frac{1}{{(2 - \gamma )^{3} }} $$

(51)

$$ \frac{{\partial \hat{\Pi }_{P}^{\emptyset } }}{\partial \gamma } = \frac{4 - 3\gamma }{{2(1 - \gamma )^{2} (2 - \gamma )^{3} }} $$

(52)

Since $0 < \gamma < 1$, we verify that $\partial \hat{\Pi }_{A}^{\emptyset } /\partial \gamma > 0$, $\partial \hat{\Pi }_{B}^{\emptyset } /\partial \gamma > 0$, and $\partial \hat{\Pi }_{P}^{\emptyset } /\partial \gamma > 0$. That is, the equilibrium profits ($\hat{\Pi }_{A}^{\emptyset }$,$\hat{\Pi }_{B}^{\emptyset }$, and $\hat{\Pi }_{P}^{\emptyset }$) for the supply chain members increase with the increase of the competition intensity ($\gamma$) between two products. Q.E.D.

1.2 Proof of Lemma 1

Under the reselling agreement, if the platform establishes an experience store to open an omni-channel retailing, we can obtain the equilibrium profits for the suppliers and platform respectively are $\hat{\Pi }_{A}^{{O\overline{E}}}$,$\hat{\Pi }_{B}^{{O\overline{E}}}$, and $\hat{\Pi }_{P}^{{O\overline{E}}}$. In order to ensure these equilibrium results are non-negative, we need $\Phi = 2(1 - \gamma^{2} ) - \alpha (\alpha + \beta \gamma ) - \beta (\beta + \alpha \gamma ) > 0$, i.e., $0 < \gamma < \hat{\gamma }^{*} (\alpha ,\beta )$. For obtaining the threshold $\hat{\gamma }^{*} (\alpha ,\beta )$, we solve the equation $(1 - \gamma^{2} ) - \alpha (\alpha + \beta \gamma ) - \beta (\beta + \alpha \gamma ) = 0$ and have

$$ \hat{\gamma }^{*} (\alpha ,\beta ) = \frac{{\sqrt {\alpha^{2} \beta^{2} + 2(2 - \alpha^{2} - \beta^{2} )} - \alpha \beta }}{2} $$

(53)

Q.E.D.

1.3 Proof of Proposition 2

When $0 < \gamma < \hat{\gamma }^{*} (\alpha ,\beta )$, We verify that $\partial \hat{\Pi }_{A}^{{O\overline{E}}} /\partial \gamma > 0$, $\partial \hat{\Pi }_{B}^{{O\overline{E}}} /\partial \gamma > 0$, $\partial \hat{\Pi }_{P}^{{O\overline{E}}} /\partial \gamma > 0$. That is, the equilibrium profits ($\hat{\Pi }_{A}^{{O\overline{E}}}$,$\hat{\Pi }_{B}^{{O\overline{E}}}$, and $\hat{\Pi }_{P}^{{O\overline{E}}}$) for the supply chain members increase with the increase of the competition intensity $\gamma$.

Next, we discuss the impact of $\alpha$ and $\beta$ on the equilibrium profits ($\hat{\Pi }_{A}^{{O\overline{E}}}$,$\hat{\Pi }_{B}^{{O\overline{E}}}$, and $\hat{\Pi }_{P}^{{O\overline{E}}}$). Due to the complicated calculation process, we draw on the proof process of Tian et al. [10] to prove the equilibrium profits ($\hat{\Pi }_{A}^{{O\overline{E}}}$,$\hat{\Pi }_{B}^{{O\overline{E}}}$, and $\hat{\Pi }_{P}^{{O\overline{E}}}$) for the supply chain members increase with the increase of the product showrooming effect $\alpha$ and $\beta$. We first examine the limiting situation ($\gamma = 0$ and $\gamma = \hat{\gamma }^{*} (\alpha ,\beta )$), and then complete the proof using a continuity argument.

When $\gamma = 0$, we obtain (53) according to (18).

$$ \hat{\Pi }_{A}^{{O\overline{E}}} = \frac{{\left\{ \begin{gathered} [2(8 - 4\alpha^{2} ) + (8\alpha - 2\alpha^{3} - 4)\beta + (6\alpha^{2} - 32)\beta^{2} + (\alpha^{3} - 4\alpha + 2)\beta^{3} + \hfill \\ (12 - \alpha^{2} )\beta^{4} ] \times [(8 - 4\alpha^{2} ) + (2 - \alpha^{3} )\beta + (4 + \alpha^{2} )\beta^{2} ] \hfill \\ \end{gathered} \right\}}}{{2(2 - \alpha^{2} - \beta^{2} )[16 - 8\alpha^{2} + (3\alpha^{2} - 8)\beta^{2} ]^{2} }} $$

(54)

We verify that $\partial \hat{\Pi }_{A}^{{O\overline{E}}} /\partial \alpha > 0$, $\partial \hat{\Pi }_{A}^{{O\overline{E}}} /\partial \beta > 0$, and have $\hat{\Pi }_{A}^{{O\overline{E}}} \left| {_{\exists \alpha \in [0,1]} } \right. \le \hat{\Pi }_{A}^{{O\overline{E}}} \left| {_{\alpha = 1} } \right.$. Further, when $\gamma = 0$, we obtain (55) according to (19).

$$ \hat{\Pi }_{B}^{{O\overline{E}}} = \frac{{\left\{ \begin{gathered} \left[ {2\left( {8 - 4\beta^{2} } \right) - \left( {4 - 8\beta + 2\beta^{3} } \right)\alpha - \left( {32 - 6\beta^{2} } \right)\alpha^{2} + \left( {2 - 4\beta + \beta^{3} } \right)\alpha^{3} + } \right. \hfill \\ \left. {\left( {12 - \beta^{2} } \right)\alpha^{4} ] \times [\left( {8 - 4\beta^{2} } \right) + \left( {2 - \beta^{3} } \right)\alpha + \left( {4 + \beta^{2} } \right)\alpha^{2} } \right] \hfill \\ \end{gathered} \right\}}}{{2\left( {2 - \alpha^{2} - \beta^{2} } \right)\left[ {16 - 8\beta^{2} + \left( {3\beta^{2} - 8} \right)\alpha^{2} } \right]^{2} }} $$

(55)

Then, we verify that $\partial \hat{\Pi }_{B}^{{O\overline{E}}} /\partial \alpha > 0$, $\partial \hat{\Pi }_{B}^{{O\overline{E}}} /\partial \beta > 0$, and have $\hat{\Pi }_{B}^{{O\overline{E}}} \left| {_{\exists \alpha \in [0,1]} } \right. \le \hat{\Pi }_{B}^{{O\overline{E}}} \left| {_{\alpha = 1} } \right.$. Similar, we verify that $\partial \hat{\Pi }_{P}^{{O\overline{E}}} /\partial \alpha > 0$, $\partial \hat{\Pi }_{P}^{{O\overline{E}}} /\partial \beta > 0$, and $\hat{\Pi }_{P}^{{O\overline{E}}} \left| {_{\exists \alpha \in [0,1]} } \right. \le \hat{\Pi }_{P}^{{O\overline{E}}} \left| {_{\alpha = 1} } \right.$.

When $\gamma = \hat{\gamma }^{*} (\alpha ,\beta )$, we verify that $\partial \hat{\Pi }_{A}^{{O\overline{E}}} /\partial \alpha > 0$, $\partial \hat{\Pi }_{A}^{{O\overline{E}}} /\partial \beta > 0$, $\partial \hat{\Pi }_{B}^{{O\overline{E}}} /\partial \alpha > 0$, $\partial \hat{\Pi }_{B}^{{O\overline{E}}} /\partial \beta > 0$, $\partial \hat{\Pi }_{P}^{{O\overline{E}}} /\partial \alpha > 0$, and $\partial \hat{\Pi }_{P}^{{O\overline{E}}} /\partial \beta > 0$ in the same manner. Thus, according to the continuity argument, if $0 < \gamma < \hat{\gamma }^{*} (\alpha ,\beta )$, the equilibrium profits ($\hat{\Pi }_{A}^{{O\overline{E}}}$,$\hat{\Pi }_{B}^{{O\overline{E}}}$, and $\hat{\Pi }_{P}^{{O\overline{E}}}$) for the supply chain members increase with the increase of the intra-product showrooming effect $\alpha$, and the product showrooming effect $\beta$. Q.E.D.

1.4 Proof of Proposition 3

According to (27), we have

$$ \frac{{\partial \tilde{\Pi }_{A}^{\emptyset } }}{\partial \gamma } = \frac{{\left[ {2 + (1 - \theta )\gamma - \theta \gamma^{2} } \right]\left[ {8(1 - \theta ) + 12(1 - \theta )\gamma - (4 - 2\theta - 2\theta^{2} )\gamma^{3} - (1 - \theta^{2} )\gamma^{4} } \right]}}{{2\left[ {8 - \left( {6 + 2\theta } \right)\gamma^{2} + (1 + \theta )\gamma^{4} } \right]}} $$

(56)

$$ \frac{{\partial \tilde{\Pi }_{B}^{\emptyset } }}{\partial \gamma } = \frac{{\left\{ \begin{gathered} (1 - \theta )[8 + 6\gamma - (3 + \theta )\gamma^{2} - (2 + \theta )\gamma^{3} ] \times \hfill \\ [48 + (48 + 32\theta )\gamma - (1 + \theta )(12\gamma^{2} + 32\gamma^{3} ) - (6 + 8\theta + 2\theta^{2} )(\gamma^{4} - \gamma^{5} ) + (2 + 3\theta + \theta^{2} )\gamma^{6} ] \hfill \\ \end{gathered} \right\}}}{{2[8 - (6 + 2\theta )\gamma^{2} + (1 + \theta )\gamma^{4} ]^{3} }} $$

(57)

Since $0 < \gamma < 1$, $\theta \in [0.05,0.25]$, we verify that

$$ 2 + (1 - \theta )\gamma - \theta \gamma^{2} > 0,8(1 - \theta ) + 12(1 - \theta )\gamma - (4 - 2\theta - 2\theta^{2} )\gamma^{3} - (1 - \theta^{2} )\gamma^{4} > 0 $$

(58)

$$ 8 - (6 + 2\theta )\gamma^{2} + (1 + \theta )\gamma^{4} > 0,8 + 6\gamma - (3 + \theta )\gamma^{2} - (2 + \theta )\gamma^{3} > 0 $$

(59)

$$ 48 + (48 + 32\theta )\gamma - (1 + \theta )(12\gamma^{2} + 32\gamma^{3} ) - (6 + 8\theta + 2\theta^{2} )(\gamma^{4} - \gamma^{5} ) + (2 + 3\theta + \theta^{2} )\gamma^{6} > 0 $$

(60)

Thus, we have $\partial \tilde{\Pi }_{A}^{\emptyset } /\partial \gamma > 0$, $\partial \tilde{\Pi }_{B}^{\emptyset } /\partial \gamma > 0$. Then, we verify that $\partial \tilde{\Pi }_{P}^{\emptyset } /\partial \gamma > 0$ in the same manner. That is, the equilibrium profits ($\tilde{\Pi }_{A}^{\emptyset }$,$\tilde{\Pi }_{B}^{\emptyset }$, and $\tilde{\Pi }_{P}^{\emptyset }$) for the supply chain members increase with the increase of the competition intensity ($\gamma$) between two products. Q.E.D.

1.5 Proof of Lemma 2

Under the hybrid retailing agreement, if the platform establish an experience store to open an omni-channel retailing strategy and supplier B does not enter the omni-channel retailing, we can obtain the equilibrium profits for the suppliers and platform respectively are $\tilde{\Pi }_{A}^{{O\overline{E}}}$,$\tilde{\Pi }_{B}^{{O\overline{E}}}$, and $\tilde{\Pi }_{P}^{{O\overline{E}}}$. In order to ensure these equilibrium results are non-negative, we need $H_{0}^{*} = [4 - (1 + \theta )\gamma^{2} ]^{2} -$ $8\alpha^{2} + 2\theta \alpha^{2} \gamma^{2} > 0$. Thus, we set $[4 - (1 + \theta )\gamma^{2} ]^{2} - 8\alpha^{2} + 2\theta \alpha^{2} \gamma^{2} = 0$ and get the positive roots as

$$ \tilde{\gamma }_{1} (\alpha ,\theta ) = \sqrt {\frac{{\left( {8 + 8\theta - 2\theta \alpha^{2} } \right) - \sqrt {\left( {8 + 8\theta - 2\theta \alpha^{2} } \right)^{2} - 4(1 + \theta )^{2} (16 - 8\alpha^{2} )} }}{{2(1 + \theta )^{2} }}} $$

(61)

$$ \tilde{\gamma }_{2} (\alpha ,\theta ) = \sqrt {\frac{{(8 + 8\theta - 2\theta \alpha^{2} ) + \sqrt {(8 + 8\theta - 2\theta \alpha^{2} )^{2} - 4(1 + \theta )^{2} (16 - 8\alpha^{2} )} }}{{2(1 + \theta )^{2} }}} $$

(62)

$\tilde{\gamma }_{1} (\alpha ,\theta ) < \tilde{\gamma }_{2} (\alpha ,\theta )$, we verify that $\partial \tilde{\gamma }_{1} (\alpha ,\theta )/\partial \alpha < 0$, $\partial \tilde{\gamma }_{1} (\alpha ,\theta )/\partial \theta < 0$, $\partial \tilde{\gamma }_{2} (\alpha ,\theta )/\partial \alpha < 0$, $\partial \tilde{\gamma }_{2} (\alpha ,\theta )/\partial \theta < 0$. That is, in the condition of $\theta \in [0.05,0.25]$ and $0 < \alpha \le 1$, the minimum positive root of $[4 - (1 + \theta )\gamma^{2} ]^{2} - 8\alpha^{2} + 2\theta \alpha^{2} \gamma^{2} = 0$ is $\tilde{\gamma }_{1} (\alpha ,\theta )\left| {_{\theta = 0.25,\alpha = 1} } \right.$. We calculate that $\tilde{\gamma }_{1} (\alpha ,\theta )\left| {_{\theta = 0.25,\alpha = 1} } \right. = 1.0049 > 1$. Thus, the equilibrium results are always non-negative in the condition of $0 < \gamma < 1$, $\theta \in [0.05,0.25]$, and $0 \le \alpha \le 1$. Q.E.D.

1.6 Proof of Lemma 3

Under the hybrid retailing agreement, if the platform establish an experience store to open an omni-channel retailing strategy and supplier B enters the omni-channel retailing, we can obtain the equilibrium profits for the suppliers and platform respectively are $\tilde{\Pi }_{A}^{OE}$,$\tilde{\Pi }_{B}^{OE}$ and $\tilde{\Pi }_{P}^{OE}$. In order to ensure these equilibrium results are non-negative, we need $H_{0} = [4 - (1 + \theta )\gamma^{2} ]^{2} -$ $2[(1 + \theta )\beta \gamma + 2\alpha ](2a + \beta \gamma - \theta \gamma \beta - \alpha \theta \gamma^{2} ) - 2\theta (2\beta + a\gamma )^{2} > 0$. Similar to the proof of Lemma 2, we get

$$ H_{0} = (16 - 4\theta \beta - 8\alpha^{2} ) - 8\alpha \beta \gamma + [4\alpha^{2} \theta - 2(1 - \theta^{2} )\beta^{2} - 2\alpha \theta - 8(1 + \theta )]\gamma^{2} + 2\alpha \beta \theta (1 + \theta )\gamma^{3} + (1 + \theta )^{2} \gamma^{4} $$

(63)

$$ \frac{{\partial H_{0} }}{\partial \gamma } = - 8\alpha \beta + 2[4\alpha^{2} \theta - 2(1 - \theta^{2} )\beta^{2} - 2\alpha \theta - 8(1 + \theta )]\gamma + 6\alpha \beta \theta (1 + \theta )\gamma^{2} + 4(1 + \theta )^{2} \gamma^{3} < 0 $$

(64)

Due to the complicated calculation process, we draw on the proof process of Tian et al. [41] to prove that the equilibrium results are non-negative if and only if $0 < \gamma < \tilde{\gamma }^{*} (\alpha ,\beta )$. We first examine the limiting situation ($\gamma = 0$ and $\gamma = 1$), and then complete the proof using a continuity argument.

When $\gamma = 0$, we have $H_{0} = (16 - 4\theta \beta - 8\alpha^{2} ) > 0$, i.e., the equilibrium results are always non-negative in the condition of $\gamma = 0$, $\theta \in [0.05,0.25]$, and $0 \le \alpha \le 1$, $0 \le \beta \le \alpha$. When $\gamma = 1$, we have $H_{0} = 8 + 4\alpha^{2} \theta + 2\alpha \beta \theta (1 + \theta ) + (1 + \theta )^{2} - 4\theta \beta - 8\alpha^{2} - 8\alpha \beta - 2(1 - \theta^{2} )\beta^{2} - 2\alpha \theta - 8\theta$, and get

$$ \left\{ \begin{gathered} \frac{{\partial H_{0} }}{\partial \alpha } = - 8\alpha (2 - \theta ) - 2\theta (1 - \beta - \theta \beta ) - 8\beta < 0 \hfill \\ \frac{{\partial H_{0} }}{\partial \beta } = - 2\theta (2 - \alpha - \alpha \theta ) - 2(1 - \theta^{2} ) - 8\alpha < 0 \hfill \\ \end{gathered} \right. $$

(65)

According to (65), we have $H_{0} > H_{0} \left| {_{\alpha = 1,\beta = 1} } \right.$, where $H_{0} \left| {_{\alpha = 1,\beta = 1} } \right. = - \theta (6 - \theta ) - 9 < 0$. According to the continuity argument, $\exists \tilde{\gamma }^{*} (\alpha ,\beta ) > 0$, if $0 < \gamma < \tilde{\gamma }^{*} (\alpha ,\beta )$, we have $H_{0} > 0$; if $\tilde{\gamma }^{*} (\alpha ,\beta ) < \gamma < 1$, we have $H_{0} < 0$. Thus, under the hybrid retailing agreement, if the platform establish an experience store to open an omni-channel retailing strategy and supplier B enters the omni-channel retailing, the equilibrium results are non-negative if and only if $0 < \gamma < \tilde{\gamma }^{*} (\alpha ,\beta )$. Q.E.D.

1.7 Proof of Proposition 4

Under the reselling agreement, suppliers cannot make the decision to enter the platform’s omni-channel retailing. Under the hybrid retailing agreement, we have

$$ \tilde{\Pi }_{B}^{{OE}} - \tilde{\Pi }_{B}^{{O\bar{E}}} = \frac{{1 - \theta }}{{[4 - (1 + \theta )\gamma ^{2} ]^{2} H_{0} ^{2} H_{0}^{{*2}} }} \times \left\{ \begin{gathered} H_{0}^{{*2}} \left[ {(2 + \gamma )H_{0} + HH_{5} + (\gamma H_{0} - H_{5} H_{{00}} )\tilde{w}_{A}^{{OE}} } \right]^{2} - \hfill \\ H_{0} ^{2} \left[ {(2 + \gamma )H_{0}^{*} + \alpha \gamma H^{*} + \left( {\gamma H_{0}^{*} - \alpha \gamma H_{{00}}^{*} } \right)\tilde{w}_{A}^{{O\bar{E}}} } \right]^{2} \hfill \\ \end{gathered} \right\} - F $$

(66)

Thus, if and only if $F < \tilde{F}(\alpha ,\beta ,\gamma ) = \frac{{(1 - \theta )\left\{ \begin{gathered} H_{0}^{*2} [(2 + \gamma )H_{0} + HH_{5} + (\gamma H_{0} - H_{5} H_{00} )\tilde{w}_{A}^{OE} ]^{2} - \hfill \\ H_{0}^{2} \left[ {(2 + \gamma )H_{0}^{*} + \alpha \gamma H^{*} + (\gamma H_{0}^{*} - \alpha \gamma H_{00}^{*} )\tilde{w}_{A}^{{O\overline{E}}} } \right]^{2} \hfill \\ \end{gathered} \right\}}}{{[4 - (1 + \theta )\gamma^{2} ]^{2} H_{0}^{2} H_{0}^{*2} }}$, we can get $\tilde{\Pi }_{B}^{OE} - \tilde{\Pi }_{B}^{{O\overline{E}}} > 0$, i.e., supplier B will choose to enter the omni-channel retailing. Further, in order to ensure $\tilde{F}(\alpha ,\beta ,\gamma )$ is non-negative, we need $H_{0} > 0$ and $H_{0}^{*} > 0$. According to Lemma 2 and Lemma 3, we have the constraint of $H_{0} > 0$ and $H_{0}^{*} > 0$ is $0 < \gamma < \tilde{\gamma }^{*} (\alpha ,\beta )$. Therefore, under the hybrid retailing agreement, when the platform establishes an experience store to open an omni-channel retailing strategy, supplier B will choose to enter the omni-channel retailing if and only if $F < \tilde{F}(\alpha ,\beta ,\gamma )$ and $0 < \gamma < \tilde{\gamma }^{*} (\alpha ,\beta )$. Q.E.D.

1.8 Proof of Proposition 5

Under the reselling agreement, suppliers cannot make the decision to enter the platform’s omni-channel retailing. Thus, the platform’s omni-channel retailing decision is not affected by two suppliers. Under the hybrid retailing agreement, we get $\partial \tilde{\Pi }_{P}^{OE} /\partial \gamma > 0$ in Lemma 3. Therefore, we will first examine the limiting situation $\gamma = 0$ and $\gamma = 1$, and then complete the proof using a continuity argument.

When $\gamma = 0$, we obtain formula (67) according to formula (49).

$$ \tilde{\Pi }_{P}^{OE} = \frac{{(2 + \theta \beta - \theta \beta^{2} )^{2} }}{{16(1 - \theta \beta^{2} )^{2} }} + \frac{{\theta (8 + 4\beta + 2\theta \beta^{2} - 10\theta \beta^{3} )^{2} }}{{64(1 - \theta \beta^{2} )(2 - \theta \beta^{2} )^{2} }} - \frac{{(2 + 3\theta \beta - \theta \beta^{2} - 2\theta^{2} \beta^{3} )^{2} }}{{8(1 - \theta \beta^{2} )(2 - \theta \beta^{2} )^{2} }} $$

(67)

We verify that $\partial \tilde{\Pi }_{P}^{OE} /\partial \beta > 0$ for $\forall \beta \in [0,1]$, so we have $\tilde{\Pi }_{P}^{{O\overline{E}}} = \tilde{\Pi }_{P}^{OE} \left| {_{\beta = 0} } \right. \le \tilde{\Pi }_{P}^{OE} \left| {_{\beta \in [0,1]} } \right.$. Because of the continuity argument, we have $\exists \tilde{\gamma }^{*} (\beta ) > 0$, if $0 < \gamma < \tilde{\gamma }^{*} (\beta )$, there is $\tilde{\Pi }_{P}^{{O\overline{E}}} < \tilde{\Pi }_{P}^{OE}$, that is, the platform’s omni-channel retailing under the hybrid retailing agreement is better off with supplier B’s entry when $0 < \gamma < \tilde{\gamma }^{*} (\beta )$.

When $\gamma = 1$ and $\theta = 0.1$, we have $H_{1} = 2 + 1.1\beta$, $H_{2} = 2.8$,$H_{3} = 1.9 + 0.9\beta$, $H_{4} = 3.1$, $H_{5} = 1 + 2\beta$, $H = 12.09 + 7.07\beta$, $H_{0} = 0.61 - 8.58\beta - 2.78\beta^{2}$, $H_{00} = 3.51 + 1.51\beta$, then we get

$$ \tilde{w}_{A}^{OE} = \frac{{24.679 - 0.29\beta - 1.421\beta^{2} }}{{14.558 - 5.104\beta - 2.842\beta^{2} }} $$

(68)

$$ \tilde{\Pi }_{P}^{OE} = \left\{ \begin{gathered} \frac{{3.1H_{0} + HH_{1} + [2H_{0} - H_{1} H_{00} - 4H_{0} + 1.1H_{0} ]\tilde{w}_{A}^{OE} }}{{8.41(0.61 - 8.58\beta - 2.78\beta^{2} )}} \times \hfill \\ \frac{{2.8H_{0} + HH_{3} - [H_{0} + (1.9 + 0.9\beta )H_{00} ]\tilde{w}_{A}^{OE} }}{{8.41(0.61 - 8.58\beta - 2.78\beta^{2} )}} \hfill \\ + \frac{{0.1[3H_{0} + (1 + 2\beta )H + H_{0} - H_{00} - 2\beta H_{00} )\tilde{w}_{A}^{OE} ]^{2} }}{{70.73(0.61 - 8.58\beta - 2.78\beta^{2} )^{2} }} \hfill \\ \end{gathered} \right\} - \frac{{(H - H_{00} \tilde{w}_{A}^{OE} )^{2} }}{{2H_{0}^{2} }} $$

(69)

According to (69), we have $\tilde{\Pi }_{P}^{{O\overline{E}}} < \tilde{\Pi }_{P}^{OE}$ when $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$ and $0 < \beta < \tilde{\beta }^{*}$, that is, the platform’s omni-channel retailing under the hybrid retailing agreement is better off with supplier B’s entry when $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$ and $0 < \beta < \tilde{\beta }^{*}$(as illustrated in Fig. 7b); $\tilde{\Pi }_{P}^{{O\overline{E}}} > \tilde{\Pi }_{P}^{OE}$ when $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$ and $\tilde{\beta }^{*} < \beta \le 1$, i.e., the platform’s omni-channel retailing under the hybrid retailing agreement is worse off with supplier B’s entry when $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$ and $\tilde{\beta }^{*} < \beta \le 1$. Q.E.D.

1.9 Proof of Proposition 6 and Corollary 1

According to Lemma 1, we have $\hat{\gamma }^{*} (\beta ) = \frac{{\sqrt {2 - \beta^{2} } - \beta }}{2}$, and the equilibrium profit $\hat{\Pi }_{P}^{{O\overline{E}}}$ is non-negative if and only if $0 < \gamma < \hat{\gamma }^{*} (\beta )$. When $0 < \gamma < \hat{\gamma }^{*} (\beta )$, we verify that $\hat{\Pi }_{P}^{{O\overline{E}}} > \hat{\Pi }_{P}^{\emptyset }$; $\hat{\gamma }^{*} (\beta ) < \gamma < 1$, we verify that $\hat{\Pi }_{P}^{{O\overline{E}}} < \hat{\Pi }_{P}^{\emptyset }$.This concludes the proof of Proposition 6.

Further, we verify that $\partial \hat{\gamma }^{*} (\beta )/\partial \beta < 0$, and thus have $\min \hat{\gamma }^{*} (\beta ) = \hat{\gamma }^{*} (\beta )\left| {_{\beta = 1} } \right. = 0$, $\max \hat{\gamma }^{*} (\beta ) = \hat{\gamma }^{*} (\beta )\left| {_{\beta = 0} } \right. = 0.707$. Based on the results of $\min \hat{\gamma }^{*} (\beta )$ and $\max \hat{\gamma }^{*} (\beta )$, we get $\hat{\Pi }_{P}^{{O\overline{E}}} < 0$ and $\hat{\Pi }_{P}^{\emptyset } > 0$ when $0.7 < \gamma < 1$.that is, when $0.7 < \gamma < 1$, the platform will not establish an experience store to open the omni-channel retailing strategy.

When $0 < \gamma < 0.7$, we verify that if $0 \le \beta < \hat{\beta }^{*}$, have $\hat{\Pi }_{P}^{{O\overline{E}}} > \hat{\Pi }_{P}^{\emptyset }$, i.e., the platform will establish an experience store to open an omni-channel retailing strategy. Where, $\hat{\beta }^{*}$ can be obtained by solve the equation $\hat{\gamma }^{*} (\beta ) = \gamma$. If $0 < \gamma < 0.7$ and $\hat{\beta }^{*} < \beta \le 1$, have $\hat{\Pi }_{P}^{{O\overline{E}}} < \hat{\Pi }_{P}^{\emptyset }$, i.e., the platform will not establish an experience store to open the omni-channel retailing strategy. This concludes the proof of Corollary 1. Q.E.D.

1.10 Proof of Proposition 7

According to Lemma 3 and proposition 5, we get $\partial \tilde{\Pi }_{P}^{OE} /\partial \gamma > 0$ under the hybrid retailing agreement, and when $\gamma = 0$, we obtain

$$ \tilde{\Pi }_{P}^{OE} = \frac{{(2 + \theta \beta - \theta \beta^{2} )^{2} }}{{16(1 - \theta \beta^{2} )^{2} }} + \frac{{\theta (8 + 4\beta + 2\theta \beta^{2} - 10\theta \beta^{3} )^{2} }}{{64(1 - \theta \beta^{2} )(2 - \theta \beta^{2} )^{2} }} - \frac{{(2 + 3\theta \beta - \theta \beta^{2} - 2\theta^{2} \beta^{3} )^{2} }}{{8(1 - \theta \beta^{2} )(2 - \theta \beta^{2} )^{2} }} $$

(70)

$$ \tilde{\Pi }_{P}^{{O\overline{E}}} = \frac{1}{4} + \frac{\theta }{4} - \frac{1}{8},\quad \tilde{\Pi }_{P}^{\emptyset } = \frac{1}{16} + \frac{\theta }{4} $$

(71)

We verify that $\tilde{\Pi }_{P}^{\emptyset } < \tilde{\Pi }_{P}^{{O\overline{E}}} \le \tilde{\Pi }_{P}^{OE}$ for $\forall \beta \in [0,1]$. Because of the continuity argument, we have $\exists \tilde{\gamma }^{*} (\beta ) > 0$, if $0 < \gamma < \tilde{\gamma }^{*} (\beta )$, there is $\tilde{\Pi }_{P}^{\emptyset } < \tilde{\Pi }_{P}^{{O\overline{E}}} \le \tilde{\Pi }_{P}^{OE}$, that is, the platform always establishes an experience store to open the omni-channel retailing with or without supplier entry when $0 < \gamma < \tilde{\gamma }^{*} (\beta )$.

When $\gamma = 1$ and $\theta = 0.1$, we have $\tilde{\Pi }_{P}^{{O\overline{E}}} = 4.143$, $\tilde{\Pi }_{P}^{\emptyset } = 0.537$ according to (68) and (69), thus, $\exists \tilde{\beta }^{*} > 0$, if $0 < \gamma < \tilde{\gamma }^{*} (\beta )$ and $0 \le \beta < \tilde{\beta }^{*}$, we get $\tilde{\Pi }_{P}^{\emptyset } < \tilde{\Pi }_{P}^{{O\overline{E}}} \le \tilde{\Pi }_{P}^{OE}$. That is, the platform always establishes an experience store to open the omni-channel retailing with or without supplier entry. When $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$, $\tilde{\beta }^{*} < \beta \le 1$, we get $\tilde{\Pi }_{P}^{OE} \le \tilde{\Pi }_{P}^{\emptyset } < \tilde{\Pi }_{P}^{{O\overline{E}}}$, i.e., the platform establishes an experience store to open the omni-channel retailing if and only if there is no supplier entry. Q.E.D.

1.11 Proof of Proposition 8

According to Formulas (8) and (28), we have

$$ \hat{\Pi }_{P}^{\emptyset } - \tilde{\Pi }_{P}^{\emptyset } = \left\{ \begin{gathered} \frac{1}{{2(1 - \gamma )(2 - \gamma )^{2} }} - \frac{{[2 + \gamma (1 + \theta ) - 2\tilde{w}_{A}^{\emptyset } + \gamma^{2} (1 + \theta )\tilde{w}_{A}^{\emptyset } ][2 + \gamma - \theta \gamma (1 + \gamma )]}}{{2[4 - (1 + \theta )\gamma^{2} ]^{2} }} - \hfill \\ \frac{{\theta [(2 + \gamma )(4 - 2\gamma^{2} + \gamma ) - \theta \gamma^{2} (1 + \gamma )]^{2} }}{{4[4 - (1 + \theta )\gamma^{2} ]^{2} (2 - \gamma^{2} )^{2} }} \hfill \\ \end{gathered} \right\} $$

(72)

where $\tilde{w}_{A}^{\emptyset } = \frac{2 + \gamma - \theta \gamma (1 + \gamma )}{{2(2 - \gamma^{2} )}}$. Further, Propositions 1 and 3 show that the equilibrium profit $\hat{\Pi }_{P}^{\emptyset }$ under the reselling agreement increases with the increase of the product competition intensity $\gamma$; under the hybrid retailing agreement, the equilibrium profit $\tilde{\Pi }_{P}^{\emptyset }$ increases with the increase of the product competition intensity $\gamma$. Thus, we can use a continuity argument to prove Proposition 8. When $\gamma = 0$, we have $\hat{\Pi }_{P}^{\emptyset } - \tilde{\Pi }_{P}^{\emptyset } = \frac{1}{16} - \frac{\theta }{4}$, and since $\theta \in [0.05,0.25]$, we verify that $\hat{\Pi }_{P}^{\emptyset } - \tilde{\Pi }_{P}^{\emptyset } \ge 0$. When $\gamma \to 1$, we verify that $\hat{\Pi }_{P}^{\emptyset } - \tilde{\Pi }_{P}^{\emptyset } > 0$ for $\forall \theta \in [0.05,0.25]$. Because of the continuity argument, we get the platform’s optimal strategic choice is reselling agreement when he does not establish an experience store to open the omni-channel retailing. Q.E.D.

1.12 Proof of Proposition 9

According to Proposition 2, Lemmas 2 and 3, we show that the equilibrium profit $\hat{\Pi }_{P}^{{O\overline{E}}}$, $\tilde{\Pi }_{P}^{{O\overline{E}}}$ and $\tilde{\Pi }_{P}^{OE}$ increases with the increase of the product competition intensity $\gamma$. Thus, we can use a continuity argument to prove Proposition 9.

When $\gamma = 0$, we have Formula (73) based on Formulas (20). i.e.,

$$ \hat{\Pi }_{P}^{{O\overline{E}}} = \left[ \begin{gathered} \frac{{(8 + 4\beta - 12\beta^{2} - 2\beta^{3} + 4\beta^{4} )^{2} + (4 + 10\beta - 2\beta^{2} - 6\beta^{3} )(4 - 10\beta - 10\beta^{2} - 7\beta^{3} + 5\beta^{4} )}}{{(2 - 2\beta^{2} )^{2} (8 - 5\beta^{2} )^{2} }} \hfill \\ - \frac{{(8 + 8\beta - 10\beta^{2} - 4\beta^{3} + 2\beta^{4} )(8 + 8\beta - 10\beta^{2} - 4\beta^{3} + 2\beta^{4} )}}{{2(2 - 2\beta^{2} )^{2} (8 - 5\beta^{2} )^{2} }} \hfill \\ \end{gathered} \right] $$

(73)

According to Formulas (39) and (49), we have $\tilde{\Pi }_{P}^{{O\overline{E}}} = \frac{1}{8} + \frac{\theta }{4}$, and get

$$ \tilde{\Pi }_{P}^{OE} = \frac{{(2 + \theta \beta - \theta \beta^{2} )^{2} }}{{16(1 - \theta \beta^{2} )^{2} }} + \frac{{\theta (8 + 4\beta + 2\theta \beta^{2} - 10\theta \beta^{3} )^{2} }}{{64(1 - \theta \beta^{2} )(2 - \theta \beta^{2} )^{2} }} - \frac{{(2 + 3\theta \beta - \theta \beta^{2} - 2\theta^{2} \beta^{3} )^{2} }}{{8(1 - \theta \beta^{2} )(2 - \theta \beta^{2} )^{2} }} $$

(74)

We verify that $\tilde{\Pi }_{P}^{{O\overline{E}}} \le \tilde{\Pi }_{P}^{OE} < \hat{\Pi }_{P}^{{O\overline{E}}}$ for $\forall \beta \in [0,1]$. Further, according to Lemma 1, we have $\hat{\gamma }^{*} (\beta ) = \frac{{\sqrt {2 - \beta^{2} } - \beta }}{2}$, and the equilibrium profit $\hat{\Pi }_{P}^{{O\overline{E}}}$ is non-negative if and only if $0 < \gamma < \hat{\gamma }^{*} (\beta )$. In order to ensure $\hat{\gamma }^{*} (\beta ) = \frac{{\sqrt {2 - \beta^{2} } - \beta }}{2} > 0$, we need $0 < \beta < \hat{\beta }^{*} < 1$. According to Lemma 3, we have the equilibrium profit $\tilde{\Pi }_{P}^{{O\overline{E}}}$ and $\tilde{\Pi }_{P}^{OE}$ is non-negative if and only if $0 < \gamma < \tilde{\gamma }^{*} (\beta )$. We verify that $\hat{\gamma }^{*} (\beta ) < \tilde{\gamma }^{*} (\beta )$ for $\forall \beta \in [0,1]$. Thus, because of the continuity argument, we have $\exists \hat{\gamma }^{*} (\beta ) > 0$ and $0 < \hat{\beta }^{*} < 1$, if $0 < \gamma < \hat{\gamma }^{*} (\beta )$ and $0 < \beta < \hat{\beta }^{*}$, there is $\tilde{\Pi }_{P}^{{O\overline{E}}} \le \tilde{\Pi }_{P}^{OE} < \hat{\Pi }_{P}^{{O\overline{E}}}$, i.e., the platform’s optimal strategic choice is reselling agreement. When $0 < \gamma < \hat{\gamma }^{*} (\beta )$ and $\hat{\beta }^{*} < \beta < \min (\tilde{\beta }^{*} ,1)$, there is $\hat{\Pi }_{P}^{{O\overline{E}}} < \tilde{\Pi }_{P}^{{O\overline{E}}} \le \tilde{\Pi }_{P}^{OE}$, i.e., the platform’s optimal strategic choice is hybrid retailing agreement with supplier entry;

When $\gamma = 1$, we have $\hat{\Pi }_{P}^{{O\overline{E}}} < 0$. Furthermore, according to Proposition 5, we have if $0 < \gamma < \tilde{\gamma }^{*} (\beta )$, there is $\tilde{\Pi }_{P}^{{O\overline{E}}} < \tilde{\Pi }_{P}^{OE}$; if $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$ and $0 < \beta < \tilde{\beta }^{*}$, there is $\tilde{\Pi }_{P}^{{O\overline{E}}} < \tilde{\Pi }_{P}^{OE}$ (as illustrated in Fig. 7b); and $\tilde{\Pi }_{P}^{{O\overline{E}}} > \tilde{\Pi }_{P}^{OE}$ when $\tilde{\gamma }^{*} (\beta ) < \gamma < 1$ and $\tilde{\beta }^{*} < \beta \le 1$. Therefore, because of the continuity argument, we have if $\hat{\gamma }^{*} (\beta ) < \gamma < 1$ and $0 < \beta < \tilde{\beta }^{*}$, the platform’s optimal strategic choice is hybrid retailing agreement with supplier entry; if $\hat{\gamma }^{*} (\beta ) < \gamma < 1$ and $\tilde{\beta }^{*} < \beta < 1$, the platform’s optimal strategic choice is hybrid retailing agreement without supplier entry. Q.E.D.

Appendix 2

See Tables 4 and 5.

Table 4 The expressions of $H^{*}$, $H_{0}^{*}$ and $H_{00}^{*}$

Table 5 The expressions of $H$, $H_{0}^{{}}$ and $H_{00}^{{}}$(including $H_{1}$ to $H_{5}$)