Optimal joint decision of information disclosure and ordering in a blockchain-enabled luxury supply chain

Abstract

This paper explores the optimal joint decision of product information disclosure and ordering in a blockchain-enabled luxury supply chain. Using analytical models, we investigate the optimal joint decision of information disclosure and ordering under three scenarios (i.e., wholesale contracts only, revenue-sharing (RS) contracts only, and a hybrid of these two types of contracts). Furthermore, we extend our study to examine the impacts of the number of competing retailers and the retailers’ fairness concerns on supply chain members’ optimal decisions. Lastly, the theoretical results are checked and illustrated by numerical examples with sensitivity analysis. The main findings are as follows: (1) As the proportion of information-sensitive consumers in the market increases, the level of product information disclosure of supply chain members increases in varying degrees, while supply chain members’ order quantities and profits first decrease and then increase in varying degrees. (2) When a RS contract is acceptable for all supply chain members, all members benefit from the cooperation between the manufacturer and retailers. (3) Although all supply chain members may benefit from an increase in the number of retailers, when the number of retailers is greater than a certain threshold, retailers would be caught in a “prisoner’s dilemma” of product information disclosure due to consumer information overload. Moreover, to maximize business profits, manufacturers should sometimes strictly limit and control the number of their reseller partners, rather than blindly expand their markets. (4) Retailers may benefit from their own fairness concerns if and only if the level of fairness concerns is sufficiently low, otherwise such concerns would be harmful to all supply chain members.

Appendices

Appendices

The proof of Theorem 1

Proof

Under Model W, substituting Eq. (2) into Eqs. (5) and (6), respectively. Thus, the profits of supply chain members are given as follows:

$$\begin{aligned} \mathop {\pi _{R1}^{W}(q_1,\alpha _1)=(p-v+g_r)\left( q_1-\beta _1\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -(c_r-v)q_1-\beta _1g_r\mu -wq_1-k\alpha _{1}^{2}}, \end{aligned}$$

(A.1)

$$\begin{aligned} \mathop {\pi _{R2}^{W}(q_2,\alpha _2)=(p-v+g_r)\left( q_2-\beta _2\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -(c_r-v)q_2-\beta _2g_r\mu -wq_2-k\alpha _{2}^{2}} \end{aligned}$$

(A.2)

and

$$\begin{aligned} \pi _{M}^{W}(\alpha _0)=\sum _{i=1}^{2}\bigg [(g_m\bigg (q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\bigg )-c_mq_i-\beta _{i}g_m\mu +wq_i \bigg ]-k\alpha _{0}^{2}. \end{aligned}$$

(A.3)

And $\beta _{1}=(1-\lambda )+\lambda (\alpha _{0}+\alpha _{1}-\gamma \alpha _{2})$, $\beta _{2}=(1-\lambda )+\lambda (\alpha _{0}+\alpha _{2}-\gamma \alpha _{1})$.

Solving this two-stage game using backward induction technology. Taking the first-order partial derivative of $\pi _{Ri}^{W}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $q_i$ and $\alpha _i$, respectively. Thus, we have

$$\begin{aligned} \frac{\partial \pi _{R1}^{W}}{\partial q_1}&=(p-v+g_r)(1-F(\frac{q_1}{\beta _1}))-(c_r-v)-w, \end{aligned}$$

(A.4)

$$\begin{aligned} \frac{\partial \pi _{R2}^{W}}{\partial q_2}&=(p-v+g_r)(1-F(\frac{q_2}{\beta _2}))-(c_r-v)-w, \end{aligned}$$

(A.5)

$$\begin{aligned} \frac{\partial \pi _{R1}^{W}}{\partial \alpha _{1}}&=\lambda (p-v+g_r)\left( \frac{q_1}{\beta _1}F(\frac{q_1}{\beta _1})-\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -\lambda g_r\mu -2k\alpha _1, \end{aligned}$$

(A.6)

$$\begin{aligned} \frac{\partial \pi _{R2}^{W}}{\partial \alpha _{2}}&=\lambda (p-v+g_r)\left( \frac{q_2}{\beta _2}F(\frac{q_2}{\beta _2})-\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -\lambda g_r\mu -2k\alpha _2. \end{aligned}$$

(A.7)

Taking the second-order partial derivative of $\mathop \pi \nolimits _{Ri}^{W}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $\alpha _i$ and $q_i$, respectively. Thus, the corresponding Hessian matrix is given as follows:

$$\begin{aligned} \mathop H_{Ri}\nolimits ^{W}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{W} }}{{\partial \mathop \alpha \nolimits _i^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{W} }}{{\partial \mathop \alpha \nolimits _i \partial q_i}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{W} }}{{\partial q_i\partial \mathop \alpha \nolimits _i }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{W} }}{{\partial \mathop q_i\nolimits ^2 }}} \end{array}} \right] \nonumber \\&=\begin{bmatrix} -\frac{\lambda ^2(p-v+g_r)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k &{} \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i})\\ \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i}) &{} -\frac{(p-v+g_r)}{\beta _i}f(\frac{q_i}{\beta _i}) \end{bmatrix}. \end{aligned}$$

(A.8)

Thus, we have

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{W} }}{{\partial \mathop \alpha \nolimits _i^2 }}&= -\frac{\lambda ^2(p-v+g_r)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k<0, \end{aligned}$$

(A.9)

$$\begin{aligned} \left| {\mathop H_{Ri}\nolimits ^{W} } \right|&= \left| {\begin{array}{*{20}{c}} -\frac{\lambda ^2(p-v+g_r)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k &{} \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i})\\ \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i}) &{} -\frac{(p-v+g_r)}{\beta _i}f(\frac{q_i}{\beta _i})\end{array}} \right| \nonumber \\&= \frac{2k(p-v+g_r)}{\beta _i}f(\frac{q_i}{\beta _i})>0. \end{aligned}$$

(A.10)

Eqs. (A.9) and (A.10) yield that the Hessian matrix is a negative definite matrix. Consequently, the profit of the retailer i ($=1$, 2) is a joint concave function of $\alpha _i$ and $q_i$, and the first-order conditions can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order conditions $\frac{{\partial \mathop \pi \nolimits _{Ri}^{W} (q_{i}, \alpha _{i})}}{{\partial \mathop \alpha \nolimits _i }} = 0$ and $\frac{{\partial \mathop \pi \nolimits _{Ri}^{W}(q_{i}, \alpha _{i}) }}{{\partial q_i}} = 0$. Thus, we can get four solutions

$$\begin{aligned} \mathop \alpha \nolimits _i^{W*}&=\frac{\lambda \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}, \end{aligned}$$

(A.11)

$$\begin{aligned} \mathop q\nolimits _i^{W}&=\left( 1-\lambda +\lambda (\alpha _0+(1-\gamma )\alpha _i)\right) A, \end{aligned}$$

(A.12)

and $A=F^{-1}\left( \frac{p-c_r-w+g_r}{p-v+g_r}\right)$.

Then, substituting Eqs. (A.11) and (A.12) into Eq. (2), and taking the first-order partial derivative of $\mathop \pi \nolimits _M^{W}(\alpha _{0})$ with respect to $\mathop \alpha \nolimits _0$, we have

$$\begin{aligned} \frac{\partial \pi _{M}^{W}(\alpha _{0})}{\partial \alpha _{0}}=2 \lambda (g_m+w-c_m)A-2\lambda g_m\bigg (\int _{0}^{A}F(a)da+\mu \bigg )-2k\alpha _{0}. \end{aligned}$$

(A.13)

Taking the second-order partial derivative of $\pi _{M}^{W}(\alpha _{0})$ with respect to $\alpha _0$, we have

$$\begin{aligned} \frac{\partial ^2 \pi _{M}^{W}(\alpha _{0})}{\partial \alpha _{0}^2}=-2k<0. \end{aligned}$$

(A.14)

Thus, the profit function of the manufacturer is a concave function of $\mathop \alpha \nolimits _0$, and the first-order condition can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order condition $\frac{{\partial \mathop \pi \nolimits _{M}^{W}(\alpha _{0}) }}{{\partial \mathop \alpha \nolimits _0 }} = 0$. Thus, we can get three solutions

$$\begin{aligned} \alpha _{0}^{W*}&=\frac{\lambda \left( (g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}, \end{aligned}$$

(A.15)

$$\begin{aligned} q_{i}^{W*}&=\left( 1-\lambda +\lambda (\alpha _{0}^{W*}+(1-\gamma )\alpha _{1}^{W*})\right) A\\&=(1-\lambda ) A+\frac{\lambda ^{2}A\left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}\nonumber \\&\quad +\frac{\lambda ^{2}(1-\gamma )A\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}\nonumber . \end{aligned}$$

(A.16)

$\square$

The proof of Theorem 2

Proof

Substituting Eqs. (A.11), (A.15) and (A.16) into Eqs. (A.1), (A.2) and (A.3), respectively. Thus, we have

$$\begin{aligned} \pi _{R1}^{W*}(q_{1}^{W*}, \alpha _{1}^{W*})&=\pi _{R2}^{W*}(q_{2}^{W*}, \alpha _{2}^{W*})\end{aligned}$$

(B.1)

$$\begin{aligned}&=(p-v+g_r)\left( q_{1}^{W*}-\frac{q_{1}^{W*}}{A}\int _{0}^{A}F(a)da\right) -(c_r-v)q_{1}^{W*}\nonumber \\&\quad -\frac{g_r\mu q_{1}^{W*}}{A}-wq_{1}^{W*}-k(\alpha _{1}^{W*})^{2}\nonumber \\&=(p-v+g_r)\left( q_{2}^{W*}-\frac{q_{2}^{W*}}{A}\int _{0}^{A}F(a)da\right) -(c_r-v)q_{2}^{W*}\nonumber \\&\quad -\frac{g_r\mu q_{2}^{W*}}{A}-wq_{2}^{W*}-k(\alpha _{2}^{W*})^{2}\nonumber \\&=\left( p-c_r-w+g_r-\frac{(p-v+g_r)\int _{0}^{A}F(a)da+g_r\mu }{A}\right) q_{1}^{W*}-k(\alpha _{1}^{W*})^{2},\nonumber \\ \pi _{M}^{W*}(\alpha _{0}^{W*})&=g_m\left( q_{1}^{W*}-\frac{q_{1}^{W*}}{A}\int _{0}^{A}F(a)da\right) -c_mq_{1}^{W*}-\frac{g_m\mu q_{1}^{W*}}{A}+wq_{1}^{W*}\nonumber \\&\quad +g_m\left( q_{2}^{W*}-\frac{q_{2}^{W*}}{A}\int _{0}^{A}F(a)da\right) -c_mq_{2}^{W*}-\frac{g_m\mu q_{2}^{W*}}{A}+wq_{2}^{W*}-k(\alpha _{0}^{W*})^{2}\\&=2\left( w+g_m-c_m-\frac{g_m(\int _{0}^{A}F(a)da+\mu )}{A}\right) q_{1}^{W*}-k(\alpha _{0}^{W*})^{2}.\nonumber \end{aligned}$$

(B.2)

$\square$

The proof of Corollary 1

Proof

Taking the first-order partial derivatives of $\alpha _{0}^{W*}$, $\alpha _{i}^{W*}$, $q_{i}^{W*}$, $\pi _{Ri}^{W*}(q_{i}^{W*}, \alpha _{i}^{W*})$ and $\pi _{M}^{W*}(\alpha _{0}^{W*})$ with respect to $\lambda$, where $i=1$, 2, respectively. Thus, we have

$$\begin{aligned} \mathop \frac{\partial \alpha _{0}^{W*}}{\partial \lambda }&=\frac{(g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )}{k}>0, \end{aligned}$$

(C.1)

$$\begin{aligned} \mathop \frac{\partial \alpha _{i}^{W*}}{\partial \lambda }&=\frac{(p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu }{2k}>0, \end{aligned}$$

(C.2)

$$\begin{aligned} \mathop \frac{\partial q_{i}^{W*}}{\partial \lambda }&=\frac{\lambda (1-\gamma )A\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad +\frac{2\lambda A\left( (g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}-A \end{aligned}$$

(C.3)

$$\begin{aligned}&=\frac{\lambda (1-\gamma ) A\left( (p-c_r+g_r)A-g_r\mu -(p-v+g_r)\int _{0}^{A}F(a)da\right) }{k}\nonumber \\&\quad +\frac{\lambda A^2\left( 2g_m+(1+\gamma )w-2c_m\right) -A\left( k+2\lambda g_m(\mu +\int _{0}^{A}F(a)da)\right) }{k},\nonumber \\ \mathop \frac{\partial \pi _{Ri}^{W*}}{\partial \lambda }&=\frac{\lambda (0.5-\gamma )\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) ^{2}}{k}\nonumber \\&\quad +\frac{2\lambda A(g_m+w-c_m)\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad -\frac{2\lambda g_m(\int _{0}^{A}F(a)da+\mu )\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad -(p-c_r-w+g_r)A+(p-v+g_r)\int _{0}^{A}F(a)da+g_r\mu , \end{aligned}$$

(C.4)

$$\begin{aligned} \mathop \frac{\partial \pi _{M}^{W*}}{\partial \lambda }&=2\lambda (1-\gamma )\left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) \nonumber \\&\quad *\frac{\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad +\frac{2\lambda \left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) ^{2}}{k}\nonumber \\&\quad -2(g_m+w-c_m)A+ 2g_m\bigg (\int _{0}^{A}F(a)da+\mu \bigg ). \end{aligned}$$

(C.5)

Thus, if $0\le \lambda \le min(\eta _1, 1)$, then $\frac{\partial q_{i}^{W*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial q_{i}^{W*}}{\partial \lambda }> 0$; If $0\le \lambda \le min(\eta _2, 1)$, then $\frac{\partial \pi _{i}^{W*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{i}^{W*}}{\partial \lambda }> 0$; And if $0\le \lambda \le min(\eta _3, 1)$, then $\frac{\partial \pi _{M}^{W*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{M}^{W*}}{\partial \lambda }> 0$, where

$\eta _{1}=\frac{k}{((1+\gamma )w+2g_m-2c_m)A-2 g_m(\mu +\int _{0}^{A}F(a)da)+(1-\gamma )\left\{ (p-c_r+g_r)A-g_r\mu -(p-v+g_r)\int _{0}^{A}F(a)da\right\} }$,

$\eta _{2}=\frac{k}{((1.5+\gamma )w+2g_m-2c_m)A-2g_m(\mu +\int _{0}^{A}F(a)da)+(0.5-\gamma )\left\{ (p-c_r+g_r)A-g_r\mu -(p-v+g_r)\int _{0}^{A}F(a)da\right\} }$,

$\eta _{3}=\frac{k}{\{\gamma w+g_m-c_m\}A-g_m(\mu +\int _{0}^{A}F(a)da)+(1-\gamma )\left\{ (p-c_r+g_r)A-g_r\mu -(p-v+g_r)\int _{0}^{A}F(a)da\right\} }$. $\square$

The proof of Theorem 3

Proof

Under Model RS, substituting Eqs. (2), (3) and (4) into Eqs. (12) and (13), respectively. Thus, the profits of the retailer i (=1, 2) and the manufacturer are given as follows:

$$\begin{aligned} \pi _{R1}^{RS}(q_1,\alpha _1)&=(\phi (p-v)+g_r)\left( q_1-\beta _1\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -(w_r+c_r-\phi v)q_1 \end{aligned}$$

(D.1)

$$\begin{aligned}&\quad -\beta _1g_r\mu -k\alpha _{1}^{2},\nonumber \\ \pi _{R2}^{RS}(q_2,\alpha _2)&=(\phi (p-v)+g_r)\left( q_2-\beta _2\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -(w_r+c_r-\phi v)q_2\nonumber \\&\quad -\beta _2g_r\mu -k\alpha _{2}^{2} \end{aligned}$$

(D.2)

and

$$\begin{aligned} \pi _{M}^{RS}(\alpha _0)&=\sum _{i=1}^{2}\bigg [(1-\phi )(p-v)+g_m)\left( q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\right) +(w_r-c_m+(1-\phi )v)q_i\nonumber \\&\quad -\beta _{i}g_m\mu \bigg ]-k\alpha _{0}^{2}. \end{aligned}$$

(D.3)

Substituting Eqs. (14) and (15) into Eqs. (D.1), (D.2) and (D.3), respectively. Thus, we have

$$\begin{aligned} \pi _{R1}^{RS}(q_1,\alpha _1)=\theta (p-v+g)\left( q_1-\beta _1\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -\theta (c-v)q_1-\beta _1\mu g_r-k\alpha _{1}^{2}, \end{aligned}$$

(D.4)

$$\begin{aligned} \pi _{R2}^{RS}(q_2,\alpha _2)=\theta (p-v+g)\left( q_2-\beta _2\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -\theta (c-v)q_2-\beta _2\mu g_r-k\alpha _{2}^{2} \end{aligned}$$

(D.5)

and

$$\begin{aligned} \pi _{M}^{RS}(\alpha _0)&=\sum _{i=1}^{2}\bigg [(1-\theta )(p-v+g)\left( q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\right) -(1-\theta )(c-v)q_i -\beta _{i}\mu g_m \bigg ]-k\alpha _{0}^{2}. \end{aligned}$$

(D.6)

Solving this two-stage game using backward induction technology. Taking the first-order partial derivative of $\pi _{Ri}^{RS}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $q_i$ and $\alpha _i$, respectively. Thus, we have

$$\begin{aligned} \frac{\partial \pi _{R1}^{RS}}{\partial q_1}&=\theta (p-v+g)(1-F(\frac{q_1}{\beta _1}))-\theta (c-v), \end{aligned}$$

(D.7)

$$\begin{aligned} \frac{\partial \pi _{R2}^{RS}}{\partial q_2}&=\theta (p-v+g)(1-F(\frac{q_2}{\beta _2}))-\theta (c-v), \end{aligned}$$

(D.8)

$$\begin{aligned} \frac{\partial \pi _{R1}^{RS}}{\partial \alpha _{1}}&=\lambda \theta (p-v+g)\left( \frac{q_1}{\beta _1}F(\frac{q_1}{\beta _1})-\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -\lambda \mu g_r-2k\alpha _1, \end{aligned}$$

(D.9)

$$\begin{aligned} \frac{\partial \pi _{R2}^{RS}}{\partial \alpha _{2}}&=\lambda \theta (p-v+g)\left( \frac{q_2}{\beta _2}F(\frac{q_2}{\beta _2})-\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -\lambda \mu g_r-2k\alpha _2. \end{aligned}$$

(D.10)

Taking the second-order partial derivative of $\mathop \pi \nolimits _{Ri}^{RS}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $\alpha _i$ and $q_i$, respectively. Thus, the corresponding Hessian matrix is given by as follows:

$$\begin{aligned} \mathop H_{Ri}\nolimits ^{RS}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{RS} }}{{\partial \mathop \alpha \nolimits _i^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{RS} }}{{\partial \mathop \alpha \nolimits _i \partial q_i}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{RS} }}{{\partial q_i\partial \mathop \alpha \nolimits _i }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{RS} }}{{\partial \mathop q_i\nolimits ^2 }}} \end{array}} \right] \nonumber \\&=\begin{bmatrix} -\frac{\lambda ^2\theta (p-v+g)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k &{} \lambda \theta (p-v+g)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i})\\ \lambda \theta (p-v+g)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i}) &{} -\frac{\theta (p-v+g)}{\beta _i}f(\frac{q_i}{\beta _i}) \end{bmatrix}. \end{aligned}$$

(D.11)

Thus, we have

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{RS} }}{{\partial \mathop \alpha \nolimits _i^2 }}&= -\frac{\lambda ^2\theta (p-v+g)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k<0, \end{aligned}$$

(D.12)

$$\begin{aligned} \left| {\mathop H_{Ri}\nolimits ^{RS} } \right|&= \left| {\begin{array}{*{20}{c}} -\frac{\lambda ^2\theta (p-v+g)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k &{} \lambda \theta (p-v+g)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i})\\ \lambda \theta (p-v+g)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i}) &{} -\frac{\theta (p-v+g)}{\beta _i}f(\frac{q_i}{\beta _i})\end{array}} \right| \\&= \frac{2\theta k(p-v+g)}{\beta _i}f(\frac{q_i}{\beta _i})>0.\nonumber \end{aligned}$$

(D.13)

Eqs. (D.12) and (D.13) yield that the Hessian matrix is a negative definite matrix. Consequently, the profit of the retailer i ($=1, 2$) is a joint concave function of $\alpha _i$ and $q_i$, and thus the first-order conditions can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order conditions $\frac{{\partial \mathop \pi \nolimits _{Ri}^{RS} (q_{i}, \alpha _{i})}}{{\partial \mathop \alpha \nolimits _i }} = 0$, $\frac{{\partial \mathop \pi \nolimits _{Ri}^{RS}(q_{i}, \alpha _{i}) }}{{\partial q_i}} = 0$, where $i=1$, 2. Thus, we can get four solutions

$$\begin{aligned} \mathop \alpha \nolimits _i^{RS*}&=\frac{\lambda \left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-\mu g_r \right) }{2k}, \end{aligned}$$

(D.14)

$$\begin{aligned} \mathop q\nolimits _i^{RS}&=\left( 1-\lambda +\lambda (\alpha _0+(1-\gamma )\alpha _1)\right) D, \end{aligned}$$

(D.15)

and $D=F^{-1}\left( \frac{p-c+g}{p-v+g}\right)$.

Then, substituting Eqs. (D.14) and (D.15) into Eq. (2), respectively, and taking the first-order partial derivative of $\mathop \pi \nolimits _M^{RS}(\alpha _{0})$ with respect to $\mathop \alpha \nolimits _0$, we have

$$\begin{aligned} \frac{\partial \pi _{M}^{RS}(\alpha _{0})}{\partial \alpha _{0}}&=2\lambda (1-\theta )\left( (p-v+g)(D-\int _{0}^{D}F(x)dx)-(c-v)D\right) -2\lambda \mu g_m-2k\alpha _{0}. \end{aligned}$$

(D.16)

Taking the second-order partial derivative of $\pi _{M}^{RS}(\alpha _{0})$ with respect to $\alpha _0$, we have

$$\begin{aligned} \frac{\partial ^2 \pi _{M}^{RS}(\alpha _{0})}{\partial \alpha _{0}^2}=-2k<0. \end{aligned}$$

(D.17)

Thus, the profit function of the manufacturer is a concave function in $\mathop \alpha \nolimits _0$, and the first-order condition can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order condition $\frac{{\partial \mathop \pi \nolimits _{M}^{RS}(\alpha _{0}) }}{{\partial \mathop \alpha \nolimits _0 }} = 0$. Thus, we can get three solutions

$$\begin{aligned} \alpha _{0}^{RS*}&=\frac{\lambda (1-\theta )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda \mu g_m}{k}, \end{aligned}$$

(D.18)

$$\begin{aligned} q_{i}^{RS*}&=(1-\lambda +\lambda (\alpha _{0}^{RS*}+(1-\gamma )\alpha _{i}^{RS*}))D\\&=(1-\lambda ) D+\frac{\lambda ^{2}(1-\gamma )D\left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-\mu g_r\right) }{2k}\nonumber \\&\quad +\frac{\lambda ^{2}(1-\theta )D\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda ^{2}\mu D g_m}{k}\nonumber \\&=(1-\lambda ) D+\frac{\lambda ^{2}(2-\theta -\theta \gamma )D\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) }{2k}\nonumber \\&\quad -\frac{\lambda ^{2}\mu D((1-\gamma )g_r+2g_m)}{2k}.\nonumber \end{aligned}$$

(D.19)

$\square$

The proof of Theorem 4

Proof

Substituting Eqs. (D.14), (D.18) and (D.19) into Eqs. (D.4), (D.5) and (D.6), respectively. Thus, we have

$$\begin{aligned} \pi _{R1}^{RS*}(q_{1}^{RS*}, \alpha _{1}^{RS*})&=\pi _{R2}^{RS*}(q_{2}^{RS*}, \alpha _{2}^{RS*}) \end{aligned}$$

(E.1)

$$\begin{aligned}&=\theta (p-v+g)\left( q_{1}^{RS*}-\frac{q_{1}^{RS*}}{D}\int _{0}^{D}F(a)da\right) -\theta (c-v)q_{1}^{RS*}\nonumber \\&\quad -\frac{\mu g_r q_{1}^{RS*}}{D}-k(\alpha _{1}^{RS*})^{2}\nonumber \\&=\theta (p-v+g)\left( q_{2}^{RS*}-\frac{q_{2}^{RS*}}{D}\int _{0}^{D}F(a)da\right) -\theta (c-v)q_{2}^{RS*}\nonumber \\&\quad -\frac{\mu g_r q_{2}^{RS*}}{D}-k(\alpha _{2}^{RS*})^{2}\nonumber \\&=\left( \theta (p-c+g)-\frac{\theta (p-v+g)\int _{0}^{D}F(a)da+\mu g_r}{D}\right) q_{1}^{RS*}-k(\alpha _{1}^{RS*})^{2},\nonumber \\ \pi _{M}^{RS*}(\alpha _{0}^{RS*})&=(1-\theta )(p-v+g)\left( q_{1}^{RS*}-\frac{q_{1}^{RS*}}{D}\int _{0}^{D}F(a)da\right) -(1-\theta )(c-v)q_{1}^{RS*}\nonumber \\&\quad -\frac{\mu g_m q_{1}^{RS*}}{D}+(1-\theta )(p-v+g)\left( q_{2}^{RS*}-\frac{q_{2}^{RS*}}{D}\int _{0}^{D}F(a)da\right) \nonumber \\&\quad -(1-\theta )(c-v)q_{2}^{RS*}-\frac{\mu g_m q_{2}^{RS*}}{D}-k(\alpha _{0}^{RS*})^{2}\\&=2\left( (1-\theta )(p-c+g)-\frac{(1-\theta )(p-v+g)\int _{0}^{D}F(a)da+\mu g_m}{D}\right) q_{1}^{RS*}\nonumber \\&\quad -k(\alpha _{0}^{RS*})^{2}.\nonumber \end{aligned}$$

(E.2)

$\square$

The proof of Corollary 2

Proof

Taking the first-order partial derivatives of $\alpha _{0}^{RS*}$, $\alpha _{i}^{RS*}$, $q_{i}^{RS*}$, $\pi _{Ri}^{RS*}(q_{i}^{RS*}, \alpha _{i}^{RS*})$ and $\pi _{M}^{RS*}(\alpha _{0}^{RS*})$ with respect to $\lambda$, where $i=1$, 2, respectively. Thus, we have

$$\begin{aligned} \mathop \frac{\partial \alpha _{0}^{RS*}}{\partial \lambda }&=\frac{(1-\theta )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\mu g_m}{k}>0, \end{aligned}$$

(F.1)

$$\begin{aligned} \mathop \frac{\partial \alpha _{i}^{RS*}}{\partial \lambda }&=\frac{\theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-\mu g_r}{2k}>0, \end{aligned}$$

(F.2)

$$\begin{aligned} \mathop \frac{\partial q_{i}^{RS*}}{\partial \lambda }&=-D+\frac{\lambda (2-\theta -\theta \gamma )D\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda \mu D((1-\gamma )g_r+2g_m)}{k}, \end{aligned}$$

(F.3)

$$\begin{aligned} \frac{\partial \pi _{Ri}^{RS*}}{\partial \lambda }&=\left( \frac{\lambda (4-3\theta -2\theta \gamma )\{(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\}-\lambda \mu ((1-2\gamma )g_r+4g_m)-2k}{2k}\right) \nonumber \\&\quad *\left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-\mu g_r\right) , \end{aligned}$$

(F.4)

$$\begin{aligned} \mathop \frac{\partial \pi _{M}^{RS*}}{\partial \lambda }&=\left( \frac{2\lambda (1-\theta \gamma )\{(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\}-2\lambda \mu ((1-\gamma )g_r+g_m)-2k}{k}\right) \nonumber \\&\quad *\left( (1-\theta )(p-c+g)D-(1-\theta )(p-v+g)\int _{0}^{D}F(a)da-\mu g_m\right) , \end{aligned}$$

(F.5)

Thus, if $0\le \lambda \le min(\eta _4, 1)$, then $\frac{\partial q_{i}^{RS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial q_{i}^{RS*}}{\partial \lambda }> 0$; If $0\le \lambda \le min(\eta _5, 1)$, then $\frac{\partial \pi _{i}^{RS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{i}^{RS*}}{\partial \lambda }> 0$; And if $0\le \lambda \le min(\eta _6, 1)$, then $\frac{\partial \pi _{M}^{RS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{M}^{RS*}}{\partial \lambda }> 0$, where

$\eta _{4}=\frac{k}{(2-\theta -\theta \gamma )\{(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\}-\mu ((1-\gamma )g_r+2g_m)}$,

$\eta _{5}=\frac{2k}{(4-3\theta -2\theta \gamma )\{(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\}-\mu ((1-2\gamma )g_r+4g_m)}$,

$\eta _{6}=\frac{k}{(1-\theta \gamma )\{(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\}-\mu ((1-\gamma )g_r+g_m)}$. $\square$

The proof of Theorem 5

Proof

Under Model WRS, substituting Eqs. (2), (3), (4), (14) and (15) into Eqs. (21), (22) and (23), respectively. Thus, the profits of the retailer i ($=1$, 2) and the manufacturer are given as follows:

$$\begin{aligned} \pi _{R1}^{WRS}(q_1,\alpha _1)=(p-v+g_r)\left( q_1-\beta _1\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -(c_r-v)q_1-\beta _1g_r\mu -wq_1-k\alpha _{1}^{2}, \end{aligned}$$

(G.1)

$$\begin{aligned} \pi _{R2}^{WRS}(q_2,\alpha _2)=\theta (p-v+g)\left( q_2-\beta _2\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -\theta (c-v)q_2-\beta _2 g_r\mu -k\alpha _{2}^{2} \end{aligned}$$

(G.2)

and

$$\begin{aligned} \pi _{M}^{WRS}(\alpha _0)&=g_m\left( q_1-\beta _1\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -c_mq_1-\beta _{1} g_m\mu +wq_1\nonumber \\&\quad +(1-\theta )(p-v+g)\left( q_2-\beta _2\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) \nonumber \\&\quad -(1-\theta )(c-v)q_2-\beta _{2}g_m\mu -k\alpha _{0}^{2}. \end{aligned}$$

(G.3)

Solving this two-stage game using backward induction technology. Taking the first-order partial derivative of $\pi _{Ri}^{WRS}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $q_i$ and $\alpha _i$, respectively. Thus, we have

$$\begin{aligned} \frac{\partial \pi _{R1}^{WRS}}{\partial q_1}&=(p-v+g_r)(1-F(\frac{q_1}{\beta _1}))-(c_r-v)-w, \end{aligned}$$

(G.4)

$$\begin{aligned} \frac{\partial \pi _{R2}^{WRS}}{\partial q_2}&=\theta (p-v+g)(1-F(\frac{q_2}{\beta _2}))-\theta (c-v), \end{aligned}$$

(G.5)

$$\begin{aligned} \frac{\partial \pi _{R1}^{WRS}}{\partial \alpha _{1}}&=\lambda (p-v+g_r)\left( \frac{q_1}{\beta _1}F(\frac{q_1}{\beta _1})-\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\right) -\lambda g_r\mu -2k\alpha _1, \end{aligned}$$

(G.6)

$$\begin{aligned} \frac{\partial \pi _{R2}^{WRS}}{\partial \alpha _{2}}&=\lambda \theta (p-v+g)\left( \frac{q_2}{\beta _2}F(\frac{q_2}{\beta _2})-\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\right) -\lambda g_r\mu -2k\alpha _2. \end{aligned}$$

(G.7)

Taking the second-order partial derivative of $\mathop \pi \nolimits _{Ri}^{WRS}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $\alpha _i$ and $q_i$, respectively. Thus, the corresponding Hessian matrices are given as follows:

$$\begin{aligned} \mathop H_{R1}\nolimits ^{WRS}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R1}^{WRS} }}{{\partial \mathop \alpha \nolimits _1^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R1}^{WRS} }}{{\partial \mathop \alpha \nolimits _1 \partial q_1}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R1}^{WRS} }}{{\partial q_1\partial \mathop \alpha \nolimits _1 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R1}^{WRS} }}{{\partial \mathop q_1\nolimits ^2 }}} \end{array}} \right] \end{aligned}$$

(G.8)

$$\begin{aligned}&=\begin{bmatrix} -\frac{\lambda ^2(p-v+g_r)q_{1}^2}{\beta _1^3}f(\frac{q_1}{\beta _1})-2k &{} \lambda (p-v+g_r)\frac{q_1}{\beta _1^2}f(\frac{q_1}{\beta _1})\\ \lambda (p-v+g_r)\frac{q_1}{\beta _1^2}f(\frac{q_1}{\beta _1}) &{} -\frac{(p-v+g_r)}{\beta _1}f(\frac{q_1}{\beta _1}) \end{bmatrix},\nonumber \\ \mathop H_{R2}\nolimits ^{WRS}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R2}^{WRS} }}{{\partial \mathop \alpha \nolimits _2^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R2}^{WRS} }}{{\partial \mathop \alpha \nolimits _2 \partial q_2}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R2}^{WRS} }}{{\partial q_2\partial \mathop \alpha \nolimits _2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R2}^{WRS} }}{{\partial \mathop q_2\nolimits ^2 }}} \end{array}} \right] \\&=\begin{bmatrix} -\frac{\lambda ^2\theta (p-v+g)q_{2}^2}{\beta _2^3}f(\frac{q_2}{\beta _2})-2k &{} \lambda \theta (p-v+g)\frac{q_2}{\beta _2^2}f(\frac{q_2}{\beta _2})\\ \lambda \theta (p-v+g)\frac{q_2}{\beta _2^2}f(\frac{q_2}{\beta _2}) &{} -\frac{\theta (p-v+g)}{\beta _2}f(\frac{q_2}{\beta _2}) \end{bmatrix}.\nonumber \end{aligned}$$

(G.9)

Thus, we have

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R1}^{WRS} }}{{\partial \mathop \alpha \nolimits _1^2 }}&= -\frac{\lambda ^2(p-v+g_r)q_{1}^2}{\beta _1^3}f(\frac{q_1}{\beta _1})-2k<0, \end{aligned}$$

(G.10)

$$\begin{aligned} \left| {\mathop H_{R1}\nolimits ^{WRS} } \right|&= \left| {\begin{array}{*{20}{c}} -\frac{\lambda ^2(p-v+g_r)q_{1}^2}{\beta _1^3}f(\frac{q_1}{\beta _1})-2k &{} \lambda (p-v+g_r)\frac{q_1}{\beta _1^2}f(\frac{q_1}{\beta _1})\\ \lambda (p-v+g_r)\frac{q_1}{\beta _1^2}f(\frac{q_1}{\beta _1}) &{} -\frac{(p-v+g_r)}{\beta _1}f(\frac{q_1}{\beta _1})\end{array}} \right| \end{aligned}$$

(G.11)

$$\begin{aligned}&= \frac{2k(p-v+g_r)}{\beta _1}f(\frac{q_1}{\beta _1})>0,\nonumber \\ \frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{R2}^{WRS} }}{{\partial \mathop \alpha \nolimits _2^2 }}&= -\frac{\lambda ^2\theta (p-v+g)q_{2}^2}{\beta _2^3}f(\frac{q_2}{\beta _2})-2k<0, \end{aligned}$$

(G.12)

$$\begin{aligned} \left| {\mathop H_{R2}\nolimits ^{WRS} } \right|&= \left| {\begin{array}{*{20}{c}} -\frac{\lambda ^2\theta (p-v+g)q_{2}^2}{\beta _2^3}f(\frac{q_2}{\beta _2})-2k &{} \lambda \theta (p-v+g)\frac{q_2}{\beta _2^2}f(\frac{q_2}{\beta _2})\\ \lambda \theta (p-v+g)\frac{q_2}{\beta _2^2}f(\frac{q_2}{\beta _2}) &{} -\frac{\theta (p-v+g)}{\beta _2}f(\frac{q_2}{\beta _2})\end{array}} \right| \\&= \frac{2\theta k(p-v+g)}{\beta _2}f(\frac{q_2}{\beta _2})>0.\nonumber \end{aligned}$$

(G.13)

Eqs. (G.10), (G.11), (G.12) and (G.13) yield that the above Hessian matrices are negative definite matrices. Consequently, the profit of the retailer i ($=1, 2$) is a joint concave function of $\alpha _i$ and $q_i$, and the first-order conditions can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order conditions $\frac{{\partial \mathop \pi \nolimits _{Ri}^{WRS} (q_{i}, \alpha _{i})}}{{\partial \mathop \alpha \nolimits _i }} = 0$, $\frac{{\partial \mathop \pi \nolimits _{Ri}^{WRS}(q_{i}, \alpha _{i}) }}{{\partial q_i}} = 0$, where $i=1, 2$. Thus, we can get four solutions

$$\begin{aligned} \mathop \alpha \nolimits _1^{WRS*}&=\frac{\lambda \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}, \end{aligned}$$

(G.14)

$$\begin{aligned} \mathop \alpha \nolimits _2^{WRS*}&=\frac{\lambda \left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-g_r\mu \right) }{2k}, \end{aligned}$$

(G.15)

$$\begin{aligned} \mathop q\nolimits _1^{WRS}&=(1-\lambda +\lambda (\alpha _0+\alpha _1-\gamma \alpha _2))A, \end{aligned}$$

(G.16)

$$\begin{aligned} \mathop q\nolimits _2^{WRS}&=(1-\lambda +\lambda (\alpha _0+\alpha _2-\gamma \alpha _1))D. \end{aligned}$$

(G.17)

Then, substituting Eqs. (G.14), (G.15), (G.16) and (G.17) into Eq. (G.3), and taking the first-order partial derivative of $\mathop \pi \nolimits _M^{WRS}(\alpha _{0})$ with respect to $\mathop \alpha \nolimits _0$, we have

$$\begin{aligned} \frac{\partial \pi _{M}^{WRS}(\alpha _{0})}{\partial \alpha _{0}}&=\lambda (g_m+w-c_m)A-\lambda g_m\left( \int _{0}^{A}F(a)da+2\mu \right) -2k\alpha _{0}\\&\quad +\lambda (1-\theta )\left( (p-v+g)(D-\int _{0}^{D}F(a)da)-(c-v)D\right) .\nonumber \end{aligned}$$

(G.18)

Taking the second-order partial derivative of $\pi _{M}^{WRS}(\alpha _{0})$ with respect to $\alpha _0$, we have

$$\begin{aligned} \frac{\partial ^2 \pi _{M}^{WRS}(\alpha _{0})}{\partial \alpha _{0}^2}=-2k<0. \end{aligned}$$

(G.19)

Thus, the profit function of the manufacturer is a concave function in $\mathop \alpha \nolimits _0$, and the first-order condition can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order condition $\frac{{\partial \mathop \pi \nolimits _{M}^{WRS}(\alpha _{0}) }}{{\partial \mathop \alpha \nolimits _0 }} = 0$. Thus, we can get three solutions

$$\begin{aligned} \alpha _{0}^{WRS*}&=\frac{\lambda (1-\theta )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda \mu g_m}{2k} \end{aligned}$$

(G.20)

$$\begin{aligned}&\quad +\frac{\lambda \left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{2k},\nonumber \\ q_{1}^{WRS*}&=(1-\lambda +\lambda (\alpha _{0}^{WRS*}+\alpha _{1}^{WRS*}-\gamma \alpha _{2}^{WRS*}))A \end{aligned}$$

(G.21)

$$\begin{aligned}&=(1-\lambda )A+\frac{\lambda ^{2}A\left( (p-c+g)A-(p-v+g)\int _{0}^{A}F(a)da-g\mu \right) }{2k}\nonumber \\&\quad +\frac{\lambda ^{2}(1-\theta -\theta \gamma )A\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda ^{2}\mu A(g_m-\gamma g_r)}{2k},\nonumber \\ q_{2}^{WRS*}&=(1-\lambda +\lambda (\alpha _{0}^{WRS*}+\alpha _{2}^{WRS*}-\gamma \alpha _{1}^{WRS*}))D \\&=(1-\lambda )D+\frac{\lambda ^{2}D\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da-g\mu \right) }{2k}\nonumber \\&\quad -\frac{\lambda ^{2}\gamma D\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}\nonumber \\&\quad +\frac{\lambda ^{2}D\left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{2k}.\nonumber \end{aligned}$$

(G.22)

$\square$

The proof of Theorem 6

Proof

Substituting Eqs. (G.14), (G.15), (G.20), (G.21) and (G.22) into Eqs. (G.1), (G.2) and (G.3), respectively. Thus, we have

$$\begin{aligned} \pi _{R1}^{WRS*}(q_{1}^{WRS*}, \alpha _{1}^{WRS*})&=(p-v+g_r)\left( q_{1}^{WRS*}-\frac{q_{1}^{WRS*}}{A}\int _{0}^{A}F(a)da\right) -(c_r-v)q_{1}^{WRS*}\nonumber \\&\quad -\frac{g_r\mu q_{1}^{WRS*}}{A}-wq_{1}^{WRS*}\nonumber \\&-k(\alpha _{1}^{WRS*})^{2} \end{aligned}$$

(H.1)

$$\begin{aligned}&=\left( p-c_r-w+g_r-\frac{(p-v+g_r)\int _{0}^{A}F(a)da+g_r\mu }{A}\right) q_{1}^{WRS*}\nonumber \\&-k(\alpha _{1}^{WRS*})^{2},\nonumber \\ \pi _{R2}^{WRS*}(q_{2}^{WRS*}, \alpha _{2}^{WRS*})&=\theta (p-v+g)\left( q_{2}^{WRS*}-\frac{q_{2}^{WRS*}}{D}\int _{0}^{D}F(a)da\right) -\theta (c-v)q_{2}^{WRS*}\nonumber \\&\quad -\frac{g_r\mu q_{2}^{WRS*}}{D}-k(\alpha _{2}^{WRS*})^{2} \end{aligned}$$

(H.2)

$$\begin{aligned}&=\left( \theta (p-c+g)-\frac{\theta (p-v+g)\int _{0}^{D}F(a)da+g_r\mu }{D}\right) q_{2}^{WRS*}\nonumber \\&-k(\alpha _{2}^{WRS*})^{2},\nonumber \\ \pi _{M}^{WRS*}(\alpha _{0}^{WRS*})&=g_m\left( q_{1}^{WRS*}-\frac{q_{1}^{WRS*}}{A}\int _{0}^{A}F(a)da\right) -c_mq_{1}^{WRS*}-\frac{g_m\mu q_{1}^{WRS*}}{A}\nonumber \\&+wq_{1}^{WRS*}\nonumber \\&\quad +(1-\theta )(p-v+g)\left( q_{2}^{WRS*}-\frac{q_{2}^{WRS*}}{D}\int _{0}^{D}F(a)da\right) -(1-\theta )(c-v)q_{2}^{WRS*}\nonumber \\&\quad -\frac{g_m\mu q_{2}^{WRS*}}{D}-k(\alpha _{0}^{WRS*})^{2}\\&=\left( (1-\theta )(p-c+g)-\frac{(1-\theta )(p-v+g)\int _{0}^{D}F(a)da+g_m\mu }{D}\right) q_{2}^{WRS*}\nonumber \\&\quad +\left( w+g_m-c_m-\frac{g_m(\int _{0}^{A}F(a)da+\mu )}{A}\right) q_{1}^{WRS*}-k(\alpha _{0}^{WRS*})^{2}.\nonumber \end{aligned}$$

(H.3)

$\square$

The proof of Corollary 3

Proof

Taking the first-order partial derivatives of $\alpha _{0}^{WRS*}$, $\alpha _{i}^{WRS*}$, $q_{i}^{WRS*}$, $\pi _{Ri}^{WRS*}(q_{i}^{WRS*}, \alpha _{i}^{WRS*})$ and $\pi _{M}^{WRS*}(\alpha _{0}^{WRS*})$ with respect to $\lambda$, where $i=1$, 2, respectively. Thus, we have

$$\begin{aligned} \mathop \frac{\partial \alpha _{0}^{WRS*}}{\partial \lambda }&=\frac{(1-\theta )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -g_m\mu }{2k}\nonumber \\&\quad +\frac{(g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )}{2k}>0, \end{aligned}$$

(I.1)

$$\begin{aligned} \frac{\partial \alpha _{1}^{WRS*}}{\partial \lambda }&=\frac{ (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu }{2k}>0, \end{aligned}$$

(I.2)

$$\begin{aligned} \frac{\partial \alpha _{2}^{WRS*}}{\partial \lambda }&=\frac{\theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-g_r\mu }{2k}>0, \end{aligned}$$

(I.3)

$$\begin{aligned} \mathop \frac{\partial q_{1}^{WRS*}}{\partial \lambda }&=-A+\frac{\lambda A\left( (p-c+g)A-(p-v+g)\int _{0}^{A}F(a)da-g\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda (1-\theta -\theta \gamma )A\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda \mu A(g_m-\gamma g_r)}{k}, \end{aligned}$$

(I.4)

$$\begin{aligned} \mathop \frac{\partial q_{2}^{WRS*}}{\partial \lambda }&=-D+\frac{\lambda D\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da-g\mu \right) }{k}\nonumber \\&\quad -\frac{\lambda \gamma D\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda D\left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}, \end{aligned}$$

(I.5)

$$\begin{aligned} \frac{\partial \pi _{R1}^{WRS*}}{\partial \lambda }&=\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) \nonumber \\&\quad *\bigg (-1+\frac{\lambda \left( (p-c+g)A-(p-v+g)\int _{0}^{A}F(a)da-g\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda (1-\theta -\theta \gamma )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda \mu (g_m-\gamma g_r)}{k}\nonumber \\&\quad -\frac{\lambda \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}\bigg ) \end{aligned}$$

(I.6)

$$\begin{aligned}&=\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) \nonumber \\&\quad *\bigg (\frac{2\lambda (1-\theta -\theta \gamma )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -2k}{2k}\nonumber \\&\quad +\frac{\lambda (2g_m+w-c_r-2c_m+p+g_r)A-\lambda (p-v+g_r+2g_m)\int _{0}^{A}F(a)da}{2k}\nonumber \\&\quad -\frac{\lambda \mu (4g_m+(1-2\gamma )g_r)}{2k}\bigg ), \nonumber \\ \frac{\partial \pi _{R2}^{WRS*}}{\partial \lambda }&=\left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-g_r\mu \right) \nonumber \\&\quad *\bigg (-1+\frac{\lambda \left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da-g\mu \right) }{k}\nonumber \\&\quad -\frac{\lambda \gamma \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda \left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}\nonumber \\&\quad -\frac{\lambda \left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-g_r\mu \right) }{2k}\bigg ) \end{aligned}$$

(I.7)

$$\begin{aligned}&=\left( \theta (p-c+g)D-\theta (p-v+g)\int _{0}^{D}F(a)da-g_r\mu \right) \nonumber \\&\quad *\bigg (\frac{\lambda \left( (2-\theta )\{(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\}-(2g_m+g_r)\mu \right) -2k}{2k}\nonumber \\&\quad -\frac{\lambda \gamma \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda \left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}\bigg ),\nonumber \\ \mathop \frac{\partial \pi _{M}^{WRS*}}{\partial \lambda }&=\left( (1-\theta )(p-c+g)D-(1-\theta )(p-v+g)\int _{0}^{D}F(a)da-g_m\mu \right) \nonumber \\&\quad *\bigg (-1+\frac{\lambda \left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da-g\mu \right) }{k}\nonumber \\&\quad -\frac{\lambda \gamma \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda \left( (g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\right) }{k}\bigg )\nonumber \\&\quad +\bigg ( (g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )\bigg )\nonumber \\&\quad \times \bigg (-1+\frac{\lambda \left( (p-c+g)A-(p-v+g)\int _{0}^{A}F(a)da-g\mu \right) }{k}\nonumber \\&\quad +\frac{\lambda (1-\theta -\theta \gamma )\left( (p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da\right) -\lambda \mu (g_m-\gamma g_r)}{k}\bigg )\nonumber \\&\quad -\frac{\lambda }{2k}\bigg ((1-\theta )(p-c+g)D-(1-\theta )(p-v+g)\int _{0}^{D}F(a)da-g_m\mu \nonumber \\&\quad +(g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )\bigg )^2. \end{aligned}$$

(I.8)

Thus, if $0\le \lambda \le min(\eta _7, 1)$, then $\frac{\partial q_{1}^{WRS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial q_{1}^{WRS*}}{\partial \lambda }> 0$; if $0\le \lambda \le min(\eta _8, 1)$, then $\frac{\partial q_{2}^{WRS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial q_{2}^{WS*}}{\partial \lambda }> 0$; If $0\le \lambda \le min(\eta _9, 1)$, then $\frac{\partial \pi _{R1}^{WRS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{R1}^{WRS*}}{\partial \lambda }> 0$; If $0\le \lambda \le min(\eta _{10}, 1)$, then $\frac{\partial \pi _{R2}^{WRS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{R2}^{WRS*}}{\partial \lambda }> 0$; And if $0\le \lambda \le min(\eta _{11}, 1)$, then $\frac{\partial \pi _{M}^{WRS*}}{\partial \lambda }\le 0$, otherwise $\frac{\partial \pi _{M}^{WRS*}}{\partial \lambda }> 0$, where

$$\begin{aligned} e_{0}&=(p-c+g)A-(p-v+g)\int _{0}^{A}F(a)da-g\mu ,\\ e_{1}&=(p-c+g)D-(p-v+g)\int _{0}^{D}F(a)da,\\ e_{2}&=(p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu ,\\ e_{3}&=(g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu ),\\ \eta _{7}&=\frac{k}{e_0+(1-\theta -\theta \gamma )e_1-(g_m-\gamma g_r)\mu },\\ \eta _{8}&=\frac{k}{e_1+e_3-\gamma e_2-g\mu },\\ \eta _{9}&=\frac{2k}{2e_0+2(1-\theta -\theta \gamma )e_1-e_2-2(g_m-\gamma g_r)\mu },\\ \eta _{10}&=\frac{2k}{(2-\theta )e_1-2\gamma e_2+2e_3-(2g_m+g_r)\mu },\\ \eta _{11}&=\frac{2((1-\theta )e_1+e_3-g_m \mu )k}{2((1-\theta )e_1-g_m\mu )(e_1-\gamma e_2+e_3-g\mu )+2e_3(e_0+(1-\theta -\theta \gamma )e_1-(g_m-\gamma g_r)\mu )-((1-\theta )e_1+e_3-g_m\mu )^2}. \end{aligned}$$

$\square$

The proof of Theorem 7

Proof

Under Model M, there are N competing retailers in the luxury market. Substituting Eqs. (2) into (32) and (33), respectively. Thus, the profits of the retailer i and the manufacturer are given as follows:

$$\begin{aligned} \mathop {\pi _{Ri}^{M}(q_i,\alpha _i)}&=(p-v+g_r)\left( q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\right) -(c_r-v)q_i-\beta _ig_r\mu -wq_i-k\alpha _{i}^{2}, \end{aligned}$$

(J.1)

$$\begin{aligned} \mathop {\pi _{M}^{M}(\alpha _0)}&=\sum _{i=1}^{N}\bigg [(g_m\left( q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\right) -c_mq_i-\beta _{i}g_m\mu +wq_i \bigg ]-k\alpha _{0}^{2}, \end{aligned}$$

(J.2)

where $\beta _{i}=(1-\lambda )+\lambda \bigg (\alpha _{0}+(1+\gamma )\alpha _{i}-\sum _{j=1}^{N}\gamma \alpha _{j} \bigg )$.

Solving this two-stage game using backward induction technology. Taking the first-order partial derivative of $\pi _{Ri}^{M}(q_{i}, \alpha _{i})$ with respect to $q_i$ and $\alpha _i$, respectively. Thus, we have

$$\begin{aligned} \frac{\partial \pi _{Ri}^{M}}{\partial q_i}&=(p-v+g_r)(1-F(\frac{q_i}{\beta _i}))-(c_r-v)-w, \end{aligned}$$

(J.3)

$$\begin{aligned} \frac{\partial \pi _{Ri}^{M}}{\partial \alpha _{i}}&=\lambda (p-v+g_r)\left( \frac{q_i}{\beta _i}F(\frac{q_i}{\beta _i})-\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\right) -\lambda g_r\mu -2k\alpha _i. \end{aligned}$$

(J.4)

Taking the second-order partial derivative of $\mathop \pi \nolimits _{Ri}^{M}(q_{i}, \alpha _{i})$ with respect to $\alpha _i$ and $q_i$, respectively. Thus, the corresponding Hessian matrix is given as follows:

$$\begin{aligned} \mathop H_{Ri}\nolimits ^{M}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{M} }}{{\partial \mathop \alpha \nolimits _i^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{M} }}{{\partial \mathop \alpha \nolimits _i \partial q_i}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{M} }}{{\partial q_i\partial \mathop \alpha \nolimits _i }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{M} }}{{\partial \mathop q_i\nolimits ^2 }}} \end{array}} \right] notag\\&=\begin{bmatrix} -\frac{\lambda ^2(p-v+g_r)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k &{} \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i})\\ \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i}) &{} -\frac{(p-v+g_r)}{\beta _i}f(\frac{q_i}{\beta _i}) \end{bmatrix}.\nonumber \end{aligned}$$

(J.5)

Thus, we have

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \mathop \pi \nolimits _{Ri}^{M} }}{{\partial \mathop \alpha \nolimits _i^2 }}&= -\frac{\lambda ^2(p-v+g_r)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k<0, \end{aligned}$$

(J.6)

$$\begin{aligned} \left| {\mathop H_{Ri}\nolimits ^{M} } \right|&= \left| {\begin{array}{*{20}{c}} -\frac{\lambda ^2(p-v+g_r)q_{i}^2}{\beta _i^3}f(\frac{q_i}{\beta _i})-2k &{} \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i})\\ \lambda (p-v+g_r)\frac{q_i}{\beta _i^2}f(\frac{q_i}{\beta _i}) &{} -\frac{(p-v+g_r)}{\beta _i}f(\frac{q_i}{\beta _i})\end{array}} \right| \\&= \frac{2k(p-v+g_r)}{\beta _i}f(\frac{q_i}{\beta _i})>0.\nonumber \end{aligned}$$

(J.7)

Eqs. (10) and (J.7) yield that this Hessian matrix is a negative definite matrix. Consequently, the profit of the retailer i is a joint concave function of $\alpha _i$ and $q_i$, and the first-order conditions can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order conditions $\frac{{\partial \mathop \pi \nolimits _{Ri}^{M} (q_{i}, \alpha _{i})}}{{\partial \mathop \alpha \nolimits _i }} = 0$, $\frac{{\partial \mathop \pi \nolimits _{Ri}^{M}(q_{i}, \alpha _{i}) }}{{\partial q_i}} = 0$. Thus, we can get the following solutions

$$\begin{aligned} \mathop \alpha \nolimits _i^{M*}&=\frac{\lambda \left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}, \end{aligned}$$

(J.8)

$$\begin{aligned} \mathop q\nolimits _i^{M}&=\{(1-\lambda )+\lambda (\alpha _0+(1-(N-1)\gamma )\alpha _i)\}A. \end{aligned}$$

(J.9)

Then, substituting Eqs. (J.8) and (J.9) into Eq. (2), and taking the first-order partial derivative of $\mathop \pi \nolimits _M^{M}(\alpha _{0})$ with respect to $\mathop \alpha \nolimits _0$, we have

$$\begin{aligned} \frac{\partial \pi _{M}^{M}(\alpha _{0})}{\partial \alpha _{0}}=N \lambda (g_m+w-c_m)A-N\lambda g_m\bigg (\int _{0}^{A}F(a)da+\mu \bigg )-2k\alpha _{0}. \end{aligned}$$

(J.10)

Taking the second-order partial derivative of $\pi _{M}^{M}(\alpha _{0})$ with respect to $\alpha _0$, we have

$$\begin{aligned} \frac{\partial ^2 \pi _{M}^{M}(\alpha _{0})}{\partial \alpha _{0}^2}=-2k<0. \end{aligned}$$

(J.11)

Thus, the profit function of the manufacturer is a concave function of $\mathop \alpha \nolimits _0$, and the first-order condition can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order condition $\frac{{\partial \mathop \pi \nolimits _{M}^{M}(\alpha _{0}) }}{{\partial \mathop \alpha \nolimits _0 }} = 0$. Thus, we can get the following solutions

$$\begin{aligned} \alpha _{0}^{M*}&=\frac{\lambda N(g_m+w-c_m)A-\lambda N g_m\left( \int _{0}^{A}F(a)da+\mu \right) }{2k}, \end{aligned}$$

(J.12)

$$\begin{aligned} q_{i}^{M*}&=\left( 1-\lambda +\lambda (\alpha _{0}^{M*}+(1-(N-1)\gamma )\alpha _{i}^{M*}\right) A\\&=(1-\lambda ) A+\frac{\lambda ^{2}N(g_m+w-c_m)A^{2}-\lambda ^{2}N g_m\left( \int _{0}^{A}F(a)da+\mu \right) A}{2k}\nonumber \\&\quad +\frac{\lambda ^{2}(1-(N-1)\gamma )A\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}\nonumber . \end{aligned}$$

(J.13)

$\square$

Substituting Eqs. (J.8), (J.12) and (J.13) into Eqs. (J.1) and (J.2), respectively. Thus, we have

$$\begin{aligned} \pi _{Ri}^{M*}(q_{i}^{M*}, \alpha _{i}^{M*})&=(p-v+g_r)\left( q_{i}^{M*}-\frac{q_{i}^{M*}}{A}\int _{0}^{A}F(a)da\right) -(c_r-v)q_{i}^{M*}\nonumber \\&\quad -\frac{g_r\mu q_{i}^{M*}}{A}-wq_{i}^{M*}-k(\alpha _{i}^{M*})^{2} \end{aligned}$$

(14)

$$\begin{aligned}&=\left( p-c_r-w+g_r-\frac{(p-v+g_r)\int _{0}^{A}F(a)da+g_r\mu }{A}\right) q_{i}^{M*}-k(\alpha _{i}^{M*})^{2},\nonumber \\ \pi _{M}^{M*}(\alpha _{0}^{M*})&=\sum _{i=1}^{N}\bigg [g_m\left( q_{i}^{M*}-\frac{q_{i}^{M*}}{A}\int _{0}^{A}F(a)da\right) -c_mq_{i}^{M*}-\frac{g_m\mu q_{i}^{M*}}{A}+wq_{i}^{M*}\bigg ]\nonumber \\&\quad -k(\alpha _{0}^{M*})^{2}\\&=N\left( w+g_m-c_m-\frac{g_m(\int _{0}^{A}F(a)da+\mu )}{A}\right) q_{i}^{M*}-k(\alpha _{0}^{M*})^{2}.\nonumber \end{aligned}$$

(15)

The proof of Corollary 4

Proof

Subtracting $\alpha _{0}^{M*}(N)$ from $\alpha _{0}^{M*}(N+1)$, subtracting $\alpha _{i}^{M*}(N)$ from $\alpha _{i}^{M*}(N+1)$, subtracting $q_{i}^{M*}(N)$ from $q_{i}^{M*}(N+1)$, subtracting $\pi _{Ri}^{M*}(q_{i}^{M*}(N), \alpha _{i}^{M*}(N))$ from $\pi _{Ri}^{M*}(q_{i}^{M*}(N+1), \alpha _{i}^{M*}(N+1))$, and subtracting $\pi _{M}^{M*}(\alpha _{0}^{M*}(N))$ from $\pi _{M}^{M*}(\alpha _{0}^{M*}(N+1))$, respectively, where i=1, 2. Thus, we have

$$\begin{aligned} \mathop {\alpha _{0}^{M*}(N+1)-\alpha _{0}^{M*}(N)}&=\frac{\lambda (g_m+w-c_m)A-\lambda g_m\left( \int _{0}^{A}F(a)da+\mu \right) }{2k}>0, \end{aligned}$$

(K.1)

$$\begin{aligned} \mathop {\alpha _{i}^{M*}(N+1)-\alpha _{i}^{M*}(N)}&=0, \end{aligned}$$

(K.2)

$$\begin{aligned} \mathop {q_{i}^{M*}(N+1)-q_{i}^{M*}(N)}\nonumber&=-\frac{\lambda ^{2}\gamma A\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) }{2k}\nonumber \\&\quad +\frac{\lambda ^{2}(g_m+w-c_m)A^{2}-\lambda ^{2} g_m\left( \int _{0}^{A}F(a)da+\mu \right) A}{2k},\end{aligned}$$

(K.3)

$$\begin{aligned} \Delta _{Ri}&=\mathop {\pi _{Ri}^{M*}(q_{i}^{M*}(N+1), \alpha _{i}^{M*}(N+1))-\pi _{Ri}^{M*}(q_{i}^{M*}(N)}, \alpha _{i}^{M*}(N))\end{aligned}$$

(K.4)

$$\begin{aligned}&=\left( (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \right) \nonumber \\ {}&\quad *\bigg (\frac{\lambda ^{2}(g_m+w-c_m)A-\lambda ^{2} g_m(\int _{0}^{A}F(a)da+\mu )}{2k}\nonumber \\&\quad -\frac{\lambda ^{2}\gamma \{ (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \}}{2k}\bigg ),\nonumber \\ \Delta _{M}&=\pi _{M}^{M*}(\alpha _{0}^{M*}(N+1))- \pi _{M}^{M}(\alpha _{0}^{M*}(N))\\&=\bigg ((g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )\bigg )\nonumber \\ {}&\quad *\bigg (1-\lambda +\frac{\lambda ^{2}(2N+1)\{(g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )\}}{4k}\nonumber \\&\quad +\frac{\lambda ^{2}(1-2N\gamma )\{ (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \}}{2k}\bigg ).\nonumber \end{aligned}$$

(K.5)

Thus, if $\gamma \in [0, min(1,\eta _{12})]$, then $\pi _{Ri}^{M*}(q_{i}^{M*}(N+1), \alpha _{i}^{M*}(N+1))\ge \pi _{Ri}^{M*}(q_{i}^{M*}(N), \alpha _{i}^{M*}(N))$ and $q_{i}^{M*}(N+1)\ge q_{i}^{M*}(N)$, otherwise $q_{i}^{M*}(N+1)< q_{i}^{M*}(N)$ and $\pi _{Ri}^{M*}(q_{i}^{M*}(N+1), \alpha _{i}^{M*}(N+1))<\pi _{Ri}^{M*}(q_{i}^{M*}(N), \alpha _{i}^{M*}(N))$. If $\Delta \ge 0$, then $\pi _{M}^{M*}(\alpha _{0}^{M*}(N+1))\ge \pi _{M}^{M}(\alpha _{0}^{M*}(N))$, if $\Delta <0$ and $N < \lfloor N^{*}\rfloor$, then $\pi _{M}^{M*}(\alpha _{0}^{M*}(N+1))\ge \pi _{M}^{M}(\alpha _{0}^{M*}(N))$; otherwise $\pi _{M}^{M*}(\alpha _{0}^{M*}(N+1))< \pi _{M}^{M}(\alpha _{0}^{M*}(N))$, where

$\Delta =(g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )-2\gamma \{ (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \}$,

$$\begin{aligned}&\eta _{12}=\frac{(g_m+w-c_m)A-g_m(\int _{0}^{A}F(a)da+\mu )}{(p-c_r-w+g_r))A-(p-v+g_r)\int _{0}^{A}F(a)da-wA-g_{r}\mu },\\&N^{*}=\frac{2\lambda ^{2}\{ (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \}+\lambda ^{2}\{(g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\}+4k(1-\lambda )}{4\lambda ^{2}\gamma \{ (p-c_r-w+g_r)A-(p-v+g_r)\int _{0}^{A}F(a)da-g_r\mu \}-2\lambda ^{2}\{(g_m+w-c_m)A- g_m(\int _{0}^{A}F(a)da+\mu )\}}. \end{aligned}$$

$\square$

The calculation of the extended model in Sect. 4.2

Proof

Under Model F, the two competing retailers are concerned about fairness. Substituting Eqs. (A.1), (A.2) and (A.3) into Eq. (37), respectively. Thus, the utility of the retailer i ($=1, 2$) is given as follows:

$$\begin{aligned} u_{R1}^{F}(q_1,\alpha _1)&=(1+f)\pi _{R1}^{F}(q_1,\alpha _1)-f\pi _{M}^{F}(\alpha _0) \end{aligned}$$

(L.1)

$$\begin{aligned}&=(1+f)\bigg [(p-v+g_r)\bigg (q_1-\beta _1\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\bigg )-(c_r-v)q_1-\beta _1g_r\mu -wq_1-k\alpha _{1}^{2}\bigg ]\nonumber \\&\quad -f\bigg [\sum _{i=1}^{2}\bigg ((g_m(q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da)-c_mq_i-\beta _{i}g_m\mu +wq_i \bigg )-k\alpha _{0}^{2}\bigg ],\nonumber \\ u_{R2}^{F}(q_2,\alpha _2)&=(1+f)\pi _{R2}^{F}(q_2,\alpha _2)-f\pi _{M}^{F}(\alpha _0)\\&=(1+f)\bigg [(p-v+g_r)\bigg (q_2-\beta _2\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\bigg )-(c_r-v)q_2-\beta _2g_r\mu -wq_2-k\alpha _{2}^{2}\bigg ]\nonumber \\&\quad -f\bigg [\sum _{i=1}^{2}\bigg ((g_m(q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da)-c_mq_i-\beta _{i}g_m\mu +wq_i \bigg )-k\alpha _{0}^{2}\bigg ].\nonumber \end{aligned}$$

(L.2)

Solving this two-stage game using backward induction technology. Taking the first-order partial derivative of $u_{Ri}^{F}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $q_i$ and $\alpha _i$, respectively. Thus, we have

$$\begin{aligned} \frac{\partial u_{R1}^{F}}{\partial q_1}&=\bigg ((1+f)(p-c_r-w+g_r)-f(g_m+w-c_m)\bigg )\nonumber \\&\quad -\bigg ((1+f)(p-v+g_r)-fg_m\bigg )F(\frac{q_1}{\beta _1}), \end{aligned}$$

(L.3)

$$\begin{aligned} \frac{\partial u_{R2}^{F}}{\partial q_2}&=\bigg ((1+f)(p-c_r-w+g_r)-f(g_m+w-c_m)\bigg )\nonumber \\&\quad -\bigg ((1+f)(p-v+g_r)-fg_m\bigg )F(\frac{q_2}{\beta _2}), \end{aligned}$$

(L.4)

$$\begin{aligned} \frac{\partial u_{R1}^{F}}{\partial \alpha _{1}}&=\lambda \bigg ((1+f)(p-v+g_r)-fg_m\bigg )\bigg (\frac{q_1}{\beta _1}F(\frac{q_1}{\beta _1})-\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\bigg ) \end{aligned}$$

(L.5)

$$\begin{aligned}&\quad +\lambda \gamma f g_m\bigg (\frac{q_2}{\beta _2}F(\frac{q_2}{\beta _2})-\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\bigg )\nonumber \\&\quad -\lambda \{(1+f)g_r-f(1-\gamma )g_m\}\mu -2k(1+f)\alpha _1,\nonumber \\ \frac{\partial u_{R2}^{F}}{\partial \alpha _{2}}&=\lambda \bigg ((1+f)(p-v+g_r)-fg_m\bigg )\bigg (\frac{q_2}{\beta _2}F(\frac{q_2}{\beta _2})-\int _{0}^{\frac{q_2}{\beta _2}}F(a)da\bigg )\\&\quad +\lambda \gamma fg_m\bigg (\frac{q_1}{\beta _1}F(\frac{q_1}{\beta _1})-\int _{0}^{\frac{q_1}{\beta _1}}F(a)da\bigg )\nonumber \\&\quad -\lambda \{(1+f)g_r-f(1-\gamma )g_m\}\mu -2k(1+f)\alpha _2,\nonumber \end{aligned}$$

(L.6)

Taking the second-order partial derivative of $\mathop u \nolimits _{Ri}^{F}(q_{i}, \alpha _{i})$ ($=1, 2$) with respect to $\alpha _i$, $q_i$, respectively. Thus, the corresponding Hessian matrices are given as follows:

$$\begin{aligned} \mathop H_{R1}\nolimits ^{F}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R1}^{F} }}{{\partial \mathop \alpha \nolimits _1^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R1}^{F} }}{{\partial \mathop \alpha \nolimits _1 \partial q_1}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R1}^{F} }}{{\partial q_1\partial \mathop \alpha \nolimits _1 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R1}^{F} }}{{\partial \mathop q_1\nolimits ^2 }}} \end{array}} \right] \end{aligned}$$

(L.7)

$$\begin{aligned}&=\begin{bmatrix} -\frac{\lambda ^2\{(1+f)(p-v+g_r)-fg_m\}q_{1}^2}{\beta _1^3}f(\frac{q_1}{\beta _1})-2(1+f)k+\Delta _{1} &{} \lambda \{(1+f)(p-v+g_r)-fg_m\}\frac{q_1}{\beta _1^2}f(\frac{q_1}{\beta _1})\\ \lambda \{(1+f)(p-v+g_r)-fg_m\}\frac{q_1}{\beta _1^2}f(\frac{q_1}{\beta _1}) &{} -\frac{\{(1+f)(p-v+g_r)-fg_m\}}{\beta _1}f(\frac{q_1}{\beta _1}) \end{bmatrix},\nonumber \\ \mathop H_{R2}\nolimits ^{F}&=\left[ {\begin{array}{*{20}{c}} {\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R2}^{F} }}{{\partial \mathop \alpha \nolimits _2^2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R2}^{F} }}{{\partial \mathop \alpha \nolimits _2 \partial q_2}}}\\ {\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R2}^{F} }}{{\partial q_2\partial \mathop \alpha \nolimits _2 }}}&{}{\frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R2}^{F} }}{{\partial \mathop q_2\nolimits ^2 }}} \end{array}} \right] \\&=\begin{bmatrix} -\frac{\lambda ^2\{(1+f)(p-v+g_r)-fg_m\}q_{2}^2}{\beta _2^3}f(\frac{q_2}{\beta _2})-2(1+f)k+\Delta _{2} &{} \lambda \{(1+f)(p-v+g_r)-fg_m\}\frac{q_2}{\beta _2^2}f(\frac{q_2}{\beta _2})\\ \lambda \{(1+f)(p-v+g_r)-fg_m\}\frac{q_2}{\beta _2^2}f(\frac{q_2}{\beta _2}) &{} -\frac{\{(1+f)(p-v+g_r)-fg_m\}}{\beta _2}f(\frac{q_2}{\beta _2}) \end{bmatrix}.\nonumber \end{aligned}$$

(L.8)

Thus, we have

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R1}^{F} }}{{\partial \mathop \alpha \nolimits _1^2 }}&=-\frac{\lambda ^2\{(1+f)(p-v+g_r)-fg_m\}q_{1}^2}{\beta _1^3}f(\frac{q_1}{\beta _1})-2(1+f)k+\Delta _{1}, \end{aligned}$$

(L.9)

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \mathop u \nolimits _{R2}^{F} }}{{\partial \mathop \alpha \nolimits _2^2 }}&=-\frac{\lambda ^2\{(1+f)(p-v+g_r)-fg_m\}q_{2}^2}{\beta _2^3}f(\frac{q_2}{\beta _2})-2(1+f)k+\Delta _{2}, \end{aligned}$$

(L.10)

$$\begin{aligned} \left| {\mathop H_{R1}\nolimits ^{F} } \right|&=\frac{(2(1+f)k-\Delta _{1})\{(1+f)(p-v+g_r)-fg_m\}}{\beta _1}f(\frac{q_1}{\beta _1}),\end{aligned}$$

(L.11)

$$\begin{aligned} \left| {\mathop H_{R2}\nolimits ^{F} } \right|&=\frac{(2(1+f)k-\Delta _{2})\{(1+f)(p-v+g_r)-fg_m\}}{\beta _2}f(\frac{q_2}{\beta _2}), \end{aligned}$$

(L.12)

and

$$\begin{aligned} \Delta _{1}=\frac{\lambda ^2\gamma ^{2}fg_m q_{2}^2}{\beta _2^3}f(\frac{q_2}{\beta _2})>0,\end{aligned}$$

(L.13)

$$\begin{aligned} \Delta _{2}=\frac{\lambda ^2\gamma ^{2}fg_m q_{1}^2}{\beta _1^3}f(\frac{q_1}{\beta _1})>0. \end{aligned}$$

(L.14)

Using the conditions $\frac{{\partial ^{2} \mathop u \nolimits _{Ri}^{F} (q_{i}, \alpha _{i})}}{{\partial \mathop \alpha ^{2}\nolimits _i }}<0$ and $\left| {\mathop H_{Ri}\nolimits ^{F} } \right| >0$, we can find that the above Hessian matrices are negative definite matrices. Consequently, the profit of the retailer i ($=1, 2$) is a joint concave function of $\alpha _i$ and $q_i$, and the first-order conditions can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order conditions $\frac{{\partial \mathop u \nolimits _{Ri}^{F} (q_{i}, \alpha _{i})}}{{\partial \mathop \alpha \nolimits _i }} = 0$ and $\frac{{\partial \mathop u \nolimits _{Ri}^{F}(q_{i}, \alpha _{i}) }}{{\partial q_i}} = 0$, where $i=1$, 2. Thus, we can get four solutions

$$\begin{aligned} \alpha _1^{F*}&=\alpha _2^{F*}=\bigg (\frac{\{(1+f)(p-c_r-w+g_r)-f(g_m+w-c_m)\}G}{(1+f)(p-v+g_r)-fg_m}-\int _{0}^{G}F(a)da\bigg )\ \end{aligned}$$

(L.15)

$$\begin{aligned}&\qquad \qquad *\frac{\lambda \{(1+f)(p-v+g_r)-(1-\gamma ) fg_m\}}{2(1+f)k}-\frac{\lambda \mu \{(1+f)g_r- (1-\gamma )fg_m\}}{2(1+f)k},\nonumber \\ \mathop q\nolimits _1^{F}&=\mathop q\nolimits _2^{F}=\{(1-\lambda )+\lambda (\alpha _0+(1-\gamma )\alpha _1)\}G, \end{aligned}$$

(L.16)

where $G=F^{-1}\left( \frac{(1+f)(p-c_r-w+g_r)-f(g_m+w-c_m)}{(1+f)(p-v+g_r)-fg_m}\right)$.

The profit of manufacturer is as follows:

$$\begin{aligned} \pi _{M}^{F}(\alpha _0)=\sum _{i=1}^{2}\bigg [g_m\bigg (q_i-\beta _i\int _{0}^{\frac{q_i}{\beta _i}}F(a)da\bigg )-c_mq_i-\beta _{i}g_m\mu +wq_i \bigg ]-k\alpha _{0}^{2}. \end{aligned}$$

(L.17)

Then, substituting Eqs. (L.15) and (L.16) into Eq. (2), and taking the first-order partial derivative of $\mathop \pi \nolimits _M^{F}(\alpha _{0})$ with respect to $\mathop \alpha \nolimits _0$, we have

$$\begin{aligned} \frac{\partial \pi _{M}^{F}(\alpha _{0})}{\partial \alpha _{0}}=2 \lambda (g_m+w-c_m)G-2\lambda g_m\bigg (\int _{0}^{G}F(a)da+\mu \bigg )-2k\alpha _{0}. \end{aligned}$$

(L.18)

Taking the second-order partial derivative of $\pi _{M}^{F}(\alpha _{0})$ with respect to $\alpha _0$, we have

$$\begin{aligned} \frac{\partial ^2 \pi _{M}^{F}(\alpha _{0})}{\partial \alpha _{0}^2}=-2k<0. \end{aligned}$$

(L.19)

Thus, the profit function of the manufacturer is a concave function of $\mathop \alpha \nolimits _0$, and the first-order condition can guarantee optimality.

Next, at the Nash equilibrium, we solve the first-order condition $\frac{{\partial \mathop \pi \nolimits _{M}^{F}(\alpha _{0}) }}{{\partial \mathop \alpha \nolimits _0 }} = 0$. Thus, we can get three solutions

$$\begin{aligned} \alpha _{0}^{F*}&=\frac{\lambda (g_m+w-c_m)G-\lambda g_m\left( \int _{0}^{G}F(a)da+\mu \right) }{k},\end{aligned}$$

(L.20)

$$\begin{aligned} q_{1}^{F*}&=q_{2}^{F*}=(1-\lambda +\lambda (\alpha _{0}^{F*}+(1-\gamma )\alpha _{1}^{F*}))G\\&\qquad \quad =(1-\lambda ) G+\frac{\lambda ^{2}(g_m+w-c_m)G^{2}-\lambda ^{2} g_m\left( \int _{0}^{G}F(a)da+\mu \right) G}{k}\nonumber \\&\qquad \qquad +\bigg (\frac{\{(1+f)(p-c_r-w+g_r)-f(g_m+w-c_m)\}G}{(1+f)(p-v+g_r)-fg_m}-\int _{0}^{G}F(a)da\bigg )\nonumber \\&\qquad \qquad *\frac{\lambda ^{2}(1-\gamma )G\{(1+f)(p-v+g_r)-(1-\gamma ) fg_m\}}{2(1+f)k}-\frac{\lambda ^{2}(1-\gamma )\mu \{(1+f)g_r- (1-\gamma )fg_m\}G}{2(1+f)k}\nonumber . \end{aligned}$$

(L.21)

$\square$