Digital showroom strategies for dual-channel supply chains in the presence of consumer webrooming behavior

Abstract

Online sellers are increasingly adopting the digital showroom service to reduce consumer valuation uncertainty and boost sales. However, such service also facilitates consumer webrooming behavior, which may reduce the online demand and intensify channel competition. In this study, we develop a theoretical supply chain model consisting of a manufacturer and a retailer to investigate the optimal digital showroom strategy for dual-channel supply chains in the presence of consumer webrooming behavior. The findings show that when the online shopping cost is relatively high, the digital showroom service can increase the profits of both players under two typical supply chain structures. However, when the online shopping cost is relatively low, the digital showroom service may decrease each player’s profit since it increases the selling prices in both channels and aggravates the double marginalization problem under the wholesale price contract. In this case, a new contract that combines a quantity discount component and a webrooming profit sharing component (abbreviated as QD-WPS) is proposed, which can perfectly coordinate the dual-channel supply chain with digital showrooms.

Appendices

Appendix A: Proofs of main results under the base model

Proof of Lemma 1

• (1) Structure A:

  We apply the backward induction to solve for the optimal price strategies of the manufacturer and the independently offline retailer. First, given wholesale price $\hat{w}^{A}$, we can derive the optimal retail prices $\hat{p}_{o}^{A}$ and $\hat{p}_{r}^{A}$. Second, we determine the optimal wholesale price $\hat{w}^{A}$.

  The manufacturer and the retailer’s profit can be written as:

  $$ \hat{\pi }_{m}^{A} = \alpha \left( {1 - \frac{{\hat{p}_{o}^{A} - \theta \hat{p}_{r}^{A} + h}}{t}} \right)\hat{p}_{o}^{A} + \left( {\alpha \theta \frac{{\hat{p}_{o}^{A} - \theta \hat{p}_{r}^{A} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - \hat{p}_{r}^{A} } \right)}}{t}} \right)\hat{w}^{A} , $$

  $\hat{\pi }_{r}^{A} = \left( {\alpha \theta \frac{{\hat{p}_{o}^{A} - \theta \hat{p}_{r}^{A} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - \hat{p}_{r}^{A} } \right)}}{t}} \right)\left( {\hat{p}_{r}^{A} - \hat{w}^{A} - c_{r} } \right)$, respectively.

  Since $\frac{{\partial^{2} \hat{\pi }_{r}^{A} }}{{\left( {\partial \hat{p}_{r}^{A} } \right)^{2} }} = - \frac{{2\theta^{2} }}{t} < 0$ and $\frac{{\partial^{2} \hat{\pi }_{m}^{A} }}{{\left( {\partial \hat{p}_{o}^{A} } \right)^{2} }} = - \frac{2\alpha }{t} < 0$, by solving $\frac{{\partial \hat{\pi }_{r}^{A} }}{{\partial \hat{p}_{r}^{A} }} = 0$ and $\frac{{\partial \hat{\pi }_{m}^{A} }}{{\partial \hat{p}_{o}^{A} }} = 0$, we can derive the equilibrium prices are:

  $$ \hat{p}_{r}^{A} = \frac{{2\left( {c_{r} + v + \hat{w}^{A} } \right)\theta + \alpha \left( {h + t - \left( {2v - \hat{w}^{A} } \right)\theta } \right)}}{{\left( {4 - \alpha } \right)\theta }}, $$

  (A1)
  $$ \hat{p}_{o}^{A} = \frac{{2t - h\left( {2 - \alpha } \right) + \left( {c_{r} + v + 3\hat{w}^{A} - v\alpha } \right)\theta }}{4 - \alpha }. $$

  (A2)

  Then, the manufacturer’s profit can be written as follow:

  $$ \hat{\pi }_{m}^{A} = \frac{{\left( {1 - \alpha } \right)}}{{t\left( {4 - \alpha } \right)^{2} }}\left( \begin{gathered} \left( {2t - h\left( {2 - \alpha } \right)} \right)^{2} \gamma + 2c_{r} \theta \gamma \left( {2t - h\left( {2 - \alpha } \right)} \right) + \theta^{2} c_{r}^{2} \gamma - 2\theta^{2} c_{r} \left( {4\hat{w}^{A} - v\alpha } \right) \hfill \\ - \theta \gamma h\left( {1 - \alpha } \right)\left( {2v\left( {2 - \alpha } \right) + \hat{w}^{A} \alpha } \right) + \theta^{2} \left( {v - \hat{w}^{A} } \right)\left( {v\left( {1 - \alpha } \right)\alpha + \hat{w}^{A} \left( {8 + \alpha } \right)} \right) \hfill \\ + t\theta \gamma \left( {4v\left( {1 - \alpha } \right) + \hat{w}^{A} \left( {8 + \alpha } \right)} \right) \hfill \\ \end{gathered} \right) $$

  Since $\frac{{\partial^{2} \hat{\pi }_{m}^{A} }}{{\left( {\partial \hat{w}^{A} } \right)^{2} }} = - \frac{{2\left( {1 - \alpha } \right)\left( {8 + \alpha } \right)\theta^{2} }}{{t\left( {4 - \alpha } \right)^{2} }} < 0$, by solving $\frac{{\partial \hat{\pi }_{m}^{A} }}{{\partial \hat{w}^{A} }} = 0$, we can derive the best response functions as follows: $\hat{w}^{A*} = \frac{{t\alpha \left( {8 + \alpha } \right) - \left( {1 - \alpha } \right)\left( {h\alpha^{2} + \left( {8c_{r} - v\left( {8 + \alpha^{2} } \right)} \right)\theta } \right)}}{{2\left( {8 - 7\alpha - \alpha^{2} } \right)\theta }}$.

  Substituting the equilibrium prices $\hat{w}^{A*}$ into (A1) and (A2), we can obtain the table of lemma 1 under structure A. Note that, to ensure the optimal decisions satisfy the assumptions (a), (b) and (c) in N scenario in Sect. 3.1, all the following calculations are conducted under such constraints: $\hat{p}_{o}^{A*} + h \le \theta v$, $\hat{p}_{r}^{A*} \le v$ and $\min \left\{ {\frac{\theta v - t}{\theta },\frac{{\hat{p}_{o}^{A*} + h - t}}{\theta }} \right\} \le \hat{p}_{r}^{A*} \le \frac{{\hat{p}_{o}^{A*} + h}}{\theta }$.

• (2) Structure B:

  We take the same procedure as Structure A to solve the problem. First, given wholesale price $\hat{w}^{B}$, we derive the optimal retail prices $\hat{p}_{o}^{B}$ and $\hat{p}_{r}^{B}$. Second, we determine the optimal wholesale price $\hat{w}^{B}$. The manufacturer and the retailer’s profit can be written as:

  $$ \hat{\pi }_{m}^{B} = \alpha \left( {1 - \frac{{\hat{p}_{o}^{B} - \theta \hat{p}_{r}^{B} + h}}{t}} \right)\hat{w}^{B} + \left( {\alpha \theta \frac{{\hat{p}_{o}^{B} - \theta \hat{p}_{r}^{B} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - \hat{p}_{r}^{B} } \right)}}{t}} \right)\left( {\hat{p}_{r}^{B} - c_{r} } \right), $$

  $\hat{\pi }_{o}^{B} = \alpha \left( {1 - \frac{{\hat{p}_{o}^{B} - \theta \hat{p}_{r}^{B} + h}}{t}} \right)\left( {\hat{p}_{o}^{B} - \hat{w}^{B} } \right)$, respectively.

  Since $\frac{{\partial^{2} \hat{\pi }_{o}^{B} }}{{\left( {\partial \hat{p}_{o}^{B} } \right)^{2} }} = - \frac{2\alpha }{t} < 0$ and $\frac{{\partial^{2} \hat{\pi }_{m}^{B} }}{{\left( {\partial \hat{p}_{r}^{B} } \right)^{2} }} = - \frac{{2\theta^{2} }}{t} < 0$, by solving $\frac{{\partial \hat{\pi }_{o}^{B} }}{{\partial \hat{p}_{o}^{B} }} = 0$ and $\frac{{\partial \hat{\pi }_{m}^{B} }}{{\partial \hat{p}_{r}^{B} }} = 0$, we can derive the equilibrium prices are:

  $$ \hat{p}_{r}^{B} = \frac{{\left( {h + t + 3\hat{w}^{B} } \right)\alpha + 2\left( {c_{r} + v - v\alpha } \right)\theta }}{{\left( {4 - \alpha } \right)\theta }}, $$

  (A3)
  $$ \hat{p}_{o}^{B} = \frac{{2t - h\left( {2 - \alpha } \right) + \hat{w}^{B} \left( {2 + \alpha } \right) + \left( {c_{r} + v - v\alpha } \right)\theta }}{4 - \alpha }. $$

  (A4)

  Then, the manufacturer’s profit can be written as follow:

  $$ \hat{\pi }_{m}^{B} = \frac{1}{{t\left( {4 - \alpha } \right)^{2} }}\left( \begin{gathered} \left( {\left( {h + t} \right)^{2} + 8h\hat{w}^{B} + 7\left( {\hat{w}^{B} } \right)^{2} } \right)\alpha^{2} + \hat{w}^{B} \left( {t + \hat{w}^{B} } \right)\alpha^{3} - 8\alpha \hat{w}^{B} \left( {h - t + \hat{w}^{B} } \right) \hfill \\ - \alpha \theta c_{r} \left( {2\left( {h + t} \right)\left( {2 - \alpha } \right) + \hat{w}^{B} \left( {1 - \alpha } \right)\alpha } \right) + \left( {c_{r} \left( {2 - \alpha } \right) - 2v\left( {1 - \alpha } \right)} \right)^{2} \theta^{2} \hfill \\ + \alpha \theta v\left( {1 - \alpha } \right)\left( {4h + 4t + \hat{w}^{B} \left( {8 + \alpha } \right)} \right) \hfill \\ \end{gathered} \right). $$

  Since $\frac{{\partial^{2} \hat{\pi }_{m}^{B} }}{{\left( {\partial \hat{w}^{B} } \right)^{2} }} = - \frac{{2\left( {1 - \alpha } \right)\left( {8 + \alpha } \right)\alpha }}{{t\left( {4 - \alpha } \right)^{2} }} < 0$, by solving $\frac{{\partial \hat{\pi }_{m}^{B} }}{{\partial \hat{w}^{B} }} = 0$, we can derive the best response functions as follows:$\hat{w}^{B*} = \frac{{8h\left( {1 - \alpha } \right) - t\left( {8 + \alpha^{2} } \right) + \left( {1 - \alpha } \right)\left( {c_{r} \alpha - v\left( {8 + \alpha } \right)} \right)\theta }}{{2\left( {7\alpha + \alpha^{2} - 8} \right)}}$.

  Substituting the equilibrium prices $\hat{w}^{B*}$ into (A3) and (A4), we can obtain the table of lemma 1 under structure B. Note that, all the subsequent calculations are conducted under such constraints: $\hat{p}_{o}^{B*} + h \le \theta v$, $\hat{p}_{r}^{B*} \le v$ and $\min \left\{ {\frac{\theta v - t}{\theta },\frac{{\hat{p}_{o}^{B*} + h - t}}{\theta }} \right\} \le \hat{p}_{r}^{B*} \le \frac{{\hat{p}_{o}^{B*} + h}}{\theta }$. □

Proof of Theorem 1

• (1) Structure A

  We apply the backward induction to solve for the optimal price strategies of the manufacturer and the independently offline retailer. First, given wholesale price $w^{A}$, we derive the optimal retail prices $p_{o}^{A}$ and $p_{r}^{A}$. Second, we determine the optimal wholesale price $w^{A}$. The manufacturer and the retailer’s profit can be written as:

  $$ \pi_{m}^{A} = \alpha \theta \left( {1 - \frac{{p_{o}^{A} - p_{r}^{A} + h}}{t}} \right)\left( {p_{o}^{A} - c} \right) + \left( {\alpha \theta \frac{{p_{o}^{A} - p_{r}^{A} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - p_{r}^{A} } \right)}}{t}} \right)w^{A} , $$

  $\pi_{r}^{A} = \left( {\alpha \theta \frac{{p_{o}^{A} - p_{r}^{A} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - p_{r}^{A} } \right)}}{t}} \right)\left( {p_{r}^{A} - w^{A} - c_{r} } \right)$, respectively.

  Since $\frac{{\partial^{2} \pi_{r}^{A} }}{{\left( {\partial p_{r}^{A} } \right)^{2} }} = - \frac{{2\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\theta }}{t} < 0$ and $\frac{{\partial^{2} \pi_{m}^{A} }}{{\left( {\partial p_{o}^{A} } \right)^{2} }} = - \frac{2\alpha \theta }{t} < 0$, by solving $\frac{{\partial \pi_{r}^{A} }}{{\partial p_{r}^{A} }} = 0$ and $\frac{{\partial \pi_{m}^{A} }}{{\partial p_{o}^{A} }} = 0$, we can derive the equilibrium prices are:

  $$ p_{r}^{A} = \frac{{2\theta \left( {1 + c_{r} + w^{A} } \right) + \alpha \left( {h + t - \theta \left( {2 - w^{A} } \right)} \right)}}{{\theta \left( {4 - \alpha } \right)}}, $$

  (A5)
  $$ p_{o}^{A} = \frac{{\left( {c_{r} + 2c - h + 2t + 3w^{A} } \right)\alpha + \left( {c_{r} + 2c - 2h + 2t + v + 3w^{A} } \right)\left( {1 - \alpha } \right)\theta }}{{3\alpha + 4\left( {1 - \alpha } \right)\theta }}. $$

  (A6)

  Then, the manufacturer’s profit can be written as follow:

  $\pi_{m}^{A} = \frac{{\left( {1 - \alpha } \right)^{3} \theta }}{{t\left( {\alpha \left( {3 - 4\theta } \right) + 4\theta } \right)^{2} }}\left( \begin{gathered} w^{A} \left( {\gamma + \theta } \right)\left( {3\gamma + 4\theta } \right)\left( {\left( {c - c_{r} + h + t} \right)\gamma - 2\left( {c_{r} - v + w^{A} } \right)\theta } \right) + \hfill \\ \gamma \left( \begin{gathered} \left( {c_{r} - c - h + 2t} \right)\left( {\gamma + \theta } \right) \hfill \\ + \left( {v - c - h - w^{A} } \right)\theta \hfill \\ \end{gathered} \right)\left( \begin{gathered} \left( {c_{r} - c - h + 2t + 3w^{A} } \right)\left( {\gamma + \theta } \right) \hfill \\ + \left( {v - c - h} \right)\theta \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right)$. Since $\frac{{\partial^{2} \pi_{m}^{A} }}{{\left( {\partial w^{A} } \right)^{2} }} = - \frac{{2\theta^{2} \left( {1 - \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}}{{t\left( {\alpha \left( {3 - 4\theta } \right) + 4\theta } \right)^{2} }} < 0$, by solving $\frac{{\partial \pi_{m}^{A} }}{{\partial w^{A} }} = 0$, we can derive the best response functions as follows:

  $$ w^{A*} = \frac{1}{2}\left( {v - c_{r} + \frac{{\left( {c_{r} + 8\left( {c + h} \right) - 9v} \right)\alpha }}{{9\alpha + 8\left( {1 - \alpha } \right)\theta }} + \frac{t\alpha }{{\theta - \alpha \theta }} - \frac{{\left( {c + h - v} \right)\alpha }}{\alpha + \theta - \alpha \theta }} \right). $$

  Substituting the equilibrium prices $w^{A*}$ into (A5) and (A6), we can obtain the table of theorem 1 under structure A. Note that, to ensure the optimal decisions satisfy the assumptions (a), (b) and (c) in S scenario in subsection 3.1, all the following results are calculated under such constraints: $p_{o}^{A*} + h \le v$, $p_{r}^{A*} \le v$ and $\min \left\{ {\frac{\theta v - t}{\theta },p_{o}^{A*} + h - t} \right\} \le p_{r}^{A*} \le p_{o}^{A*} + h$.

• (2) Structure B

  We take the same procedure as Structure A to solve this problem. First, given wholesale price $w^{B}$, we derive the optimal retail prices $p_{o}^{B}$ and $p_{r}^{B}$. Second, we determine the optimal wholesale price $w^{B}$. The manufacturer and the retailer’s profit can be written as:

  $$ \pi_{m}^{B} = \alpha \left( {1 - \frac{{p_{o}^{B} - p_{r}^{B} + h}}{t}} \right)w^{B} + \left( {\alpha \theta \frac{{p_{o}^{B} - p_{r}^{B} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - p_{r}^{B} } \right)}}{t}} \right)\left( {p_{r}^{B} - c_{r} } \right), $$

  $\pi_{o}^{B} = \alpha \left( {1 - \frac{{p_{o}^{B} - p_{r}^{B} + h}}{t}} \right)\left( {p_{o}^{B} - w^{B} - c} \right)$, respectively. Since $\frac{{\partial^{2} \pi_{o}^{B} }}{{\left( {\partial p_{o}^{B} } \right)^{2} }} = - \frac{2\alpha \theta }{t} < 0$ and $\frac{{\partial^{2} \pi_{m}^{B} }}{{\left( {\partial p_{r}^{B} } \right)^{2} }} = - \frac{{2\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\theta }}{t} < 0$, by solving $\frac{{\partial \pi_{o}^{B} }}{{\partial p_{o}^{B} }} = 0$ and $\frac{{\partial \pi_{m}^{B} }}{{\partial p_{r}^{B} }} = 0$, we can derive the equilibrium prices are:

  $$ p_{r}^{B} = \frac{{\left( {2c_{r} + c + h + t + 3w^{B} } \right)\alpha + 2\theta \left( {v + c_{r} } \right)\left( {1 - \alpha } \right)}}{{3\alpha + 4\theta \left( {1 - \alpha } \right)}}, $$

  (A7)
  $$ p_{o}^{B} = \frac{{\left( {c_{r} + 2c - h + 2t + 3w^{B} } \right)\alpha + \left( {c_{r} + 2c - 2h + 2t + v + 2w^{B} } \right)\left( {1 - \alpha } \right)\theta }}{{3\alpha + 4\theta \left( {1 - \alpha } \right)}}. $$

  (A8)

  Let $\gamma = \frac{\alpha }{1 - \alpha }$. Then the manufacturer’s profit can be written as follows

  $$ \pi_{m}^{B} = \frac{{\left( {1 - \alpha } \right)^{3} }}{{t\left( {\alpha \left( {3 - 4\theta } \right) + 4\theta } \right)^{2} }}\left( \begin{gathered} w^{B} \theta \gamma \left( {3\gamma + 4\theta } \right)\left( {\left( {c_{r} - c - h + 2t} \right)\left( {\gamma + \theta } \right) + \left( {v - c - h - 2w^{B} } \right)\theta } \right) + \hfill \\ \theta \left( {\left( {c_{r} - c - h - t - 3w^{B} } \right)\gamma + 2\left( {c_{r} - v} \right)\theta } \right)\left( \begin{gathered} \gamma \left( {c_{r} - c - h - t} \right)\left( {\gamma + \theta } \right) \hfill \\ + \left( {2c_{r} - 2v + w^{B} } \right)\gamma \theta + 2\left( {c_{r} - v} \right)\theta^{2} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right) $$

  Since $\frac{{\partial^{2} \pi_{m}^{B} }}{{\left( {\partial w^{B} } \right)^{2} }} = - \frac{{2\alpha \theta^{2} \left( {1 - \alpha } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}}{{t\left( {\alpha \left( {3 - 4\theta } \right) + 4\theta } \right)^{2} }} < 0$, by solving $\frac{{\partial \pi_{m}^{B} }}{{\partial w^{B} }} = 0$, we can derive the best response functions as follows:

  $$ w^{B*} = \frac{{9t\alpha^{2} - \left( {c_{r} + 8\left( {c + h - 2t} \right) - 9v} \right)\left( {1 - \alpha } \right)\alpha \theta - 8\left( {c + h - t - v} \right)\left( {1 - \alpha } \right)^{2} \theta^{2} }}{{2\theta \left( {1 - \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}}. $$

  Substituting the equilibrium prices $w^{B*}$ into (B7) and (B8), we can obtain the table of theorem 1 under structure B. Note that, all the following results are calculated under such constraints: $p_{o}^{B*} + h \le v$, $p_{r}^{B*} \le v$ and $\min \left\{ {\frac{\theta v - t}{\theta },p_{o}^{B*} + h - t} \right\} \le p_{r}^{B*} \le p_{o}^{B*} + h$. □

Proof of Corollary 1

• (1) According to the equilibrium decisions of Structure A in Table 4, we have

  $$ \frac{{\partial p_{o}^{A*} }}{\partial h} = \frac{{ - 7\alpha - 8\theta \left( {1 - \alpha } \right)}}{{2\left( {9\alpha + 8\theta - 8\alpha \theta } \right)}} < 0,\;\frac{{\partial p_{r}^{A*} }}{\partial h} = \frac{1}{2}\alpha \left( {\frac{{5\alpha + 4\left( {1 - \alpha } \right)\theta }}{{\left( {\alpha + \theta - \alpha \theta } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}}} \right) > 0, $$
  $$ \frac{{\partial w^{A*} }}{\partial h} = - \frac{{\alpha^{2} }}{{2\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}} < 0; $$

• (2) According to the equilibrium decisions of Structure B in Table 4, we have.

  $$ \frac{{\partial p_{o}^{B*} }}{\partial h} = - \frac{{\alpha \left( {7 - 6\theta } \right) + 6\theta }}{{8\theta + \alpha \left( {9 - 8\theta } \right)}} < 0,\frac{{\partial p_{r}^{B*} }}{\partial h} = - \frac{\alpha }{{9\alpha + 8\left( {1 - \alpha } \right)\theta }} < 0,\frac{{\partial w^{B*} }}{\partial h} = - \frac{{4\left( {\alpha + \theta - \alpha \theta } \right)}}{{9\alpha + 8\left( {1 - \alpha } \right)\theta }} < 0. $$

  □

Proof of Proposition 1

• (1) Structure A.

  Taking the difference of the online selling price with and without digital showrooms, we have $p_{o}^{A*} - \hat{p}_{o}^{A*} = \frac{\begin{gathered} \left( {8 + \alpha } \right)\left( {t\gamma \left( {1 - \theta } \right)\left( {9\gamma + 8\theta } \right) + \theta c\left( {11\gamma + 8\theta } \right)} \right) \hfill \\ - \theta \left( {1 - \theta } \right)\left( {2c_{r} \left( {\gamma \left( {8 + \alpha } \right) + 8\theta } \right) - 16h\alpha - 9v\gamma \left( {8 + \alpha } \right) - 8v\left( {10 - \alpha } \right)\theta } \right) \hfill \\ \end{gathered} }{{2\theta \left( {8 + \alpha } \right)\left( {9\gamma + 8\theta } \right)}}$.

  Since $\frac{{\partial \left( {p_{o}^{A*} - \hat{p}_{o}^{A*} } \right)}}{\partial h} = \frac{{8\alpha \left( {1 - \theta } \right)\left( {1 - \alpha } \right)}}{{\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta - 8\alpha \theta } \right)}} > 0$, by setting $p_{o}^{A*} - \hat{p}_{o}^{A*} = 0$, we can derive.

  $$ \begin{aligned} h_{1} & = \frac{1}{16\alpha }\left( {8\left( {2c_{r} - v\left( {10 - \alpha } \right)} \right)\theta - \frac{{\alpha \left( {8t + 9v - 2c_{r} } \right)\left( {8 + \alpha } \right)}}{1 - \alpha } - \frac{{9t\alpha^{2} \left( {8 + \alpha } \right)}}{{\left( {1 - \alpha } \right)^{2} \theta }} - \frac{{c\left( {8 + \alpha } \right)\left( {11\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}}{{\left( {1 - \alpha } \right)\left( {1 - \theta } \right)}}} \right) \\ & < \left( \begin{gathered} \theta \left( {1 - \alpha } \right)\left( {2c_{r} \left( {1 - \theta } \right)\left( {\alpha^{2} + 8\alpha \left( {1 - \theta } \right) + 8\theta } \right) - v\left( {1 - \theta } \right)\left( {\alpha \left( {72 - 88\theta } \right) + 80\theta + \alpha^{2} \left( {9 + 8\theta } \right)} \right)} \right) \hfill \\ - \left( {8 + \alpha } \right)\left( {t\alpha \left( {1 - \theta } \right)\left( {8\theta + \alpha \left( {9 - 8\theta } \right)} \right) + c\theta \left( {1 - \alpha } \right)\left( {8\theta + \alpha \left( {11 - 8\theta } \right)} \right)} \right) \hfill \\ \end{gathered} \right) \\ & < - \frac{{\left( {1 - \alpha } \right)\theta }}{{\alpha \left( {1 - \theta } \right) + \theta }}\left( \begin{gathered} v\left( {1 - \alpha } \right)\left( {1 - \theta } \right)\theta \left( {3\left( {8 - 5\alpha } \right)\alpha + 16\left( {1 - \alpha } \right)^{2} \theta } \right) + \theta c\left( {8 + \alpha } \right)\left( {1 - \alpha } \right)\left( {19\alpha + 8\left( {1 - \alpha } \right)\theta } \right) \hfill \\ + 2c_{r} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {1 - \theta } \right)\left( {3\alpha \left( {8 + \alpha } \right) + 4\left( {1 - \alpha } \right)\left( {6 + \alpha } \right)\theta } \right) + \alpha^{2} c\left( {8 + \alpha } \right)\left( {12 - \theta } \right) \hfill \\ \end{gathered} \right) \\ & < 0. \\ \end{aligned} $$

  To guarantee $w^{A*} \ge 0$, that is,$0 < h \le 9v - 8c_{r} - c + 17t - \frac{{8\left( {2c_{r} - t - 2v} \right)\theta }}{\gamma } - \frac{{8\left( {c_{r} - v} \right)\theta^{2} }}{{\gamma^{2} }} + \frac{9t\gamma }{\theta }$. Hence, we have $p_{o}^{A*} > \hat{p}_{o}^{A*}$.

  Taking the difference of the offline selling price with and without digital showrooms, we have.

  $$ p_{r}^{A*} - \hat{p}_{r}^{A*} = \frac{\alpha }{4}\left( {\frac{{7v + c_{r} + 2v\alpha }}{8 + \alpha } - \frac{{2h\left( {4 + \alpha } \right)}}{{\theta \left( {8 + \alpha } \right)}} + \frac{{9v - c_{r} - 8c - 8h}}{{9\alpha + 8\theta \left( {1 - \alpha } \right)}} - \frac{{2\left( {v - c - h} \right)}}{\alpha + \theta - \alpha \theta }} \right). $$

  Since $\frac{{\partial \left( {p_{r}^{A*} - \hat{p}_{r}^{A*} } \right)}}{\partial h} = - \frac{{\alpha^{2} \left( {1 - \theta } \right)\left( {28\theta + 4\alpha \left( {9 - 5\theta } \right) + \alpha^{2} \left( {9 - 8\theta } \right)} \right)}}{{2\theta \left( {8 + \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {8\theta + \alpha \left( {9 - 8\theta } \right)} \right)}} < 0$, by setting $p_{r}^{A*} - \hat{p}_{r}^{A*} = 0$, we can derive.

  $$ h_{1}^{A} = \frac{{\theta \left( {c\left( {8 + \alpha } \right)\left( {5\gamma + 4\theta } \right) - 4c_{r} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {1 - \theta } \right) - v\left( {1 - \theta } \right)\left( {1 - \alpha } \right)\left( {9\gamma \left( {4 + \alpha } \right) + 4\left( {7 + 2\alpha } \right)\theta } \right)} \right)}}{{\alpha \left( {1 - \theta } \right)\left( {9\gamma \left( {4 + \alpha } \right) + 4\left( {7 + 2\alpha } \right)\theta } \right)}}. $$

  Hence, if $h \le h_{1}^{A}$, we have $p_{r}^{A*} - \hat{p}_{r}^{A*} \ge 0$; otherwise, $p_{r}^{A*} - \hat{p}_{r}^{A*} < 0$.

  Taking the difference of the wholesale price with and without digital showrooms, we have:

  $w^{A*} - \hat{w}^{A*} = \frac{\alpha }{2}\left( {\frac{{v - c_{r} - v\alpha }}{8 + \alpha } + \frac{h\alpha }{{\theta \left( {8 + \alpha } \right)}} - \frac{{9v - c_{r} - 8c - 8h}}{{9\alpha + 8\theta \left( {1 - \alpha } \right)}} + \frac{v - c - h}{{\alpha + \theta - \alpha \theta }}} \right)$ and $\frac{{\partial \left( {w^{A*} - \hat{w}^{A*} } \right)}}{\partial h} = - \frac{{\alpha^{2} \left( {1 - \theta } \right)\left( {\theta \left( {1 - \alpha } \right)^{2} - 9\alpha^{2} } \right)}}{{2\theta \left( {8 + \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {8\theta + \alpha \left( {9 - 8\theta } \right)} \right)}}$.

  If $\theta \le \frac{{9\alpha^{2} }}{{8\left( {1 - \alpha } \right)^{2} }}$, then $\frac{{\partial \left( {w^{A*} - \hat{w}^{A*} } \right)}}{\partial h} > 0$. By setting $w^{A*} - \hat{w}^{A*} = 0$, we can derive.

  $$ h_{2}^{A} = \frac{{\theta \left( {c\gamma \left( {8 + \alpha } \right) - 8c_{r} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {1 - \theta } \right) - v\left( {1 - \theta } \right)\left( {1 - \alpha } \right)^{2} \left( {9\gamma^{2} - 8\theta } \right)} \right)}}{{\alpha \left( {1 - \theta } \right)\left( {1 - \alpha } \right)\left( {9\gamma^{2} - 8\theta } \right)}}. $$

  Hence, when $h \le h_{2}^{A}$, we have $w^{A*} - \hat{w}^{A*} \le 0$; otherwise, $w^{A*} - \hat{w}^{A*} > 0$.

  If $\theta > \frac{{9\alpha^{2} }}{{8\left( {1 - \alpha } \right)^{2} }}$, then $\frac{{\partial \left( {w^{A*} - \hat{w}^{A*} } \right)}}{\partial h} < 0$. Hence, when $h \le h_{2}^{A}$, we have $w^{A*} - \hat{w}^{A*} \ge 0$; otherwise, $w^{A*} - \hat{w}^{A*} < 0$.

  In summary, when $\theta \le \frac{{9\gamma^{2} }}{8} \wedge h \ge h_{2}^{A}$ or $\theta \ge \frac{{9\gamma^{2} }}{8} \wedge h \ge h_{2}^{A}$, $w^{A*} - \hat{w}^{A*} \le 0$; otherwise, $w^{A*} - \hat{w}^{A*} > 0$.

• (2) Structure B.

  Taking the difference of the online selling price with and without digital showrooms, we have the following derivations.

  $$ \begin{aligned} p_{o}^{B*} - \hat{p}_{o}^{B*} & = \frac{{\left( {1 - \alpha } \right)\left( {2h\left( {6 + \alpha } \right) - \theta c_{r} \left( {4 + \alpha } \right) - v\theta \left( {8 + \alpha } \right)} \right) - t\left( {12 - \alpha \left( {2 + \alpha } \right)} \right)}}{{2\left( {1 - \alpha } \right)\left( {8 + \alpha } \right)}} \\ & \; + \frac{{9t\gamma^{2} + \left( {5c_{r} + 4c - 14h + 22t + 9v} \right)\gamma \theta + 4\left( {c_{r} + c - 3h + 3t + 2v} \right)\theta^{2} }}{{2\theta \left( {9\gamma + 8\theta } \right)}}. \\ \end{aligned} $$

  Since $\frac{{\partial \left( {p_{o}^{B*} - \hat{p}_{o}^{B*} } \right)}}{\partial h} = - \frac{{2\alpha \left( {1 - \alpha } \right)\left( {1 - \theta } \right)}}{{\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} < 0$, by setting $p_{o}^{B*} - \hat{p}_{o}^{B*} = 0$, we can derive $h_{2} = \frac{1}{4\alpha }\left( \begin{gathered} \left( {\left( {5c_{r} + 9v} \right)\left( {8 + \alpha } \right) + 4t\left( {17 + \alpha } \right)} \right)\gamma + 4c\left( {8 + \alpha } \right)\frac{\gamma + \theta }{{\left( {1 - \theta } \right)}} \hfill \\ + \frac{{9t\gamma^{2} \left( {8 + \alpha } \right)}}{\theta } + 8\left( {c_{r} \left( {4 + \alpha } \right) + v\left( {8 + \alpha } \right)} \right)\theta \hfill \\ \end{gathered} \right)$.

  To guarantee $w^{B*} \ge 0$, that is $h \le \frac{{9t\alpha^{2} - \left( {c_{r} + 8c - 16t - 9v} \right)\left( {1 - \alpha } \right)\alpha \theta + 8\left( {v - c + t} \right)\left( {1 - \alpha } \right)^{2} \theta^{2} }}{{8\left( {1 - \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\theta }} = \widehat{h}$.

  Then we have.

  $$ \widehat{h} - h_{2} = \frac{{ - \left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)\left( {1 - \alpha } \right)^{2} }}{{8\alpha \theta \left( {1 - \alpha } \right)^{2} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {1 - \theta } \right)}}\left( \begin{gathered} \left( {1 - \theta } \right)\left( {t\gamma + v\theta } \right)\left( {3\gamma \left( {5 + \alpha } \right) + 2\left( {8 + \alpha } \right)\theta } \right) \hfill \\ + c_{r} \theta \left( {1 - \theta } \right)\left( {\gamma \left( {9 + \alpha } \right) + 2\left( {4 + \alpha } \right)\theta } \right) + 8c\theta \left( {\gamma + \theta } \right) \hfill \\ \end{gathered} \right) < 0. $$

  Hence, $p_{o}^{B*} - \hat{p}_{o}^{B*} > 0$ can always be satisfied.

  Taking the difference of the offline selling price with and without digital showrooms, we have the following derivations.

  $$ p_{r}^{B*} - \hat{p}_{r}^{B*} = \frac{{\alpha \left( {\left( {1 - \theta } \right)\left( {9\alpha \left( {h - t} \right) + 8\theta c_{r} \left( {1 - \alpha } \right)} \right) - \theta c\left( {8 + \alpha } \right)} \right)}}{{\theta \left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}}. $$

  Since $\frac{{\partial \left( {p_{r}^{B*} - \hat{p}_{r}^{B*} } \right)}}{\partial h} = \frac{{9\alpha^{2} \left( {1 - \theta } \right)}}{{\theta \left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} > 0$, by setting $p_{r}^{B*} - \hat{p}_{r}^{B*} = 0$, we can derive $h_{1}^{B} = t + \frac{{\theta \left( {c\left( {8 + \alpha } \right) - 8c_{r} \left( {1 - \alpha } \right)\left( {1 - \theta } \right)} \right)}}{{9\alpha \left( {1 - \theta } \right)}}$. Hence, when $h \le h_{1}^{B}$, we have $p_{r}^{B*} - \hat{p}_{r}^{B*} \le 0$; otherwise, $p_{r}^{B*} - \hat{p}_{r}^{B*} > 0$.

  Taking the difference of the wholesale price with and without digital showrooms, we have the following derivations:

  $$ w^{B*} - \hat{w}^{B*} = \frac{1}{2}\left( \begin{gathered} \frac{{t\left( {8 + \alpha^{2} } \right) - \left( {1 - \alpha } \right)\left( {8h + c_{r} \alpha \theta - v\theta \left( {8 + \alpha } \right)} \right)}}{{7\alpha + \alpha^{2} - 8}} \hfill \\ + \frac{{9t\gamma^{2} - \left( {c_{r} + 8\left( {c + h - 2t} \right) - 9v} \right)\gamma \theta - 8\left( {c + h - t - v} \right)\theta^{2} }}{{\theta \left( {9\gamma + 8\theta } \right)}} \hfill \\ \end{gathered} \right). $$

  Since $\frac{{\partial \left( {w^{B*} - \hat{w}^{B*} } \right)}}{\partial h} = \frac{{4\alpha \left( {1 - \alpha } \right)\left( {1 - \theta } \right)}}{{\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} > 0$, by setting $w^{B*} - \hat{w}^{B*} = 0$, we can derive.

  $$ h_{2}^{B} = \frac{\begin{gathered} \gamma \left( {1 - \theta } \right)\left( {\theta c_{r} \left( {8 + \alpha + 8\left( {1 - \alpha } \right)\theta } \right) - 9t\gamma \left( {8 + \alpha } \right) - 8t\left( {7 + 2\alpha } \right)\theta } \right) \hfill \\ + \theta \left( {8 + \alpha } \right)\left( {8c\gamma + 8c\theta - v\left( {1 - \theta } \right)\left( {9\gamma + 8\theta } \right)} \right) \hfill \\ \end{gathered} }{{8\alpha \left( {1 - \theta } \right)\theta }}. $$

  Hence, when $h_{2}^{B} \le 0$, $w^{B*} - \hat{w}^{B*} \le 0$; otherwise, $w^{B*} - \hat{w}^{B*} > 0$. □

Proof of Proposition 2

• (1) Structure A.

  Taking the difference of the online demand with and without digital showrooms, we have the following derivations.

  $$ D_{o}^{A*} - \hat{D}_{o}^{A*} = \frac{\alpha }{2t}\left( \begin{gathered} \frac{{h\left( {8 - \alpha - \alpha^{2} } \right) - t\left( {8 + \alpha } \right) - \left( {6c_{r} + v\left( {2 - \alpha - \alpha^{2} } \right)} \right)\theta }}{8 + \alpha } \hfill \\ + \frac{{\theta \left( \begin{gathered} 3\left( {2c_{r} - 2\left( {c + h} \right) + 3t} \right)\gamma^{2} \hfill \\ + \left( {12c_{r} - 15\left( {c + h} \right) + 17t + 3v} \right)\gamma \theta + 2\left( {3c_{r} - 4\left( {c + h - t} \right) + v} \right)\theta^{2} \hfill \\ \end{gathered} \right)}}{{\left( {\gamma + \theta } \right)\left( {9\gamma + 8\theta } \right)}} \hfill \\ \end{gathered} \right). $$

  Since.

  $$ \begin{aligned} \frac{{\partial \left( {D_{o}^{A*} - \hat{D}_{o}^{A*} } \right)}}{\partial h} & = - \frac{{\alpha \left( \begin{gathered} 9\alpha^{2} \left( { - 8 + \alpha + \alpha^{2} } \right) - \alpha \left( {136 - \alpha \left( {201 + \left( {6 - 17\alpha } \right)\alpha } \right)} \right)\theta \hfill \\ - \left( {1 - \alpha } \right)\left( {64 - \alpha \left( {192 + \alpha \left( {15 - 8\alpha } \right)} \right)} \right)\theta^{2} + 8\left( {1 - \alpha } \right)^{2} \left( {8 + \alpha } \right)\theta^{3} \hfill \\ \end{gathered} \right)}}{{2t\left( {8 + \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {8\theta + \alpha \left( {9 - 8\theta } \right)} \right)}} \\ & > \frac{{\alpha^{2} \left( {1 - \theta } \right)\left( {9\alpha \left( {8 - \alpha - \alpha^{2} } \right) + \left( {1 - \alpha } \right)\left( {136 + \alpha \left( {7 - 8\alpha } \right)} \right)\theta } \right)}}{{2t\left( {8 + \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {8\theta + \alpha \left( {9 - 8\theta } \right)} \right)}} > 0, \\ \end{aligned} $$

  by setting $D_{o}^{A*} - \hat{D}_{o}^{A*} = 0$, we can derive.

  $$ h_{o}^{A} = \frac{{\left( {1 - \theta } \right)\left( \begin{gathered} \left( {\gamma + \theta } \right)\left( {t\left( {8 + \alpha } \right)\left( {9\gamma + 8\theta } \right) + 6\theta \alpha c_{r} } \right) \hfill \\ + \theta \alpha v\left( {9\gamma \left( {2 + \alpha } \right) + 2\left( {5 + 4\alpha } \right)\theta } \right) \hfill \\ \end{gathered} \right) + \theta c\left( {8 + \alpha } \right)\left( {6\gamma^{2} + 15\gamma \theta + 8\theta^{2} } \right)}}{{\left( {1 - \theta } \right)\left( {9\gamma^{2} \left( {8 - \alpha - \alpha^{2} } \right) + \gamma \left( {136 + \alpha \left( {7 - 8\alpha } \right)} \right)\theta + 8\left( {8 + \alpha } \right)\theta^{2} } \right)}}. $$

  Hence, when $h \le h_{o}^{A}$, $D_{o}^{A*} - \hat{D}_{o}^{A*} \le 0$; otherwise, $D_{o}^{A*} - \hat{D}_{o}^{A*} > 0$.

  Taking the difference of the offline demand with and without digital showrooms, we have the following derivations.

  $$ D_{r}^{A*} - \hat{D}_{r}^{A*} = \frac{{\theta \left( {2 + \alpha } \right)\left( {\theta c_{r} - h\alpha - v\theta \left( {1 - \alpha } \right)} \right)}}{{t\left( {8 + \alpha } \right)}} + \frac{{\theta \left( {3\gamma + 2\theta } \right)\left( {1 - \alpha } \right)\left( {\left( {c - c_{r} + h} \right)\gamma - \left( {c_{r} - v} \right)\theta } \right)}}{9t\gamma + 8t\theta }. $$

  Since $\frac{{\partial \left( {D_{r}^{A*} - \hat{D}_{r}^{A*} } \right)}}{\partial h} = \frac{{6\alpha^{2} \theta \left( {1 - \alpha } \right)\left( {1 - \theta } \right)}}{{t\left( {8 + \alpha } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}} > 0$, by setting $D_{r}^{A*} - \hat{D}_{r}^{A*} = 0$, we can derive.

  $$ h_{r}^{A} = \frac{{c_{r} \left( {1 - \theta } \right)\left( {3\gamma \left( {8 + \alpha } \right) + 2\left( {11 + \alpha } \right)\theta } \right) - 6v\left( {1 - \alpha } \right)\left( {1 - \theta } \right)\theta - c\left( {8 + \alpha } \right)\left( {3\gamma + 2\theta } \right)}}{{6\alpha \left( {1 - \theta } \right)}}. $$

  Hence, when $h \le h_{r}^{A}$, we have $D_{r}^{A*} - \hat{D}_{r}^{A*} \le 0$; otherwise, $D_{r}^{A*} - \hat{D}_{r}^{A*} > 0$.

  Taking the difference of the total market demand with and without digital showrooms, we have the following derivations.

  $$ D_{m}^{A*} - \hat{D}_{m}^{A*} = \frac{1}{2t}\left( \begin{gathered} \frac{{\theta \left( {1 - \alpha } \right)\left( \begin{gathered} 9t\gamma^{3} - \left( {4c_{r} + 5c + 5h - 17t - 9v} \right)\gamma^{2} \theta \hfill \\ - 4\left( {2c_{r} + c + h - 2t - 3v} \right)\gamma \theta^{2} - 4\left( {c_{r} - v} \right)\theta^{3} \hfill \\ \end{gathered} \right)}}{{\left( {9\gamma + 8\theta } \right)\left( {\gamma + \theta } \right)}} \hfill \\ - \frac{{t\alpha \left( {8 + \alpha } \right) + 6c_{r} \alpha \theta - h\alpha \left( {8 - \alpha - \alpha^{2} } \right) + \theta \left( {2 + \alpha } \right)\left( {2h\alpha - 2c_{r} \theta + v\left( {1 - \alpha } \right)\left( {\alpha + 2\theta } \right)} \right)}}{8 + \alpha } \hfill \\ \end{gathered} \right). $$

  Since$\frac{{\partial \left( {D_{m}^{A*} - \hat{D}_{m}^{A*} } \right)}}{\partial h} = \frac{{\alpha \left( {1 - \theta } \right)\left( \begin{gathered} 9\alpha^{2} \left( {8 - \alpha - \alpha^{2} } \right) \hfill \\ + \alpha \left( {1 - \alpha } \right)\left( {136 + \alpha \left( {19 - 8\alpha } \right)} \right)\theta + 4\left( {1 - \alpha } \right)^{2} \left( {16 + 5\alpha } \right)\theta^{2} \hfill \\ \end{gathered} \right)}}{{2t\left( {8 + \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}} > 0$, by setting $D_{m}^{A*} - \hat{D}_{m}^{A*} = 0$, we can derive

  $$ h_{3}^{A} = \frac{{\left( \begin{gathered} \left( {\gamma + \theta } \right)\left( {1 - \theta } \right)\left( {t\left( {8 + \alpha } \right)\left( {9\gamma + 8\theta } \right) + 2\theta c_{r} \left( {27\gamma + 2\left( {11 + \alpha } \right)\theta } \right)} \right) + + \theta^{2} c\left( {8 + \alpha } \right)\left( {5\gamma + 4\theta } \right) \hfill \\ + \theta v\left( {1 - \theta } \right)\left( {1 - \alpha } \right)\left( {9\gamma^{2} \left( {2 + \alpha } \right) - 2\gamma \left( {1 - 4\alpha } \right)\theta - 12\theta^{2} } \right) \hfill \\ \end{gathered} \right)}}{{\left( {1 - \theta } \right)\left( {9\gamma^{2} \left( {8 - \alpha - \alpha^{2} } \right) + \gamma \left( {136 + \alpha \left( {19 - 8\alpha } \right)} \right)\theta + 4\left( {16 + 5\alpha } \right)\theta^{2} } \right)}} $$

  Hence, when $h \le h_{3}^{A}$, we have $D_{m}^{A*} - \hat{D}_{m}^{A*} \le 0$; otherwise, $D_{m}^{A*} - \hat{D}_{m}^{A*} > 0$.

• (2) Structure B.

  Taking the difference of the online demand with and without digital showrooms, we have the following derivations:

  $$ D_{o}^{B*} - \hat{D}_{o}^{B*} = \frac{{\alpha \left( {2 + \alpha } \right)\left( {h - t - \theta c_{r} } \right)}}{{t\left( {8 + \alpha } \right)}} + \frac{{\alpha \theta \left( {c_{r} - c - h + t} \right)\left( {3\alpha + 2\theta \left( {1 - \alpha } \right)} \right)}}{{9t\alpha + 8t\theta \left( {1 - \alpha } \right)}}. $$

  Since $\frac{{\partial \left( {D_{o}^{B*} - \hat{D}_{o}^{B*} } \right)}}{\partial h} = \frac{{\alpha \left( {1 - \theta } \right)\left( {9\alpha \left( {2 + \alpha } \right) + 2\left( {1 - \alpha } \right)\left( {8 + \alpha } \right)\theta } \right)}}{{t\left( {8 + \alpha } \right)\left( {9\alpha + 8\left( {1 - \alpha } \right)\theta } \right)}} > 0$, by setting $D_{o}^{B*} - \hat{D}_{o}^{B*} = 0$, we can derive $h_{o}^{B} = t - \frac{{\theta \left( {6c_{r} \alpha \left( {1 - \alpha } \right)\left( {1 - \theta } \right) - c\left( {8 + \alpha } \right)\left( {3\alpha + 2\theta \left( {1 - \alpha } \right)} \right)} \right)}}{{\left( {1 - \theta } \right)\left( {9\alpha \left( {2 + \alpha } \right) + 2\theta \left( {1 - \alpha } \right)\left( {8 + \alpha } \right)} \right)}}$. Hence, when $h \le h_{o}^{B}$, we have $D_{o}^{B*} - \hat{D}_{o}^{B*} \le 0$; otherwise, $D_{o}^{B*} - \hat{D}_{o}^{B*} > 0$.

  Taking the difference of the offline demand with and without digital showrooms, we have the following derivations.

  $$ D_{r}^{B*} - \hat{D}_{r}^{B*} = \frac{{3\alpha \theta \left( {\left( {8 + \alpha } \right)\left( {\alpha c - \alpha c_{r} \left( {1 - \theta } \right) + \theta c\left( {1 - \alpha } \right)} \right) - \left( {1 - \theta } \right)\left( {1 - \alpha } \right)\left( {\alpha \left( {h - t} \right) + 8\theta c_{r} } \right)} \right)}}{{t\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}}. $$

  Since $\frac{{\partial \left( {D_{r}^{B*} - \hat{D}_{r}^{B*} } \right)}}{\partial h} = - \frac{{3\alpha^{2} \theta \left( {1 - \alpha } \right)\left( {1 - \theta } \right)}}{{t\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} < 0$, by setting $D_{r}^{B*} - \hat{D}_{r}^{B*} = 0$, we can derive $h_{r}^{B} = \frac{{t\alpha - 8\theta c_{r} }}{\alpha } - \frac{{\left( {8 + \alpha } \right)\left( {\alpha c_{r} \left( {1 - \theta } \right) - c\left( {\alpha + \theta \left( {1 - \alpha } \right)} \right)} \right)}}{{\alpha \left( {1 - \alpha } \right)\left( {1 - \theta } \right)}}$. Hence, when $h \le h_{r}^{B}$, we have $D_{r}^{B*} - \hat{D}_{r}^{B*} \ge 0$; otherwise, $D_{r}^{B*} - \hat{D}_{r}^{B*} < 0$.

  Taking the difference of the total market demand with and without digital showrooms, we have the following derivations:

  $$ D_{m}^{B*} - \hat{D}_{m}^{B*} = \frac{{\alpha \left( {\left( {1 - \theta } \right)\left( \begin{gathered} \left( {h - t} \right)\left( {9\alpha \left( {2 + \alpha } \right) + \theta \left( {16 - \alpha } \right)\left( {1 - \alpha } \right)} \right) \hfill \\ - 3\theta c_{r} \left( {3\alpha \left( {2 + \alpha } \right) + 8\theta \left( {1 - \alpha } \right)} \right) \hfill \\ \end{gathered} \right) + \theta^{2} c\left( {1 - \alpha } \right)\left( {8 + \alpha } \right)} \right)}}{{t\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}}. $$

  Since $\frac{{\partial \left( {D_{m}^{B*} - \hat{D}_{m}^{B*} } \right)}}{\partial h} = \frac{{\alpha \left( {1 - \theta } \right)\left( {9\alpha \left( {2 + \alpha } \right) + \theta \left( {16 - \alpha } \right)\left( {1 - \alpha } \right)} \right)}}{{t\left( {8 + \alpha } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} > 0$, by setting $D_{m}^{B*} - \hat{D}_{m}^{B*} = 0$, we can derive.

  $$ h_{3}^{B} = t - \frac{{\theta \left( {c\theta \left( {1 - \alpha } \right)\left( {8 + \alpha } \right) - 3c_{r} \left( {1 - \theta } \right)\left( {3\alpha \left( {2 + \alpha } \right) + 8\theta \left( {1 - \alpha } \right)} \right)} \right)}}{{\left( {1 - \theta } \right)\left( {9\alpha \left( {2 + \alpha } \right) + \theta \left( {16 - \alpha } \right)\left( {1 - \alpha } \right)} \right)}}. $$

  Hence, when $h_{3}^{B} \le 0$, we have $D_{m}^{B*} - \hat{D}_{m}^{B*} \le 0$; otherwise, $D_{m}^{B*} - \hat{D}_{m}^{B*} > 0$. □

Proof of Corollary 2

• (1) Structure A.

  $$ \frac{{\partial \omega_{{}}^{A} }}{\partial h} = \frac{{\alpha \theta \left( {6\alpha^{2} + 15\alpha \theta \left( {1 - \alpha } \right) + 8\left( {1 - \alpha } \right)^{2} \theta^{2} } \right)}}{{2t\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} > 0. $$
  $$ \frac{{\partial \omega_{{}}^{A} }}{\partial t} = \frac{{\alpha \theta \left( {6\left( {c_{r} - c - h} \right)\alpha^{2} + 3\left( {4c_{r} - 5\left( {c + h} \right) + v} \right)\left( {1 - \alpha } \right)\alpha \theta + 2\left( {3c_{r} - 4\left( {c + h} \right) + v} \right)\left( {1 - \alpha } \right)^{2} \theta^{2} } \right)}}{{2t^{2} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {9\alpha - 8\left( {1 - \alpha } \right)\theta } \right)}} $$

  and $\frac{{\partial \left( {\frac{{\partial \omega_{{}}^{A} }}{\partial t}} \right)}}{\partial h} = - \frac{{\alpha \theta \left( {6\alpha^{2} + 15\alpha \theta \left( {1 - \alpha } \right) + 8\theta^{2} \left( {1 - \alpha } \right)^{2} } \right)}}{{2t^{2} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} < 0$. By solving $\frac{{\partial \omega_{{}}^{A} }}{\partial t} = 0$, we can obtain $\overline{h}_{{}}^{A} = \frac{{6\left( {c_{r} - c} \right)\alpha^{2} + 3\left( {v + 4c_{r} - 5c} \right)\left( {1 - \alpha } \right)\alpha \theta + 2\left( {v + 3c_{r} - 4c} \right)\left( {1 - \alpha } \right)^{2} \theta^{2} }}{{6\alpha^{2} + 15\left( {1 - \alpha } \right)\alpha \theta + 8\left( {1 - \alpha } \right)^{2} \theta^{2} }}$.

  Hence, when $h \le \overline{h}_{{}}^{A}$, we have $\frac{{\partial \omega_{{}}^{A} }}{\partial t} \ge 0$; otherwise, $\frac{{\partial \omega_{{}}^{A} }}{\partial t} < 0$.

• (2) Structure B

  $$ \frac{{\partial \omega_{{}}^{B} }}{\partial h} = \frac{{\alpha \theta \left( {3\alpha + 2\theta \left( {1 - \alpha } \right)} \right)}}{{t\left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}} > 0. $$

  $\frac{{\partial \omega_{{}}^{B} }}{\partial t} = \frac{{\alpha \theta \left( {c_{r} - c - h} \right)\left( {3\alpha + 2\theta \left( {1 - \alpha } \right)} \right)}}{{t^{2} \left( {9\alpha + 8\theta \left( {1 - \alpha } \right)} \right)}}$. Hence when $h \le \overline{h}^{B} = c_{r} - c$, we have $\frac{{\partial \omega_{{}}^{B} }}{\partial t} \ge 0$; otherwise, $\frac{{\partial \omega_{{}}^{B} }}{\partial t} < 0$. □

Proof of Proposition 3

• (1) Structure A.

  Taking the difference of the retailer’s profit between with and without digital showrooms, we have the following derivations.

  $$ \pi_{r}^{A*} - \hat{\pi }_{r}^{A*} = \frac{{\left( {1 - \alpha } \right)}}{t}\left( {\frac{{\theta \left( {3\gamma + 2\theta } \right)^{2} \left( {c\gamma + h\gamma + v\theta - c_{r} \left( {\gamma + \theta } \right)} \right)^{2} }}{{\left( {\gamma + \theta } \right)\left( {9\gamma + 8\theta } \right)^{2} }} - \frac{{\left( {2 + 3\gamma } \right)^{2} \left( {h\gamma - \left( {c_{r} - v + c_{r} \gamma } \right)\theta } \right)^{2} }}{{\left( {1 + \gamma } \right)\left( {8 + 9\gamma } \right)^{2} }}} \right). $$

  Since $\begin{gathered} \frac{{\partial^{2} \left( {\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} } \right)}}{{\partial h^{2} }} = - \frac{{2\alpha^{3} \left( \begin{gathered} 81\alpha^{2} \left( {2 + \alpha } \right)^{2} + 9\alpha \left( {36 - \alpha \left( {16 + \alpha \left( {76 + 25\alpha } \right)} \right)} \right)\theta \hfill \\ + 4\left( {1 - \alpha } \right)\left( {16 - \alpha \left( {48 + \alpha \left( {159 + 52\alpha } \right)} \right)} \right)\theta^{2} \hfill \\ - 4\left( {1 - \alpha } \right)^{2} \left( {16 + \alpha \left( {49 + 16\alpha } \right)} \right)\theta^{3} \hfill \\ \end{gathered} \right)}}{{t\left( {8 + \alpha } \right)^{2} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {\alpha \left( {9 - 8\theta } \right) + 8\theta } \right)^{2} }} \\ < - \frac{{18\alpha^{4} \left( {1 - \theta } \right)\left( {9\alpha \left( {2 + \alpha } \right)^{2} + 4\left( {1 - \alpha } \right)\left( {9 + 2\alpha \left( {7 + 2\alpha } \right)} \right)\theta } \right)}}{{t\left( {8 + \alpha } \right)^{2} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {\alpha \left( {9 - 8\theta } \right) + 8\theta } \right)^{2} }} < 0 \\ \end{gathered}$, by setting $\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} = 0$, we can derive.

  $$ h_{4}^{A} = \frac{{M + \theta \left( {1 + \gamma } \right)\left( {c\left( {8 + 9\gamma } \right)^{2} \left( {3\gamma + 2\theta } \right)^{2} - 12c_{r} \left( {1 - \theta } \right)\left( {\gamma + \theta } \right)\left( {16\theta + 3\gamma \left( {7 + 9\gamma + 7\theta } \right)} \right)} \right)}}{{\gamma \left( {1 - \theta } \right)\left( {81\gamma^{2} \left( {2 + 3\gamma } \right)^{2} + 36\gamma \left( {9 + \gamma \left( {32 + 27\gamma } \right)} \right)\theta + 4\left( {16 + 81\gamma \left( {1 + \gamma } \right)} \right)\theta^{2} } \right)}} - \frac{v\theta }{\gamma }, $$

  wherein $M = \left( {2 + 3\gamma } \right)\left( {8 + 9\gamma } \right)\left( {3\gamma + 2\theta } \right)\left( {9\gamma + 8\theta } \right)\sqrt {\left( {1 + \gamma } \right)\left( {c - c_{r} \left( {1 - \theta } \right)} \right)^{2} \theta \left( {\gamma + \theta } \right)}$. Thus, when $h \le h_{4}^{A}$, we have $\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} \ge 0$; otherwise, $\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} < 0$.

  Taking the difference of the manufacturer’s profit between with and without digital showrooms, we have $\frac{{\partial^{2} \left( {\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} } \right)}}{{\partial h^{2} }} = - \frac{{\alpha \left( {1 - \theta } \right)\left( \begin{gathered} 9\alpha^{2} \left( {8 - \alpha \left( {3 + \alpha } \right)} \right) \hfill \\ + \alpha \left( {1 - \alpha } \right)\left( {136 - \alpha \left( {11 + 8\alpha } \right)} \right)\theta + 8\left( {1 - \alpha } \right)^{2} \left( {8 + \alpha } \right)\theta^{2} \hfill \\ \end{gathered} \right)}}{{2t\left( {8 + \alpha } \right)\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {8\theta + \alpha \left( {9 - 8\theta } \right)} \right)}} < 0$, by setting $\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} = 0$, we can derive.

  $$ h_{5}^{A} = C - \sqrt {\begin{gathered} C + 2\left( {1 + \gamma } \right)\left( {8 + 9\gamma } \right)\left( {\gamma + \theta } \right)\left( {9\gamma + 8\theta } \right)\left( {t^{2} \left( {1 - \theta } \right) + 2\theta c_{r} } \right) \hfill \\ - 2v^{2} \theta^{2} \left( {1 - \theta } \right)\left( {28\theta + 9\gamma \left( {4 + 5\gamma + 4\theta } \right)} \right) - 16v\theta^{2} c_{r} \left( {1 - \theta } \right)\left( {\gamma + \theta } \right)\left( {1 + \gamma } \right) \hfill \\ - 2c\theta \left( {1 + \gamma } \right)\left( {8 + 9\gamma } \right)\left( {c\left( {4\gamma^{2} + 13\gamma \theta + 8\theta^{2} } \right) - 8c_{r} \left( {\gamma + \theta } \right)^{2} - 2v\theta \left( {5\gamma + 4\theta } \right)} \right) \hfill \\ - 8c_{r}^{2} \theta \left( {1 - \theta } \right)\left( {\gamma + \theta } \right)\left( {1 + \gamma } \right)\left( {7\theta + 8\gamma + 9\gamma^{2} + 8\theta \gamma } \right). \hfill \\ \end{gathered} }$$

  where $C = \frac{{\left( \begin{gathered} \left( {1 - \theta } \right)\left( {\gamma + \theta } \right)\left( {1 + \gamma } \right)\left( {t\left( {8 + 9\gamma } \right)\left( {9\gamma + 8\theta } \right) + 4\theta \gamma c_{r} } \right) \hfill \\ + \theta c\left( {1 + \gamma } \right)\left( {8 + 9\gamma } \right)\left( {4\gamma^{2} + 13\gamma \theta + 8\theta^{2} } \right) + \theta v\gamma \left( {1 - \theta } \right)\left( {28\theta + 9\gamma \left( {4 + 5\gamma + 4\theta } \right)} \right) \hfill \\ \end{gathered} \right)}}{{\left( {1 - \theta } \right)\left( {9\gamma^{2} \left( {8 + \gamma \left( {13 + 4\gamma } \right)} \right) + \gamma \left( {136 + 9\gamma \left( {29 + 13\gamma } \right)} \right)\theta + 8\left( {1 + \gamma } \right)\left( {8 + 9\gamma } \right)\theta^{2} } \right)}}$.

  Hence, if $h \le h_{5}^{A}$, we have $\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} \le 0$; otherwise, $\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} > 0$.

  Since $\frac{{\partial^{2} \left( {\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} } \right)}}{{\partial h^{2} }}{ + }\frac{{\partial^{2} \left( {\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} } \right)}}{{\partial h^{2} }} < 0$, by setting $\left( {\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} } \right) + \left( {\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} } \right) = 0$, we can derive the threshold.

  $$ h_{6}^{A} = E - \sqrt {E - F\left( \begin{gathered} \left( {t^{2} - \theta t^{2} + 2c_{r} \theta } \right)\left( {8 + \alpha } \right)^{2} \left( {\gamma + \theta } \right)\left( {9\gamma + 8\theta } \right)^{2} - 12\theta^{4} \left( {128 - \alpha \left( {384 - 15\alpha - 52\alpha^{2} } \right)} \right) \hfill \\ + 48\theta^{5} \left( {1 - \alpha } \right)\left( {32 - \alpha - 4\alpha^{2} } \right) - 27\gamma \theta^{3} \left( {2 - \alpha } \right)\left( {56 - 58\alpha - 25\alpha^{2} } \right) \hfill \\ - \left( {8 + \alpha } \right)^{2} \left( {72\theta c^{2} \gamma^{3} + \gamma^{2} \theta^{2} c\left( {197c - 18} \right) + 4\gamma \theta^{3} c\left( {43c - 14} \right) + 32\theta^{4} c\left( {2c - 1} \right)} \right) \hfill \\ - 8\theta c_{r} \left( {8 + \alpha } \right)^{2} \left( {\gamma + \theta } \right)\left( {9c_{r} \left( {1 - \theta } \right)\gamma^{2} - c\left( {18\gamma^{2} + 29\gamma \theta + 12\theta^{2} } \right)} \right) + 81\theta^{2} \gamma^{2} \alpha \left( {20 - \alpha - 3\alpha^{2} } \right) \hfill \\ - 4c_{r}^{2} \theta^{2} \left( {\gamma + \theta } \right)\left( {1 - \theta } \right)\left( {\gamma \left( {2036 + \alpha \left( {527 + 29\alpha } \right)} \right) + 4\left( {224 + \alpha \left( {61 + 3\alpha } \right)} \right)\theta } \right) \hfill \\ + 8c_{r} \theta^{2} \left( {\gamma + \theta } \right)\left( {1 - \alpha } \right)\left( {1 - \theta } \right)\left( {9\gamma \left( {20 + 7\alpha } \right) + 4\left( {32 + 13\alpha } \right)\theta } \right) - 1296\theta^{2} \gamma^{2} \hfill \\ \end{gathered} \right)} , $$

  where

  $$ E = F\left( \begin{gathered} t\left( {8 + \alpha } \right)^{2} \left( {\gamma + \theta } \right)\left( {1 - \theta } \right)\left( {9\gamma + 8\theta } \right)^{2} + 16\left( {4c\left( {8 + \alpha } \right)^{2} - 32\left( {3 - c_{r} } \right)\alpha + \alpha^{2} \left( {3 + 13c_{r} + 12\alpha } \right)} \right)\theta^{4} \hfill \\ + 9\theta \gamma^{3} \left( {8c\left( {8 + \alpha } \right)^{2} - 4c_{r} \left( {1 - \alpha } \right)\left( {20 + 7\alpha } \right) + 9\left( {16 - \alpha \left( {20 - \alpha - 3\alpha^{2} } \right)} \right)} \right) \hfill \\ + \gamma^{2} \left( {197c\left( {8 + \alpha } \right)^{2} + 27\left( {2 - \alpha } \right)\left( {56 - \alpha \left( {58 + 25\alpha } \right)} \right) - 4c_{r} \left( {308 - \alpha \left( {373 + 178\alpha } \right)} \right)} \right)\theta^{2} \hfill \\ + 4\gamma \left( {384 + 48c\left( {8 + \alpha } \right)^{2} - 3\alpha \left( {384 - \alpha \left( {15 + 52\alpha } \right)} \right) - c_{r} \left( {128 - \alpha \left( {384 + 167\alpha } \right)} \right)} \right)\theta^{3} \hfill \\ \end{gathered} \right), $$
  $$ F = \frac{{\left( {1 - \alpha } \right)}}{\begin{gathered} \left( {1 - \alpha } \right)\left( {81\gamma^{3} \left( {64 + \alpha^{2} \left( {5 + 3\alpha } \right)} \right) - 64\left( {8 + \alpha } \right)^{2} \theta^{4} } \right) + 9\theta \gamma^{2} \left( {1600 - \alpha \left( {2112 + \alpha \left( {3 + \alpha \left( {58 + 75\alpha } \right)} \right)} \right)} \right) \hfill \\ + \theta^{2} \gamma \left( {13312 - \alpha \left( {25920 + \alpha \left( {2112 + \alpha \left( {613 + 624\alpha } \right)} \right)} \right)} \right) + 64\left( {64 - \alpha \left( {256 + \alpha \left( {43 + \alpha \left( {5 + 3\alpha } \right)} \right)} \right)} \right)\theta^{3} \hfill \\ \end{gathered} }. $$

  Hence, if $h \le h_{6}^{A}$, we have $\left( {\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} } \right) + \left( {\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} } \right) \ge 0$; otherwise, $\left( {\pi_{r}^{A*} - \hat{\pi }_{r}^{A*} } \right) + \left( {\pi_{m}^{A*} - \hat{\pi }_{m}^{A*} } \right) < 0$.

• (2) Structure B.

  Taking the difference of the retailer’s profit between with and without digital showrooms, we have $\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} = \frac{\alpha }{t}\left( {\frac{{\left( {c_{r} - c - h + t} \right)^{2} \theta \left( {3\gamma + 2\theta } \right)^{2} }}{{\left( {9\gamma + 8\theta } \right)^{2} }} - \frac{{\left( {2 + 3\gamma } \right)^{2} \left( {h - t - \theta c_{r} } \right)^{2} }}{{\left( {8 + 9\gamma } \right)^{2} }}} \right)$.

  Since $\frac{{\partial^{2} \left( {\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} } \right)}}{{\partial h^{2} }} = \frac{{2\gamma \left( \begin{gathered} 4\left( {1 - \alpha } \right)\left( {8 + \alpha } \right)^{2} \theta^{3} - 81\alpha \gamma \left( {2 + \alpha } \right)^{2} \hfill \\ - 9\gamma \theta \left( {64\left( {1 - \alpha - \alpha^{2} } \right) - 17\alpha^{3} } \right) \hfill \\ - 4\theta^{2} \left( {64 - \alpha \left( {192 + \alpha \left( {96 + 19\alpha } \right)} \right)} \right) \hfill \\ \end{gathered} \right)}}{{t\left( {8 + \alpha } \right)^{2} \left( {9\gamma + 8\theta } \right)^{2} }} < 0$, by setting $\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} = 0$, we can derive.

  $$ h_{o}^{B} = \frac{{\left( \begin{gathered} t\left( {1 - \theta } \right)\left( {729\gamma^{4} + 256\theta^{2} - 32\gamma \left( {29\gamma^{2} + 18\theta } \right)\left( {1 + \theta } \right) + 36\gamma^{2} \left( {9 + \theta \left( {41 + 9\theta } \right)} \right)} \right) \hfill \\ + \left( {8 + 9\gamma } \right)\left( {3\gamma + 2\theta } \right)\left( {\theta c\left( {8 + 9\gamma } \right)\left( {3\gamma + 2\theta } \right) - \left( {2 + 3\gamma } \right)\left( {9\gamma + 8\theta } \right)\sqrt {\left( {c - c_{r} \left( {1 - \theta } \right)} \right)^{2} \theta } } \right) \hfill \\ - 12\theta c_{r} \gamma \left( {1 - \theta } \right)\left( {16\theta + 3\gamma \left( {7 + 9\gamma + 7\theta } \right)} \right) \hfill \\ \end{gathered} \right)}}{{\left( {1 - \theta } \right)\left( {729\gamma^{4} + 256\theta^{2} - 32\gamma \left( {29\gamma^{2} + 18\theta } \right)\left( {1 + \theta } \right) + 36\gamma^{2} \left( {9 + \theta \left( {41 + 9\theta } \right)} \right)} \right)}}. $$

  Hence, when $h \le h_{o}^{B}$, we have $\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} \le 0$; otherwise, $\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} > 0$.

  Taking the difference of the manufacturer’s profit between with and without digital showrooms, we have: $\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} = \frac{{\alpha \left( \begin{gathered} \left( {9\theta \gamma^{2} - 9\gamma \left( {1 + \gamma } \right) - 8\theta } \right)\left( {h - t} \right)^{2} + \theta^{2} c_{r}^{2} \left( {8 - 9\gamma^{2} } \right) - 8c_{r}^{2} \left( {1 + \gamma } \right)\theta^{3} \hfill \\ + 2\theta \gamma \left( {c_{r} + 8c + 9\gamma c - \theta c_{r} } \right)\left( {h - t} \right) + \theta \left( {8 + 9\gamma } \right)\left( {\left( {c + h - t} \right)^{2} \theta - 2c_{r} c\theta + \mu^{2} \gamma } \right) \hfill \\ \end{gathered} \right)}}{{t\left( {8 + 9\gamma } \right)\left( {9\gamma + 8\theta } \right)}}$. Since $\frac{{\partial^{2} \left( {\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} } \right)}}{{\partial h^{2} }} = - \frac{{2\alpha \left( {1 - \theta } \right)\left( {8\theta + 9\gamma \left( {1 + \gamma + \theta } \right)} \right)}}{{t\left( {8 + 9\gamma } \right)\left( {9\gamma + 8\theta } \right)}} < 0$, by setting $\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} = 0$, we can derive:

  $$ h_{m}^{B} = t + \frac{{c_{r} \theta \gamma \left( {1 - \theta } \right) + \theta c\left( {8 + 9\gamma } \right)\left( {\gamma + \theta } \right) - \sqrt {\theta \left( {\gamma + \theta } \right)\left( {9\gamma + 8\theta } \right)\left( {1 + \gamma } \right)\left( {8 + 9\gamma } \right)\left( {c - c_{r} \left( {1 - \theta } \right)} \right)^{2} } }}{{\left( {1 - \theta } \right)\left( {8\theta + 9\gamma \left( {1 + \gamma + \theta } \right)} \right)}}. $$

  Hence, when $h \le h_{m}^{B}$, we have $\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} \le 0$; otherwise, $\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} > 0$.

  Since $\frac{{\partial^{2} \left( {\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} } \right)}}{{\partial h^{2} }} + \frac{{\partial^{2} \left( {\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} } \right)}}{{\partial h^{2} }} < 0$, by setting $\left( {\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} } \right) + \left( {\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} } \right) = 0$, we can derive.

  $$ h_{4}^{B} = t + \frac{{\left( \begin{gathered} \theta c\left( {8 + 9\gamma } \right)^{2} \left( {18\gamma^{2} + 29\gamma \theta + 12\theta^{2} } \right) - \theta \gamma c_{r} \left( {1 - \theta } \right)\left( {128\theta + 9\gamma \left( {27\gamma + 20\left( {1 + \theta } \right)} \right)} \right) \hfill \\ - \left( {9\gamma + 8\theta } \right)\left( {8 + 9\gamma } \right)\sqrt {\theta \left( {12 + \gamma \left( {29 + 18\gamma } \right)} \right)\left( {c - c_{r} \left( {1 - \theta } \right)} \right)^{2} \left( {18\gamma^{2} + 29\gamma \theta + 12\theta^{2} } \right)} \hfill \\ \end{gathered} \right)}}{{3\left( {1 - \theta } \right)\left( {27\gamma^{2} \left( {12 + \gamma \left( {29 + 18\gamma } \right)} \right) + 9\gamma \left( {64 + \gamma \left( {148 + 87\gamma } \right)} \right)\theta + 4\left( {8 + 9\gamma } \right)^{2} \theta^{2} } \right)}}. $$

  Hence, when $h \le h_{4}^{B}$, we have $\left( {\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} } \right) + \left( {\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} } \right) \le 0$; otherwise,$\left( {\pi_{o}^{B*} - \hat{\pi }_{o}^{B*} } \right) + \left( {\pi_{m}^{B*} - \hat{\pi }_{m}^{B*} } \right) > 0$.

Proof of Lemma 2

The profit of centralized dual-channel supply chain can be written as.

$$ \begin{aligned} \pi_{{}}^{C} & = D_{o} \left( {p_{o}^{C} - c} \right) + D_{r} \left( {p_{r}^{C} - c_{r} } \right) \\ & = \left( {1 - \frac{{p_{o}^{C} - p_{r}^{C} + h}}{t}} \right)\left( {p_{o}^{C} - c} \right) + \left( {\alpha \theta \frac{{p_{o}^{C} - p_{r}^{C} + h}}{t} + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - p_{r}^{C} } \right)}}{t}} \right)\left( {p_{r}^{C} - c_{r} } \right). \\ \end{aligned} $$

Since $\frac{{\partial^{2} \pi^{C} }}{{\left( {\partial p_{r}^{C} } \right)^{2} }} = - \frac{{2\left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\theta }}{t} < 0$ and $\frac{{\partial^{2} \pi^{C} }}{{\left( {\partial p_{o}^{C} } \right)^{2} }} = - \frac{2\alpha \theta }{t} < 0$, by solving $\frac{{\partial \pi^{C} }}{{\partial p_{r}^{C} }} = 0$ and $\frac{{\partial \pi^{C} }}{{\partial p_{o}^{C} }} = 0$, we can derive the equilibrium prices as follows: $p_{o}^{C*} = \frac{1}{2}\left( {c - h + t + v + \frac{t\gamma }{\theta }} \right)$ and $p_{r}^{C*} = \frac{1}{2}\left( {c_{r} + v + \frac{t\gamma }{\theta }} \right)$. □

Proof of Theorem 2

• (1) Under structure A, a dual-channel supply chain can achieve coordination when both the online selling price and the offline selling price are equal to those in the centralized supply chain system. Based on Lemma 2, let $p_{fr}^{A*} = p_{r}^{C*}$ and $p_{fo}^{A*} = p_{o}^{C*}$. Then we have $w_{f}^{A*} = \frac{{\gamma \left( {t\gamma - \left( {c + h - t - v} \right)\theta } \right)}}{{2\theta \left( {\gamma + \theta } \right)}}$.

  To guarantee that the manufacturer is willing to provide digital showrooms to consumers and achieve both “win–win coordination” and “supply chain system coordination”, we have two conditions: $\pi_{fm}^{A*} - \pi_{m}^{A*} \ge 0$ and $\pi_{fm}^{A*} - \hat{\pi }_{m}^{A*} \ge 0$. Thus, the fixed compensation needs to satisfy: $F \ge \underline {F} = \max \left\{ {F_{1} ,F_{2} } \right\}$, where $F_{1} = \frac{{\theta \left( {1 - \alpha } \right)\left( {\left( {c_{r} - c - h} \right)\gamma + \left( {c_{r} - v} \right)\theta } \right)^{2} }}{9t\gamma + 8t\theta }$ and.

  $$ F_{2} = \frac{1}{{4t\left( {8 + \alpha } \right)\left( {\gamma + \theta } \right)}}\left( \begin{gathered} \left( {\gamma + \theta } \right)\left( {t\left( {8 + \alpha } \right)\left( {t\gamma + 2v\alpha \theta - 2h\alpha } \right) + 8\alpha h^{2} + \alpha \left( {3 + \alpha } \right)\left( {v\theta - h^{2} \alpha } \right)} \right) \hfill \\ + 4c_{r}^{2} \theta^{2} \left( {\gamma + \theta } \right)^{2} - 4\theta \left( {\gamma + \theta } \right)\left( {c_{r} + v} \right) + v\theta^{2} \left( {1 - \alpha } \right)\left( {\gamma + \theta } \right)\left( {v\left( {2 + \alpha } \right)^{2} - 8c_{r} } \right) \hfill \\ - \alpha \left( {8 + \alpha } \right)\left( {t\gamma - \left( {c + h - t - v} \right)\theta } \right)^{2} \hfill \\ \end{gathered} \right). $$

  Similarly, we also need to guarantee that the retailer is willing to accept such contract, thus, we have two conditions: $\pi_{fr}^{A*} - \pi_{r}^{A*} \ge 0$ and $\pi_{fr}^{A*} - \hat{\pi }_{r}^{A*} \ge 0$. Thus, the fixed compensation needs to satisfy $F \le \overline{F} = \min \left\{ {F_{3} ,F_{4} } \right\}$, where.

  $$ F_{3} = \frac{{3\theta \left( {1 - \alpha } \right)\left( {5\gamma + 4\theta } \right)\left( {3\gamma + 4\theta } \right)\left( {\left( {c_{r} - c - h} \right)\gamma + \left( {c_{r} - v} \right)\theta } \right)^{2} }}{{4t\left( {\gamma + \theta } \right)\left( {9\gamma + 8\theta } \right)^{2} }}, $$
  $$ F_{4} = \frac{1}{4t}\left( {\frac{{\theta \left( {1 - \alpha } \right)\left( {\left( {c_{r} - c - h} \right)\gamma + \left( {c_{r} - v} \right)\theta } \right)^{2} }}{\gamma + \theta } - \frac{{4\left( {2 + \alpha } \right)^{2} \left( {h\alpha - \theta c_{r} + \theta v\left( {1 - \alpha } \right)} \right)^{2} }}{{\left( {8 + \alpha } \right)^{2} }}} \right). $$

  Hence, the two-part tariff contract $\left( {w_{f}^{A*} ,F^{A} } \right)$ with $w_{f}^{A*} = \frac{{\gamma \left( {t\gamma - \left( {c + h - t - v} \right)\theta } \right)}}{{2\theta \left( {\gamma + \theta } \right)}}$ and $\underline {F} \le F^{A} \le \overline{F}$ can achieve both “win–win coordination” and “supply chain system coordination”.

• (2) (2) Under structure B, let $p_{fr}^{B*} = p_{r}^{C*}$ and $p_{fo}^{B*} = p_{o}^{C*}$, then we have $w_{f}^{B*} = \frac{1}{2}\left( {v - c_{r} + \frac{t\gamma }{\theta }} \right)$.

  To guarantee that the manufacturer is willing to provide digital showrooms to consumers and achieve both “win–win coordination” and “supply chain system coordination”, we have two conditions $\pi_{fm}^{B*} - \pi_{m}^{B*} \ge 0$ and $\pi_{fm}^{B*} - \hat{\pi }_{m}^{B*} \ge 0$.

  Since $\pi_{fm}^{B*} - \pi_{m}^{B*} = - F^{B} - \frac{{\alpha \left( {c_{r} - c - h + t} \right)^{2} \left( {\gamma + \theta } \right)\theta }}{9t\gamma + 8t\theta } < 0$ and $\pi_{fm}^{B*} - \hat{\pi }_{m}^{B*} = - F^{B} - \frac{{\alpha \left( {h - t - c_{r} \theta } \right)^{2} }}{{t\left( {8 + \alpha } \right)}} < 0$. Thus, under structure B, the two-part tariff contract cannot achieve either supply chain coordination. □

Proof of Theorem 3

• (1) Under structure A, according to the Eqs. (10) and (11), we have

  $\frac{{\partial^{2} \pi_{qr}^{A} }}{{\left( {\partial p_{qr}^{A} } \right)^{2} }} = - \frac{{2\theta \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\left( {t - \delta_{q}^{A} \left( {\alpha \left( {1 - \theta } \right) + \theta } \right)\theta } \right)}}{{t^{2} }} < 0$ and.

  $\frac{{\partial^{2} \pi_{qm}^{A} }}{{\left( {\partial p_{qo}^{A} } \right)^{2} }} = - \frac{{2\alpha \theta \left( {t + \delta_{q}^{A} \alpha \theta } \right)}}{{t^{2} }} < 0$, by solving $\frac{{\partial \pi_{qr}^{A} }}{{\partial p_{qr}^{A} }} = 0$ and $\frac{{\partial \pi_{qm}^{A} }}{{\partial p_{qo}^{A} }} = 0$, we can derive the equilibrium prices as follows:

  $$ p_{qr}^{A} = \frac{{t\left( \begin{gathered} \left( {2ct + \left( {1 - \lambda^{A} } \right)\left( {c + h + t} \right) + \left( {3 - \lambda^{A} } \right)w_{q}^{A} } \right)\gamma + 2\theta t\left( {c_{r} + v + w_{q}^{A} } \right) \hfill \\ + 2\theta \delta_{q}^{A} \left( {1 - \alpha } \right)\left( {\gamma \left( {\gamma + \theta } \right)\left( {c_{r} - c - h - t} \right) - v\gamma \theta \left( {2 - \lambda^{A} } \right) - 2v\theta^{2} } \right) \hfill \\ \end{gathered} \right)}}{{\left( {3 - \lambda^{A} } \right)\left( {1 - \lambda^{A} } \right)t\gamma + 4t\theta - 4\delta_{q}^{A} \theta^{2} \left( {\theta + \alpha \left( {1 - \lambda^{A} - \theta } \right)} \right)}}, $$
  $$ p_{qo}^{A} = \frac{{\left( \begin{gathered} t\left( {ct\left( {1 + F} \right) + \left( {1 - F} \right)\left( {2c - \left( {1 - F} \right)h + 2t} \right) + \left( {3 - F} \right)w_{q}^{A} } \right)\gamma + \hfill \\ \theta t\left( {c_{r} + 2c + Fc_{r} - 2h + 2t + v + Fv + \left( {3 + F} \right)w_{q}^{A} } \right) + 2\theta \delta_{q}^{A} \left( {1 - \alpha } \right)\left( {c_{r} - c - h - t} \right)\gamma^{2} \hfill \\ + 2\theta \delta_{q}^{A} \left( {1 - \alpha } \right)\left( {\left( {2\left( {c_{r} - c - Fh - t} \right) - \left( {2 - F} \right)v} \right)\gamma \theta + \left( {c_{r} - c + h - t - \left( {2 + F} \right)v} \right)\theta^{2} } \right) \hfill \\ \end{gathered} \right)}}{{\left( {3 - F} \right)\left( {1 - F} \right)t\gamma + 4t\theta - 4\delta_{q}^{A} \theta^{2} \left( {\theta + \alpha \left( {1 - F - \theta } \right)} \right)}}. $$

  To coordinate the supply chain, the manufacturer needs to let $w_{q}^{A}$ and $\delta_{q}^{A}$ satisfy $p_{qo}^{A*} = p_{o}^{C*}$ and $p_{qr}^{A*} = p_{r}^{C*}$. Thus, we have.

  $w_{q}^{A*} = \frac{{\gamma \left( {1 - \lambda^{A} } \right)\left( {\left( {2 - \lambda^{A} } \right)\theta v - \left( {c_{r} - c - h - t} \right)\gamma } \right) - c_{r} \gamma \theta \left( {3 - \lambda^{A} } \right) + \gamma \theta \left( {1 + \lambda^{A} } \right)\left( {c + h + t} \right) - 2\left( {c_{r} - v} \right)\theta^{2} }}{{2\theta \left( {\theta + \gamma \left( {1 - \lambda^{A} } \right)} \right)}}$ and $\delta_{q}^{A*} = \frac{{t\left( {\gamma \left( {1 - \lambda^{A} } \right)\left( {3 - \lambda^{A} } \right) + 4\theta } \right)}}{{4\theta^{2} \left( {1 - \alpha } \right)\left( {\theta + \gamma \left( {1 - \lambda^{A} } \right)} \right)}}$.

  To guarantee that the manufacturer is willing to provide digital showrooms to consumers and achieve both “win–win coordination” and “supply chain system coordination”, two conditions should be satisfied: $\pi_{qm}^{A*} - \pi_{m}^{A*} \ge 0$ and $\pi_{qm}^{A*} - \hat{\pi }_{m}^{A*} \ge 0$. Since $\frac{{\partial^{2} \left( {\pi_{qm}^{A*} - \pi_{m}^{A*} } \right)}}{{\partial \lambda^{2} }} > 0$ and $\frac{{\partial^{2} \left( {\pi_{qm}^{A*} - \hat{\pi }_{m}^{A*} } \right)}}{{\partial \lambda^{2} }} > 0$, thus by solving $\pi_{qm}^{A*} - \pi_{m}^{A*} = 0$, we can obtain $\hat{\lambda }_{1}$ and $\hat{\lambda }_{2}$; by solving $\pi_{qm}^{A*} - \hat{\pi }_{m}^{A*} = 0$, we can obtain $\hat{\lambda }_{3}$ and $\hat{\lambda }_{4}$. Thus, it needs to satisfy $\lambda^{A} \le \lambda_{1}^{A}$ or $\lambda^{A} \ge \lambda_{3}^{A}$, where $\lambda_{1}^{A} = \max \left\{ {\hat{\lambda }_{1} ,\hat{\lambda }_{3} } \right\}$ and $\lambda_{3}^{A} = \min \left\{ {\hat{\lambda }_{2} ,\hat{\lambda }_{4} } \right\}$.

  Similarly, we also need to guarantee that the retailer is willing to accept such contract, thus two conditions should be satisfied:$\pi_{qr}^{A*} - \pi_{r}^{A*} \ge 0$ and $\pi_{qr}^{A*} - \hat{\pi }_{r}^{A*} \ge 0$. Since $\frac{{\partial^{2} \left( {\pi_{qr}^{A*} - \pi_{r}^{A*} } \right)}}{{\partial \lambda^{2} }} < 0$ and $\frac{{\partial^{2} \left( {\pi_{qr}^{A*} - \hat{\pi }_{r}^{A*} } \right)}}{{\partial \lambda^{2} }} < 0$, then by solving $\pi_{qr}^{A*} - \pi_{r}^{A*} = 0$, we can obtain $\tilde{\lambda }_{1}$ and $\tilde{\lambda }_{2}$; by solving $\pi_{qr}^{A*} - \hat{\pi }_{r}^{A*} = 0$, we can obtain $\tilde{\lambda }_{3}$ and $\tilde{\lambda }_{4}$. Thus, it needs to satisfy $\tilde{\lambda }_{1} \le \lambda^{A} \le \tilde{\lambda }_{2}$ or $\tilde{\lambda }_{3} \le \lambda^{A} \le \tilde{\lambda }_{4}$, where $\lambda_{2}^{A} = \max \left\{ {\tilde{\lambda }_{1} ,\tilde{\lambda }_{3} } \right\}$ and $\lambda_{4}^{A} = \min \left\{ {\tilde{\lambda }_{2} ,\tilde{\lambda }_{4} } \right\}$.

  In summary, if $\left\{ {\lambda_{1}^{A} \le \lambda^{A} \le \lambda_{2}^{A} } \right\} \cup \left\{ {\lambda_{3}^{A} \le \lambda^{A} \le \lambda_{4}^{A} } \right\}$, the QD-WPS contract can achieve both “win–win coordination” and “supply chain system coordination”.

• (2) Under structure B, according to the Eqs. (10) and (11), we have $\frac{{\partial^{2} \pi_{qm}^{B} }}{{\left( {\partial p_{qr}^{B} } \right)^{2} }} = \frac{{2\alpha \theta \left( { - \lambda^{B} t + \delta_{{}}^{B} \alpha \theta } \right)}}{{t^{2} }} < 0$ and $\frac{{\partial^{2} \pi_{qo}^{B} }}{{\left( {\partial p_{qo}^{B} } \right)^{2} }} = - \frac{2\alpha \theta }{t} < 0$.

  By solving $\frac{{\partial \pi_{qo}^{B} }}{{\partial p_{qo}^{B} }} = 0$ and $\frac{{\partial \pi_{qm}^{B} }}{{\partial p_{qr}^{B} }} = 0$, we can derive the equilibrium prices as follows:

  $$ p_{qr}^{B} = \frac{{\left( \begin{gathered} t\left( {2c_{r} + \left( {1 - \lambda^{B} } \right)\left( {c + h + t} \right) + \left( {3 - \lambda^{B} } \right)w_{q}^{B} + 2\theta \delta^{B} \alpha \left( {2 - \lambda^{B} } \right)} \right)\gamma - 2\theta \delta^{B} \left( {c_{r} - c - h} \right)\alpha \gamma \hfill \\ + 2\theta \left( {c_{r} + v} \right)\left( {t - \delta^{B} \alpha \theta } \right) \hfill \\ \end{gathered} \right)}}{{\left( {3 - \lambda^{B} } \right)\left( {1 - \lambda^{B} } \right)t\gamma + 4t\theta - 4\delta^{B} \alpha \theta^{2} }} $$

  and

  $$ p_{qo}^{B} = \frac{{\left( \begin{gathered} t\left( {c_{r} \left( {1 + \lambda^{B} } \right) + \left( {1 - \lambda^{B} } \right)\left( {2c - \left( {1 - \lambda^{B} } \right)h + 2t} \right) + \left( {3 - \lambda^{B} } \right)w_{q}^{B} } \right)\gamma - 2\delta^{B} \left( {c_{r} - 2h + 2t + v} \right)\alpha \theta^{2} + \hfill \\ \left( {t\left( {\left( {c_{r} + v} \right)\left( {1 + \lambda^{B} } \right) + 2c - 2h + 2t + 2w_{q}^{B} } \right) - 2\delta^{B} \left( {c_{r} - c - h + \left( {2 - \lambda^{B} } \right)t} \right)\alpha \gamma } \right)\theta \hfill \\ \end{gathered} \right)}}{{\left( {3 - \lambda^{B} } \right)\left( {1 - \lambda^{B} } \right)t\gamma + 4t\theta - 4\delta^{B} \alpha \theta^{2} }}. $$

  In order to coordinate the supply chain, the manufacturer needs to let $w_{q}^{B}$ and $\delta^{B}$ satisfy $p_{qo}^{B*} = p_{o}^{C*}$ and $p_{qr}^{B*} = p_{r}^{C*}$. Thus, we have.

  $$ w_{q}^{B*} { = }\frac{1}{2}\left( {1 - \lambda^{B} } \right)(c_{r} + v - \frac{{\left( {c - c_{r} + h - \left( {2 - \lambda^{B} } \right)t} \right)\gamma }}{\theta }) - \left( {c + h - t} \right), $$

  and $\delta^{B*} = \frac{{t\left( {\gamma \left( {1 - \lambda^{B} } \right)\left( {3 - \lambda^{B} } \right) + 4\theta } \right)}}{{4\alpha \theta^{2} }}$.

  To guarantee that the manufacturer is willing to provide digital showrooms to consumers and achieve both “win–win coordination” and “supply chain system coordination”, two conditions should be satisfied:$\pi_{qm}^{B*} - \pi_{m}^{B*} \ge 0$ and $\pi_{qm}^{B*} - \hat{\pi }_{m}^{B*} \ge 0$. Since $\frac{{\partial^{2} \left( {\pi_{qm}^{B*} - \pi_{m}^{B*} } \right)}}{{\partial \lambda^{2} }} > 0$ and $\frac{{\partial^{2} \left( {\pi_{qm}^{B*} - \hat{\pi }_{m}^{B*} } \right)}}{{\partial \lambda^{2} }} > 0$, then we can obtain $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{1}$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{2}$ by solving $\pi_{qm}^{B*} - \pi_{m}^{B*} = 0$ and $\pi_{qm}^{B*} - \hat{\pi }_{m}^{B*} = 0$. Thus, it needs to satisfy $\lambda^{B} \le \lambda_{2}^{B}$, where $\lambda_{2}^{B} = \min \left\{ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{1} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{2} } \right\}$.

  Similarly, we also need to guarantee that the retailer is willing to accept such contract, thus, two conditions should be satisfied: $\pi_{qo}^{B*} - \pi_{o}^{B*} \ge 0$ and $\pi_{qo}^{B*} - \hat{\pi }_{o}^{B*} \ge 0$. Since $\frac{{\partial^{2} \left( {\pi_{qo}^{B*} - \pi_{o}^{B*} } \right)}}{{\partial \lambda^{2} }} < 0$ and $\frac{{\partial^{2} \left( {\pi_{qo}^{B*} - \hat{\pi }_{o}^{B*} } \right)}}{{\partial \lambda^{2} }} < 0$, then we can obtain $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{3}$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{4}$ by solving $\pi_{qo}^{B*} - \pi_{o}^{B*} = 0$ and $\pi_{qo}^{B*} - \hat{\pi }_{o}^{B*} = 0$. Thus, it needs to satisfy $\lambda^{B} \ge \lambda_{1}^{B}$, where $\lambda_{1}^{B} = \min \left\{ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{3} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\lambda }_{4} } \right\}$.

  In summary, if $\lambda_{1}^{B} \le \lambda^{B} \le \lambda_{2}^{B}$, the QD-WPS contract can achieve both “win–win coordination” and “supply chain system coordination”. □

Appendix B: channel structure D

2.1 The benchmark without digital showrooms

The profit functions of the online channel, the offline retailer, and the manufacturer under structure D can be expressed as.

$$ \hat{\pi }_{o}^{D} = \hat{D}_{o}^{D} \left( {\hat{p}_{o}^{D} - \hat{w}^{D} } \right), $$

(B1)

$$ \hat{\pi }_{r}^{D} = \hat{D}_{r}^{D} \left( {\hat{p}_{r}^{D} - \hat{w}^{D} - c_{r} } \right), $$

(B2)

$$ \hat{\pi }_{m}^{D} = \left( {\hat{D}_{o}^{D} + \hat{D}_{r}^{D} } \right)\hat{w}^{D} . $$

(B3)

By substituting $\hat{D}_{o}$ and $\hat{D}_{r}$ into Eqs. (B1) and (B2) and using the first-order conditions, $\frac{{\partial \hat{\pi }_{r}^{D} }}{{\partial \hat{p}_{r}^{D} }} = 0$ and $\frac{{\partial \hat{\pi }_{o}^{D} }}{{\partial \hat{p}_{o}^{D} }} = 0$, we can derive the best response functions as follows:

$$ \hat{p}_{r}^{D} = \frac{{\left( {h + \hat{p}_{o}^{D} } \right)\alpha + \left( {c_{r} + v + \hat{w}^{D} - v\alpha } \right)\theta }}{2\theta }, $$

(B4)

$$ \hat{p}_{o}^{D} = \frac{1}{2}\left( { - h + t + \hat{w}^{D} + \hat{p}_{r}^{D} \theta } \right). $$

(B5)

We can verify that the second-order derivatives of the profit functions are negative. Solving these two equations yields the equilibrium prices:

$$ \hat{p}_{r}^{D} = \frac{{\left( {h + t + \hat{w}^{D} } \right)\alpha + 2\left( {c_{r} + v + \hat{w}^{D} - v\alpha } \right)\theta }}{{\left( {4 - \alpha } \right)\theta }}, $$

(B6)

$$ \hat{p}_{o}^{D} = - \frac{{2\left( {t + \hat{w}^{D} } \right) + h\left( { - 2 + \alpha } \right) + \left( {c_{r} + v + \hat{w}^{D} - v\alpha } \right)\theta }}{ - 4 + \alpha }. $$

(B7)

Substituting the equilibrium prices into Eq. (B3) and using the first-order conditions, $\frac{{\partial \hat{\pi }_{m}^{D} }}{{\partial \hat{w}^{D} }} = 0$, we can derive the best response functions as follows:

$$ \hat{w}^{D*} = \frac{{t\alpha \left( {2 + \theta } \right) - h\alpha \left( {2 - \alpha - \theta } \right) + \theta \left( {v\left( {1 - \alpha } \right)\left( {\alpha + 2\theta } \right) + c_{r} \left( {\alpha - \left( {2 - \alpha } \right)\theta } \right)} \right)}}{{2\left( {\left( {2 - \alpha } \right)\alpha - 2\alpha \theta + \left( {2 - \alpha } \right)\theta^{2} } \right)}}. $$

(B8)

Substituting the equilibrium prices $\hat{w}^{D*}$ into (B6), (B7) and Eqs. (B1)-(B3) yields the equilibrium retail prices and profits:

Structure	Optimal decisions
Structure D	$\hat{w}^{D*} = \frac{{t\alpha \left( {2 + \theta } \right) - h\alpha \left( {2 - \alpha - \theta } \right) + \theta \left( {v\left( {1 - \alpha } \right)\left( {\alpha + 2\theta } \right) + c_{r} \left( {\alpha - \left( {2 - \alpha } \right)\theta } \right)} \right)}}{{2\left( {\left( {2 - \alpha } \right)\alpha - 2\alpha \theta + \left( {2 - \alpha } \right)\theta^{2} } \right)}}$
	$\hat{p}_{o}^{D*} = \frac{{\left( \begin{gathered} 2\left( {3 - \alpha } \right)\left( {1 - \alpha } \right)\left( {2t\gamma - h\gamma \left( {2 - \alpha } \right) + \gamma \theta \left( {c_{r} + v - v\alpha } \right) + v\theta^{3} } \right) + \alpha \theta \left( {8h - 3h\alpha - 4t} \right) \hfill \\ - \left( {8h - 4\left( {2t + v} \right) + c_{r} \left( {4 + \alpha } \right) + \alpha \left( {3t + v\left( {7 - 3\alpha } \right) - h\left( {9 - 2\alpha } \right)} \right)} \right)\theta^{2} + \theta^{3} c_{r} \left( {2 - \alpha } \right) \hfill \\ \end{gathered} \right)}}{{2\left( {4 - \alpha } \right)\left( {\alpha \left( {2 - \alpha } \right) - 2\alpha \theta + \left( {2 - \alpha } \right)\theta^{2} } \right)}}$
	$\hat{p}_{r}^{D*} = \frac{{\left( \begin{gathered} 2\left( {3 - \alpha } \right)\left( {t\alpha^{2} + t\alpha \theta^{2} + h\alpha \theta^{2} + 2v\left( {3 - \alpha } \right)\theta^{3} } \right) + \alpha \theta v\left( {1 - \alpha } \right)\left( {8 - 3\alpha - 4\theta } \right) \hfill \\ + h\alpha \left( {2\alpha - \alpha^{2} - 4\theta - \alpha \theta } \right) + t\alpha \theta \left( {4 - 3\alpha } \right) \hfill \\ + \theta c_{r} \left( {\left( {8 - 3\alpha } \right)\alpha - \left( {8 - \alpha } \right)\alpha \theta + 2\left( {2 - \alpha } \right)\theta^{2} } \right) \hfill \\ \end{gathered} \right)}}{{2\theta \left( {4 - \alpha } \right)\left( {\left( {2 - \alpha } \right)\alpha - 2\alpha \theta + \left( {2 - \alpha } \right)\theta^{2} } \right)}}$

2.2 Optimal decisions in the presence of digital showrooms

The profit functions of the online channel, the offline retailer, and the manufacturer can be expressed as.

$$ \pi_{o}^{D} = D_{o}^{D} \left( {p_{o}^{D} - w^{D} - c} \right), $$

(B9)

$$ \pi_{r}^{D} = D_{r}^{D} \left( {p_{r}^{D} - w^{D} - c_{r} } \right), $$

(B10)

$$ \pi_{m}^{D} = \left( {D_{o}^{D} + D_{r}^{D} } \right)w^{D} . $$

(B11)

By substituting $D_{o}^{D}$ and $D_{r}^{D}$ into Eqs. (B9) and (B10) and using the first-order conditions, $\frac{{\partial \pi_{r}^{D} }}{{\partial p_{r}^{D} }} = 0$ and $\frac{{\partial \pi_{m}^{D} }}{{\partial p_{o}^{D} }} = 0$, we can derive the best response functions as follows:

$$ p_{r}^{D} = \frac{{\left( {c_{r} + h + p_{o}^{D} + w^{D} } \right)\alpha + \left( {c_{r} + v + w^{D} } \right)\left( {1 - \alpha } \right)\theta }}{{2\alpha \left( {1 - \theta } \right) + 2\theta }}, $$

(B12)

$$ p_{o}^{D} = \frac{1}{2}\left( {c - h + p_{r}^{D} + t + w^{D} } \right). $$

(B13)

We can verify that the second-order derivatives of the profit functions are negative. Solving these two equations yields the equilibrium prices:

$$ p_{r}^{D} = \frac{{\left( {2c_{r} + c + h + t + 3w^{D} } \right)\alpha + 2\left( {c_{r} + v + w^{D} } \right)\left( {1 - \alpha } \right)\theta }}{{3\alpha + 4\left( {1 - \alpha } \right)\theta }}, $$

(B14)

$$ p_{o}^{D} = \frac{{\left( {c_{r} + 2c - h + 2t + 3w^{D} } \right)\alpha + \left( {c_{r} + 2c - 2h + 2t + v + 3w^{D} } \right)\left( {1 - \alpha } \right)\theta }}{{3\alpha + 4\left( {1 - \alpha } \right)\theta }}. $$

(B15)

Substituting the equilibrium prices into Eq. (B11) and using the first-order conditions, $\frac{{\partial \pi_{m}^{D} }}{{\partial w^{D} }} = 0$, we can derive the best response functions as follows:

$$ w^{D*} = \frac{{3t\alpha^{2} - \left( {2c_{r} + c + h - 3\left( {t + v} \right)} \right)\left( {1 - \alpha } \right)\alpha \theta - 2\left( {c_{r} - v} \right)\left( {1 - \alpha } \right)^{2} \theta^{2} }}{{2\theta \left( {1 - \alpha } \right)\left( {3\alpha + 2\left( {1 - \alpha } \right)\theta } \right)}}. $$

(B16)

Substituting the equilibrium prices $w^{ND*}$ into (B14), (B15) and Eqs. (B9-B11) yields the equilibrium retail prices and profits:

Structure	Optimal decisions
Structure D	$w^{D*} = \frac{{3t\gamma^{2} - \left( {2c_{r} + c + h - 3\left( {t + v} \right)} \right)\gamma \theta - 2\left( {c_{r} - v} \right)\theta^{2} }}{{2\theta \left( {3\gamma + 2\theta } \right)}}$
	$p_{o}^{D*} = \frac{{4\left( {c - h + t} \right) - c_{r} + 5v}}{8} + \frac{\gamma }{8}\left( {\frac{{2\left( {c_{r} - c - h + t} \right)}}{3\gamma + 2\theta } + \frac{{c_{r} + 2\left( {c + h + t} \right) - 3v}}{3\gamma + 4\theta } + \frac{4t}{\theta }} \right)$
	$p_{r}^{D*} = \frac{{3t\gamma^{2} + \left( {2c_{r} + c + h + 5t + 3v} \right)\gamma \theta + 2\left( {c_{r} + 3v} \right)\theta^{2} }}{{2\theta \left( {3\gamma + 4\theta } \right)}}$

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Appendix C: Digital showrooms partially resolve match uncertainty

In this section, we first introduce the extension part base on the main model. Then, we derive the equilibrium price decisions in this case. Last, we compare the equilibrium profits of supply chain members with/without digital showrooms to check the robustness of our main results.

When digital showrooms can only partially solve consumers’ valuation uncertainty. Following Gao and Su (2017) and Sun et al. (2020), we assume that consumer will receive a signal $S$ if they view digital showrooms, where $S$ is binary, i.e., $s = \{ 0,1\}$. Specifically, some consumers believe that the product fits them well and remain interested in the product (i.e., $s = 1$), while remaining consumers discover that the product does not fit them and leave the market (i.e., $s = 0$). Moreover, we also assume that a fraction $\rho$ ($\rho \in \left[ {0,1} \right]$) of misfit-type consumers learn their true types, where the parameter $\rho$ can be interpreted as the degree of informativeness of digital showrooms. Note that if $\rho = 1$, digital showrooms offer a perfect signal to consumer, which is the assumption of our main model. Then, the probabilities that fit-type and misfit-type consumers receive the signal $s = 1$ are $P\left( {s \, = \, 1|Y} \right) \, = \, 1$ and $P\left( {s \, = \, 1|F} \right){ = }1 - \rho$, respectively, where $Y$ and $F$ denote fit-type and misfit-type consumers, respectively, with $P\left( Y \right) = \theta$ and $P\left( F \right) = 1 - \theta$. Following the Bayesian updating process, the posterior belief of a remaining consumer on the probability of fitness (i.e., $\theta^{P}$) can be computed as

$$ \theta^{P} = P\left( {Y|s = 1} \right) = \frac{{P\left( {s = 1|Y} \right)P\left( Y \right)}}{{P\left( {s = 1|Y} \right)P\left( Y \right) + P\left( {s = 1|F} \right)P\left( F \right)}} = \frac{\theta }{{1 - \rho \left( {1 - \theta } \right)}}. $$

(C1)

Note, we use superscript “P” to denote the case when the digital showrooms can only partly resolve consumer match uncertainty.

Then the net utility of a dual-channel consumer purchasing the product from the online and offline channels can be characterized as $EU_{o}^{P} = \theta^{P} v - p_{o}^{P} - h$ and $EU_{r}^{P} = \theta^{p} \left( {v - p_{o}^{P} } \right) - tx$, respectively. To avoid trivial situations, we make several assumptions in this model: (1)$p_{o}^{P} + h \le \theta^{P} v$, to avoid negative customer online utility for any online selling price; (2) $p_{r}^{P} \le v$, to avoid negative customer offline utility for any offline selling price; and (3) $\min \left\{ {\frac{\theta v - t}{\theta },\frac{{p_{o}^{P} + h - t}}{{\theta^{P} }}} \right\} \le p_{r}^{P} \le \frac{{p_{o}^{P} + h}}{{\theta^{P} }}$, in case of one of the shopping channel always dominate other channel. To maximize their respective utilities, dual-channel consumers with $EU_{o}^{P} \ge EU_{r}^{P}$ will purchase the product from the online channel, and those with $EU_{o}^{P} < EU_{r}^{P}$ will choose to visit the physical store. In this case, the amount of dual-channel consumers that purchase the product from the online and offline channels can be formulated as.

$$ D_{o}^{P} = \alpha \left( {1 - \rho \left( {1 - \theta } \right)} \right)\left( {1 - \frac{{p_{o}^{P} - \theta^{P} p_{r}^{P} + h}}{t}} \right), $$

(C2)

$$ D_{r}^{P} = \alpha \theta^{P} \left( {1 - \rho \left( {1 - \theta } \right)} \right)\left( {\frac{{p_{o}^{P} - \theta^{P} p_{r}^{P} + h}}{t}} \right) + \theta \left( {1 - \alpha } \right)\frac{{\theta \left( {v - p_{r}^{P} } \right)}}{t}. $$

(C3)

Under structure A, the profit functions of the manufacturer and the offline retailer can be expressed as.

$$ \pi_{pm}^{A} = \left( {p_{po}^{A} - c} \right)D_{po}^{A} + w_{p}^{A} D_{pr}^{A} , $$

(C4)

$$ \pi_{pr}^{A} = \left( {p_{pr}^{A} - w_{p}^{A} - c_{r} } \right)D_{pr}^{A} . $$

(C5)

Under structure B, the profit functions of the manufacturer and the online retailer can be expressed as.

$$ \pi_{pm}^{B} = \left( {p_{pr}^{B} - c_{r} } \right)D_{pr}^{B} + w_{p}^{B} D_{po}^{B} , $$

(C6)

$$ \pi_{po}^{B} = \left( {p_{po}^{B} - w_{p}^{B} - c} \right)D_{po}^{B} . $$

(C7)

Similar to the proof process of Theorem 1, we can obtain the equilibrium pricing decisions for the member firms under each structure, which are summarized in following Table 5.

Table 5 Equilibrium decisions under structures A and B in the presence of digital showrooms

According to above results, we can compare the equilibrium profits of supply chain members with/without digital showrooms. We define $\mu = \frac{{\left( {2 + \alpha } \right)}}{{\left( {8 + \alpha } \right)}}$.

• (1) Under structure A, we define $\mu = \frac{{\left( {2 + \alpha } \right)}}{{\left( {8 + \alpha } \right)}}$, then we have

  $$ \pi_{pr}^{A*} - \hat{\pi }_{r}^{A*} = \frac{{\theta \left( {1 - \alpha } \right)\left( {3\theta^{P} \gamma + 2\theta } \right)^{2} \left( {c\gamma - \theta^{P} c_{r} \gamma + h\gamma - \theta c_{r} + v\theta } \right)^{2} }}{{t\left( {\theta^{P} \gamma + \theta } \right)\left( {9\theta^{P} \gamma + 8\theta } \right)^{2} }} - \frac{{\mu^{2} \left( {h\alpha - \left( {c_{r} - v\left( {1 - \alpha } \right)} \right)\theta } \right)^{2} }}{t} $$

  and $\frac{{\partial^{2} \left( {\pi_{pr}^{A*} - \hat{\pi }_{r}^{A*} } \right)}}{{\partial h^{2} }} = - \frac{2\alpha }{t}\left( {\frac{{\alpha \mu^{2} \left( {\theta^{P} \gamma + \theta } \right)\left( {9\theta^{P} \gamma + 8\theta } \right)^{2} - \gamma \theta \left( {3\theta^{P} \gamma + 2\theta } \right)^{2} }}{{\left( {\theta^{P} \gamma + \theta } \right)\left( {9\theta^{P} \gamma + 8\theta } \right)^{2} }}} \right) < 0$.

  By setting $\pi_{pr}^{A*} - \hat{\pi }_{r}^{A*} = 0$, we can derive.

  $$ h_{pr}^{A} = \frac{{\left( \begin{gathered} \mu \left( {\theta^{P} \gamma + \theta } \right)\left( {9\theta^{P} \gamma + 8\theta } \right)\left( {3\theta^{P} \gamma + 2\theta } \right)\sqrt {\frac{{\theta \left( {1 - \alpha } \right)\left( {c\alpha \gamma - \theta^{P} c_{r} \alpha \gamma - \theta c_{r} \left( {\alpha - \gamma } \right)} \right)^{2} }}{{\left( {\theta^{P} \gamma + \theta } \right)}}} + \hfill \\ \theta \mu^{2} \left( {c_{r} - v\left( {1 - \alpha } \right)} \right)\left( {81\left( {\theta^{P} } \right)^{3} \gamma^{3} + 208\theta^{P} \gamma \theta^{2} + 225\left( {\theta^{P} } \right)^{2} \gamma^{2} \theta + 64\theta^{3} } \right) \hfill \\ + \theta \left( {4\theta^{2} \left( {c\gamma - \theta \left( {c_{r} - v} \right)} \right) - 9\left( {\theta^{P} } \right)^{3} \gamma^{3} c_{r} + \theta^{P} \left( {4\gamma \theta + 3\theta^{P} \gamma^{2} } \right)\left( {3c\gamma - \theta \left( {7c_{r} - 3v} \right)} \right)} \right) \hfill \\ \end{gathered} \right)}}{{\alpha \mu^{2} \left( {81\left( {\theta^{P} } \right)^{3} \gamma^{3} + 225\left( {\theta^{P} } \right)^{2} \gamma^{2} \theta + 208\theta^{P} \gamma \theta^{2} + 64\theta^{3} } \right) - \gamma \theta \left( {4\theta^{2} + 12\theta^{P} \gamma \theta + 9\left( {\theta^{P} } \right)^{2} \gamma^{2} } \right)}}. $$

  Hence, when $h \le h_{pr}^{A}$, we have $\pi_{pr}^{A*} - \hat{\pi }_{r}^{A*} \ge 0$; otherwise, $\pi_{pr}^{A*} - \hat{\pi }_{r}^{A*} < 0$.

  Similarly, we have.

  $$ \frac{{\partial^{2} \left( {\pi_{pm}^{A*} - \hat{\pi }_{m}^{A*} } \right)}}{{\partial h^{2} }} = \frac{{\left( \begin{gathered} 2\alpha \theta \left( {8 + \alpha } \right)\left( {3\left( {\theta^{P} } \right)^{2} \left( {7 - 4\theta^{P} } \right)\gamma^{2} + 8\theta^{P} \left( {4 - 3\theta^{P} } \right)\gamma \theta + 12\left( {1 - \theta^{P} } \right)\theta^{2} } \right) \hfill \\ - \alpha \theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right)^{2} \left( {8 - \alpha \left( {3 + \alpha } \right)} \right) \hfill \\ \end{gathered} \right)}}{{t\theta^{P} \left( {8 + \alpha } \right)\left( {9\theta^{P} \gamma + 8\theta } \right)^{2} }} < 0, $$

  then we can derive one root $h_{pm}^{A}$ by setting $\pi_{pm}^{A*} - \hat{\pi }_{m}^{A*} = 0$. Hence, when $h \le h_{pm}^{A}$, $\pi_{pm}^{A*} - \hat{\pi }_{m}^{A*} \le 0$; otherwise, $\pi_{pm}^{A*} - \hat{\pi }_{m}^{A*} > 0$.

• (2) Under structure B, we have

  $$ \pi_{po}^{B*} - \hat{\pi }_{o}^{B*} = \frac{1}{t}\left( {\frac{{\alpha \theta \left( {c - \theta^{P} c_{r} + h - t} \right)^{2} \left( {3\theta^{P} \gamma + 2\theta } \right)^{2} }}{{\theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right)^{2} }} - \alpha \mu^{2} \left( {h - t - c_{r} \theta } \right)^{2} } \right) $$

and $\frac{{\partial^{2} \left( {\pi_{po}^{B*} - \hat{\pi }_{o}^{B*} } \right)}}{{\partial h^{2} }} = \frac{{2\alpha \theta \left( {3\theta^{P} \gamma + 2\theta } \right)^{2} - 2\alpha \mu^{2} \theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right)^{2} }}{{t\theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right)^{2} }} < 0$.

By setting $\pi_{po}^{B*} - \hat{\pi }_{o}^{B*} = 0$, we can derive.

$$ h_{po}^{B} = \frac{{\left( \begin{gathered} \mu^{2} \left( {81\left( {\theta^{P} } \right)^{3} t\gamma^{2} + 4\theta^{3} 16\theta^{P} c_{r} + 9\theta \gamma \left( {\theta^{P} } \right)^{2} \left( {16t + 9\theta^{P} c_{r} \gamma } \right) + {16}\theta^{P} \theta^{2} \left( {{4}t + {9}\theta^{P} c_{r} \gamma } \right)} \right) \hfill \\ { + }\theta \left( {c - \theta^{P} c_{r} - t} \right)\left( {{12}\theta^{P} \theta \gamma { + }9\left( {\theta^{P} } \right)^{2} \gamma^{{2}} { + }4\theta^{{2}} } \right) - \mu \sqrt {\theta^{P} \theta \left( {3\theta^{P} \gamma + 2\theta } \right)^{2} \left( {9\theta^{P} \gamma + 8\theta } \right)^{2} \left( {c - c_{r} \left( {\theta^{P} - \theta } \right)} \right)^{2} } \hfill \\ \end{gathered} \right)}}{{\mu^{2} \left( {81\left( {\theta^{P} } \right)^{3} \gamma^{2} + 144\left( {\theta^{P} } \right)^{2} \gamma \theta + 64\theta^{P} \theta^{2} } \right) - \theta \left( {12\theta^{P} \theta \gamma + 4\theta^{2} + 9\left( {\theta^{P} } \right)^{2} \gamma^{2} } \right)}}. $$

Hence, when $h \le h_{po}^{B}$, we have $\pi_{po}^{B*} - \hat{\pi }_{o}^{B*} \le 0$; otherwise, $\pi_{po}^{B*} - \hat{\pi }_{o}^{B*} > 0$.

Similarly, we have $\frac{{\partial^{2} \left( {\pi_{pm}^{B*} - \hat{\pi }_{pm}^{B*} } \right)}}{{\partial h^{2} }} = - \frac{2\alpha }{t}\left( {\frac{{\theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right) - \theta \left( {8 + \alpha } \right)\left( {\theta^{P} \gamma + \theta } \right)}}{{\theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right)\left( {8 + \alpha } \right)}}} \right) < 0$. By setting $\pi_{pm}^{B*} - \hat{\pi }_{pm}^{B*} = 0$, we can derive.

$$ h_{pm}^{B} = G - \sqrt {G^{2} - \alpha \left( \begin{gathered} \alpha \left( {\theta^{P} } \right)^{2} \left( {8 + \alpha } \right)\left( {9t^{2} \gamma + 2\theta \left( {18tv - 4cc_{r} - 5c_{r} t + 2\theta^{P} c_{r}^{2} } \right)} \right) \hfill \\ + 4\theta \left( {1 - \alpha } \right)\left( {8 + \alpha } \right)\left( {\left( {c - t} \right)^{2} \theta + \theta^{P} \left( {3t^{2} \gamma - 2tc\gamma - 2t\theta c_{r} + c^{2} \gamma - 2cc_{r} \theta } \right)} \right) \hfill \\ + 2\theta c_{r} \left( {\theta^{P} } \right)^{2} \left( {9\alpha t\left( {4 + \alpha } \right) + 2c_{r} \left( {8 - \alpha \left( {16 + \alpha } \right)} \right)\theta } \right) \hfill \\ - t^{2} \theta^{P} \left( {2 + \alpha } \right)^{2} \left( {9\theta^{P} \gamma + 8\theta } \right) - 16\theta^{P} \theta^{2} c_{r} \left( {1 - \alpha } \right)\left( {2\theta c_{r} - 4t - \alpha t} \right) \hfill \\ \end{gathered} \right)} . $$

where $G = \frac{{\theta^{P} \left( {9\theta^{P} t\gamma + \theta \left( {8t + 9\theta^{P} c_{r} \gamma } \right) + 8\theta^{2} c_{r} } \right) + \theta \left( {\theta^{P} \gamma + \theta } \right)\left( {c - \theta^{P} c_{r} - t} \right)\left( {8 + \alpha } \right)}}{{\theta^{P} \left( {9\theta^{P} \gamma + 8\theta } \right) - \theta \left( {8 + \alpha } \right)\left( {\theta^{P} \gamma + \theta } \right)}}$.

Hence, when $h \le h_{pm}^{B}$, we have $\pi_{pm}^{B*} - \hat{\pi }_{pm}^{B*} \le 0$; otherwise, $\pi_{pm}^{B*} - \hat{\pi }_{pm}^{B*} > 0$.

In summary, in the case of digital showrooms partially resolve match uncertainty, the effects of the digital showrooms on the profits of supply chain members are as shown in Table 6.

Table 6 Effects of digital showrooms on the players’ profits

By comparing the results in Table 6 and those in Proposition 3, we can find that the main results in our base model remain robust when digital showrooms partially solve match uncertainty. □