Channel structure selection in a competitive supply chain under consideration of marketing effort strategy

Abstract

We study the optimal decisions of a supplier in terms of channel structure selection and marketing effort strategy when facing a competitor offering substitutable products. By employing game theoretic models for different retail competition scenarios, we show that equilibrium channel structures are primarily determined by the intensity of retail competition along with the marketing effort level. Channel structure selection and marketing effort strategy are interdependent and interactive. Our results show that the supplier has an incentive to make marketing effort and sell products directly to consumers in most cases. When the retail competition becomes fiercer, the supplier can benefit from selecting a distribution strategy based on customers’ sensitivity to the supplier’s marketing effort. In contrast, the supplier selling products through the downstream retailer will not make marketing effort under certain market conditions (e.g., low competition and high customer sensitivity to the supplier’s effort performance), while the competitor adopts the direct-sales strategy. Generally, the supplier will prefer making marketing effort relative to undertaking product distribution. Finally, we also examine the strategic effects of channel structure selection and marketing effort decisions on the competitor’s profit in the retail competition market.

Appendices

Appendix A Symbols and functions

A.1. The following symbols are used to simplify the results, where $i=1,2$ and $i\ne j$.

$$\begin{aligned} A= & {} (2 + \gamma )a + (2 + \gamma )(\gamma - 1)c;\\ B= & {} (2 + \gamma )a + (2 - {\gamma ^2})c;\\ S= & {} 2{\gamma ^2} + \gamma - 4;\\ T= & {} 2{\gamma ^2} - 4 - \gamma ;\\ \tau ({t_i},{t_j})= & {} (2 + \gamma )a + 2\beta \sqrt{{t_i}} + \beta \gamma \sqrt{{t_j}} ;\\ \phi= & {} 4(4 - {\gamma ^2}) + ({\gamma ^2} - 4){\beta ^2};\\ \varphi= & {} 2(4 - 2{\gamma ^2} + \gamma ) + (\gamma - 2){\beta ^2};\\ M= & {} 4(2 - {\gamma ^2})(4 - {\gamma ^2}) - (8 - 3{\gamma ^2}){\beta ^2};\\ N= & {} 2(4 - {\gamma ^2}) + (\gamma - 2){\beta ^2}. \end{aligned}$$

A.2. The solutions ${\beta _i}$ and ${\gamma _j}$ presented in all propositions and corollaries are determined by ${Z_i}({\beta _i}\mid {{\gamma _j}}) = 0$, where the functions ${Z_i}({\beta _i}\mid {{\gamma _j}} )$ are given as

$$\begin{aligned} {Z_1}({\beta _1}\mid \gamma )= & {} 4{(4 - {\gamma ^2})^{1/2}}{(2 - {\gamma ^2})^{3/2}} \\&- 2(4{\gamma ^4} - 17{\gamma ^2} + 16)\\&+ [8 -3{\gamma ^2} - 2{(4 - {\gamma ^2})^{1/2}}{(2 - {\gamma ^2})^{1/2}}]{\beta ^2};\\ {Z_2}({\beta _2}\mid \gamma )= & {} [4(2 - {\gamma ^2}) - (4 - {\gamma ^2}){\beta ^2}]{M^2}\\&- \phi {[2ST - (8 - 3{\gamma ^2}){\beta ^2}]^2};\\ {Z_3}({\beta _3}\mid \gamma )= & {} 2({\gamma ^2} - 4)ST + 8(8 - 3{\gamma ^2})(2 - {\gamma ^2})\\&- (4 - {\gamma ^2})(8 - 3{\gamma ^2}){\beta ^2};\\ {Z_{4,5}}({\beta _4},{\beta _5}\mid \gamma )= & {} [4(2 - {\gamma ^2}) - (4 - {\gamma ^2}){\beta ^2}](2M - \phi {\beta ^2})^2\\&- {\varphi ^2}(4 - {\beta ^2})(4 - {\gamma ^2}){[2(\gamma + 2{\gamma ^2} - 4)} \\&{+ (\gamma + 2){\beta ^2}]^2};\\ {Z_6}({\beta _6}\mid \gamma )= & {} [2\gamma (2\gamma - 3) + (2 - \gamma ){\beta ^2}]\\&[2(2{\gamma ^2} - 4 + \gamma )+ (2 + \gamma ){\beta ^2}];\\ {Z_7}({\beta _7}\mid \gamma )= & {} - (2 - {\gamma ^2})(4 - {\gamma ^2}){\beta ^6} \\&+ (64 - 56{\gamma ^2} + 11{\gamma ^4}){\beta ^4}\\&+8(2 - {\gamma ^2})({\gamma ^4} - 8){\beta ^2} \\&+ 16{\gamma ^2}{(2 - {\gamma ^2})^2}(4 - {\gamma ^2});\\ {Z_8}({\beta _8}\mid \gamma )= & {} 2(2{\gamma ^2} - 4 + \gamma )(2{\gamma ^2} - 4 - \gamma )(3 - {\gamma ^2})\\&+ (\gamma - 3)( -{\gamma ^3} - 3{\gamma ^2} + 3\gamma + 8){\beta ^2};\\ {Z_9}({\beta _9}\mid \gamma )= & {} 2(4 - {\gamma ^2}){(4 - {\gamma ^2} - {\beta ^2})^2}[8 - 4{\gamma ^2}\\&- (4 - {\gamma ^2}){\beta ^2}] -{[2M - \phi {\beta ^2}]^2}. \end{aligned}$$

Appendix B Proofs

Proof of Theorem 1

For the four channel structures under Model I, we give the proof as follows.

(i) Channel structure CC

The demands of $S_{1}$ and $S_{2}$ are ${q_1} = a - {p_1} + \gamma {p_2}$ and ${q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}}$, respectively. The profit functions of $S_{1}$ and $S_{2}$ can be presented by $\Pi _{S1}^{CC} = ({p_1} - c){q_1}$ and $\Pi _{S2}^{CC} = ({p_2} - c){q_2} - {e_2}$, respectively.

The optimization problem of chains i is ${\mathop {\max }\limits _{{p_i}}}\ \Pi _{Si}^{CC}({p_i},{p_j})$. Because ${{\partial \Pi _{S1}^{CC}} /{\partial {p_1}}} = a - 2{p_1} + \gamma {p_2} + c$ and ${{{{\partial ^2}\prod _{S1}^{CC}} / {\partial {p_1}}}^2} = - 2 < 0$ , $\Pi _{S1}^{CC}$ is concave over ${p_1}$. Similarly, we can show that $\Pi _{S1}^{CC}$ is concave over ${p_2}$. From the first-order conditions, we have

$$\begin{aligned} \begin{aligned}&{{\partial \Pi _{S1}^{CC}} / {\partial {p_1}}} = a - 2{p_1} + \gamma {p_2} + c = 0\ \mathrm {and} \\&{{\partial \Pi _{S2}^{CC}} / {\partial {p_2}}} = a - 2{p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} + c = 0. \end{aligned} \end{aligned}$$

(B1)

Because ${{{{\partial ^2}\Pi _{S2}^{CC}} / {\partial {e_2}}}^2} < 0$, $\Pi _{S2}^{CC}$ is concave over ${e_2}$ and we have

$$\begin{aligned} {{\partial \Pi _{S2}^{CC}}/ {\partial {e_2}}} = {{({p_2} - c)\beta } / {(2\sqrt{{e_2}} }}) - 1 = 0. \end{aligned}$$

(B2)

From Eqs. (B1) and (B2), the equilibrium solutions of decision and profit are given in Table 2.

(ii) Channel structure CD

The demand of $S_{1}$ is ${q_1} = a - {p_1} + \gamma {p_2}$. The profit function of $S_{1}$ is $\Pi _{S1}^{CD} = ({p_1} - c){q_1}$. The optimization problem is ${\mathop {\max }\limits _{{p_1}}} \ \Pi _{S1}^{CD}({p_1},{p_2})$. Similarly to channel structure CC, we can prove that the function $\Pi _{S1}^{CD}$ is concave over the price ${p_1}$. We have

$$\begin{aligned} {{\partial \Pi _{S1}^{CD}} / {\partial {p_1}}} = a - 2{p_1} + \gamma {p_2} + c = 0. \end{aligned}$$

(B3)

The demand of $S_{2}$ is ${q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}}$. The profit function of $R_{2}$ is $\Pi _{R2}^{CD} = ({p_2} - {w_2}){q_2}$. Given the unit wholesale price ${w_2}$ offered by its upstream $S_{2}$, $R_{2}$’s optimal problem is written as ${\mathop {\max }\limits _{{p_2}}}\ \Pi _{R2}^{CD}({p_2}\mid {{w_2},{p_1}})$. Obviously, ${{\partial \Pi _{R2}^{CD}} / \partial }{p_2} = a - 2{p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} + {w_2}$ and ${{{\partial ^2}\Pi _{R2}^{CD}} / \partial }{p_2}^2 = - 2 < 0$; therefore, we have $\Pi _{R2}^{CD}$ is concave over $p_{2}$. We have

$$\begin{aligned} {{\partial \Pi _{R2}^{CD}} / \partial }{p_2} = a - 2{p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} + {w_2} = 0. \end{aligned}$$

(B4)

From Eqs. (B3) and (B4), we can derive

$$\begin{aligned} p_1^{*CD}= & {} \frac{{(2 + \gamma )a + 2c + \beta \gamma \sqrt{e_2^{CD}} + \gamma w_2^{CD}}}{{4 - {\gamma ^2}}}, \end{aligned}$$

(B5)

$$\begin{aligned} p_2^{*CD}= & {} \frac{{(2 + \gamma )a + 2c + 2\beta \sqrt{e_2^{CD}} + 2w_2^{CD}}}{{4 - {\gamma ^2}}}. \end{aligned}$$

(B6)

The profit function of $S_{2}$ is $\Pi _{S2}^{CD} = ({w_2} - c){q_2} - {e_2}$. According to Eqs. (B5) and (B6), the optimization problem of $S_{2}$ is ${\mathop {\max }\limits _{w_2}} \ \Pi _{S2}^{CD}({w_2},{p_2},{p_1})$.

From ${{\partial \Pi _{S2}^{CD}} / \partial }{w_2} = [(2 + \gamma )a + 2\beta \sqrt{{e_2}} + (2{\gamma ^2} - 4){w_2}{{ + (\gamma - {\gamma ^2} + 2)c]} / {(4 - {\gamma ^2})}}$ and ${{{\partial ^2}\Pi _{S2}^{CD}} / \partial }{w_2}^2 = 2{\gamma ^2} - 4 < 0$, the function $\Pi _{S2}^{CD}$ is concave over $w_{2}$. Then, from the first-order condition, we have

$$\begin{aligned} \begin{aligned}&\partial \Pi _{S2}^{CD} / \partial {w_2}\\&\quad = \frac{(2 + \gamma )a + 2\beta \sqrt{{e_2}} + (2{\gamma ^2} - 4){w_2}+ (\gamma - {\gamma ^2} + 2)c}{4 - {\gamma ^2}}\\&\quad =0 \end{aligned} \end{aligned}$$

(B7)

Since ${{{{\partial ^2}\prod _{S2}^{CD}}/ {\partial {e_2}}}^2} < 0$, the optimal marketing effort level of $S_{2}$ should satisfy the first-order condition for the given prices, which is written as follows.

$$\begin{aligned} {{({w_2} - c)\beta } / {(2\sqrt{{e_2}} }}) - 1 = 0. \end{aligned}$$

(B8)

From Eqs. (B7) and (B8), we can derive the equilibrium solutions as shown in Table 2.

(iii) Channel structure DC

The demand of $S_{1}$ is ${q_1} = a - {p_1} + \gamma {p_2}$. The profit function of $R_{1}$ is $\Pi _{R1}^{DC} = ({p_1} - {w_1}){q_1}$. Given the unit wholesale price determined by the upstream $S_{1}$, the optimal problem of $R_{1}$ is ${\mathop {\max }\limits _{{p_1}}}\ \Pi _{R1}^{DC}( {{p_1}} \mid {w_1},{p_2})$. Since ${{\partial \prod _{R1}^{DC}} / \partial }{p_1} = a - 2{p_1} + \gamma {p_2} + {w_1}$ and ${{{\partial ^2}\Pi _{R1}^{DC}} / \partial }{p_1}^2 = - 2 < 0$, we have that the function $\Pi _{R1}^{DC}$ is concave over $p_{1}$. From the first-order condition of ${{\partial \Pi _{R1}^{DC}} / {\partial {p_1}}}$, we have

$$\begin{aligned} a - 2{p_1} + \gamma {p_2} + {w_1} = 0. \end{aligned}$$

(B9)

The demand of $S_{2}$ is ${q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}}$. The profit of $S_{2}$ is written as ${\mathop {\max }\limits _{{p_2}}} \ \Pi _{S2}^{DC} = ({p_2} - c){q_2}$. Similarly, we can show that $\Pi _{S2}^{DC}$ is concave over $p_{2}$. We have

$$\begin{aligned} {{\partial \Pi _{S2}^{DC}} / {\partial {p_2}}} = a - 2{p_2} + \gamma {p_1} + \beta \sqrt{e{}_2} + c = 0. \end{aligned}$$

(B10)

From Eqs. (B9) and (B10), we can derive the following results.

$$\begin{aligned} p_1^{*DC}= & {} \frac{{(2 + \gamma )a + \gamma c + \beta \gamma \sqrt{e_2^{DC}} + 2w_1^{DC}}}{{4 - {\gamma ^2}}}, \end{aligned}$$

(B11)

$$\begin{aligned} p_2^{*DC}= & {} \frac{{(2 + \gamma )a + 2c + 2\beta \sqrt{e_2^{DC}} + \gamma w_2^{DC}}}{{4 - {\gamma ^2}}}. \end{aligned}$$

(B12)

According to Eqs. (B11) and (B12), the optimal problem of $S_{1}$ is written as ${\mathop {\max }\limits _{{w_1}}} \ \Pi _{S1}^{DC}({w_1},{p_1},{p_2})$. Since $\displaystyle \frac{{\partial \Pi _{S1}^{DC}}}{{\partial {w_1}}}=$ $\displaystyle \frac{{(2 + \gamma )a + \gamma \beta \sqrt{{e_2}} + 2({\gamma ^2} - 2){w_1} + (\gamma - {\gamma ^2} + 2)c}}{{4 - {\gamma ^2}}}$ and $\displaystyle \frac{{{\partial ^2}\Pi _{S1}^{DC}}}{{\partial {w_1}^2}} = 2({\gamma ^2} - 2) < 0$, the function $\Pi _{S1}^{DC}$ is concave over $w_{1}$. From the first-order condition, we have

$$\begin{aligned} w_1^{*DC} = \frac{{(2 + \gamma )a + (\gamma - {\gamma ^2} + 2)c + \gamma \beta \sqrt{e_2^{DC}} }}{{4 - 2{\gamma ^2}}}. \end{aligned}$$

(B13)

Similarly, the optimal problem of $S_{2}$ is ${\mathop {\max }\limits _{e_2}} \ \Pi _{S2}^{DC}({e_2},{p_1},{p_2})$. Since ${{{{\partial ^2}\Pi _{S2}^{DC}} / {\partial {e_2}}}^2} < 0$, the optimal marketing effort level of $S_{2}$ should satisfy the first-order condition ${{\partial \Pi _{S2}^{DC}} / {\partial {e_2}}} = 0$ for the given prices and we have

$$\begin{aligned} e_2^{*DC} = {{{{(p_2^{DC} - c)}^2}{\beta ^2}} / 4}. \end{aligned}$$

(B14)

According to Eqs. (B12), (B13) and (B14), we can derive the equilibrium solutions as shown in Table 2.

(iv) Channel structure DD

The demand of $S_{1}$ is ${q_1} = a - {p_1} + \gamma {p_2}$ and that of $S_{2}$ is ${q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}}$. The profit functions of $S_{1}$ and $R_{1}$ are $\Pi _{S1}^{DD} = ({p_1} - {w_1}){q_1}$ and $\Pi _{R1}^{DD} = ({w_1} - c){q_1}$, respectively. Similarly, it is easy to show that $\Pi _{S1}^{DD}$ is concave over $p_{1}$ and $\Pi _{R1}^{DD}$ is concave over $w_{1}$. Given $w_{1}$ offered by the upstream $S_{1}$, $R_{1}$’s optimal problem is ${\mathop {\max } \limits _{{p_1}}} \ \Pi _{R1}^{DD}({p_1},{w_1},{p_2})$. We have

$$\begin{aligned} {{\partial \Pi _{R1}^{DD}} / {\partial {p_1}}} = a - 2{p_1} + \gamma {p_2} + {w_1} = 0. \end{aligned}$$

(B15)

For supply chain 2, the profit functions of $S_{2}$ and $R_{2}$ are $\Pi _{S2}^{DD} = ({p_2} - {w_2}){q_2} - {e_2}$ and $\Pi _{R2}^{DD} = ({w_2} - c){q_2}$, respectively. Similarly, it is easy to show $\Pi _{S2}^{DD}$ and $\Pi _{R2}^{DD}$ are concave over $p_{2}$ and $w_{2}$, respectively. Given the unit wholesale price ($w_{2}$) determined by the upstream $S_{2}$, the optimal problem of $R_{2}$ is ${\mathop {\max } \limits _{{p_2}}} \ \Pi _{R2}^{DD}({p_2},{w_2},{p_1})$. We have

$$\begin{aligned} {{\partial \Pi _{R2}^{DD}} / {\partial {p_2}}} = a - 2{p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} + {w_2} = 0. \end{aligned}$$

(B16)

From Eqs. (B15) and (B16), we can obtain the best responses as follows.

$$\begin{aligned} p_1^{*DD}= & {} \frac{{(2 + \gamma )a + \gamma \beta \sqrt{e_2^{DD}} + 2w_1^{DD} + \gamma w_2^{DD}}}{{4 - {\gamma ^2}}}, \end{aligned}$$

(B17)

$$\begin{aligned} p_2^{*DD}= & {} \frac{{(2 + \gamma )a + 2\beta \sqrt{e_2^{DD}} + 2w_2^{DD} + \gamma w_1^{DD}}}{{4 - {\gamma ^2}}}. \end{aligned}$$

(B18)

Based on Eqs. (B17) and (B18), the profit of $S_{i}$ is written as ${\mathop {\max } \limits _{w_i}} \ \Pi _{Si}^{DD}({w_i}\mid {{p_i},{p_j}} )$, where $i,j = 1,2$ and $i \ne j$. We have ${{\partial \Pi _{S1}^{DD}} / \partial }{w_1} = {{[a - {p_1} + \gamma {p_2} + ({\gamma ^2} - 2)({w_1} - c)]} }{/ {(4 - {\gamma ^2})}} = 0$ and ${{\partial \Pi _{S2}^{DD}} / \partial }{w_2} = [a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} {{ + ({\gamma ^2} - 2)({w_2} - c)]} / {(4 - {\gamma ^2})}} = 0$. Therefore, we obtain

$$\begin{aligned} w_1^{*DD}= & {} {{[2\gamma (3 - {\gamma ^2})\beta \sqrt{e_2^{DD}} - TB]} / {(ST)}}, \end{aligned}$$

(B19)

$$\begin{aligned} w_2^{*DD}= & {} {{[(8 - 3{\gamma ^2})\beta \sqrt{e_2^{DD}} - TB]}/ {(ST)}}. \end{aligned}$$

(B20)

Similar to (ii), we decide the optimal marketing effort level of $S_{2}$ according to the first-order condition (i.e., ${{\partial \Pi _{S2}^{DD}} / {\partial {e_2}}} = {{({w_2} - c)\beta } / {(2\sqrt{{e_2}} }}) - 1 = 0$) and have

$$\begin{aligned} e_2^{*DD} = {{{{(w_2^{DD} - c)}^2}{\beta ^2}} / 4}. \end{aligned}$$

(B21)

From Eqs. (B19–B21), we can derive the equilibrium solutions as shown in Table 2. $\square$

Proof of Corollary 1

(i) From Theorem 1, we have

$$\begin{aligned}&p_1^{*CC} - p_2^{*CC} = \{ ({\beta ^2} - 4 - 2\gamma )a + ({\beta ^2}- 4 - 2\gamma \\&\quad + \gamma {\beta ^2})c + 2[(2 + \gamma )a + (2 + \gamma - {\beta ^2})c]\} /[2({\gamma ^2} + {\beta ^2} - 4)]. \end{aligned}$$

Here, we have $2({\gamma ^2} + {\beta ^2} - 4) < 0$, and $({\beta ^2} - 4 - 2\gamma )a + ({\beta ^2} - 4 - 2\gamma + \gamma {\beta ^2})c + 2[(2 + \gamma )a + (2 + \gamma - {\beta ^2})c] = {\beta ^2}[a + (\gamma - 1)c]$. Meanwhile, since $\beta ,\gamma \in [0,1]$, we obtain ${\beta ^2}[a + (\gamma - 1)c] \ge 0$ and ${\beta ^2}[a + (\gamma - 1)c] = 0$ if and only if $\beta = 0$. Therefore, we have $p_1^{*CC} \le p_2^{*CC}$.

Similarly, from Theorem 1, we have $q_1^{*CC} - q_2^{*CC} = {\beta ^2}[a + (\gamma - 1)c]/[2({\gamma ^2} + {\beta ^2} - 4)]$. It is easy to find $q_1^{*CC} - q_2^{*CC} \le 0$. Obviously, $q_1^{*CC} - q_2^{*CC} = 0$ if and only if $\beta = 0$.

Meanwhile, from Theorem 1, we have

$$\begin{aligned}&\Pi _{S1}^{*CC} - \Pi _{S2}^{*CC}\\&\quad = {{{A^2}{{({\beta ^2} - 4 - 2\gamma )}^2}} / {[4{{(2 + \gamma )}^2}{{({\gamma ^2} + {\beta ^2} - 4)}^2}]}}\\&\qquad - {{{A^2}(4 - {\beta ^2})} / {[4{{({\gamma ^2} + {\beta ^2} - 4)}^2}]}}\\&\quad = {{{\beta ^2}({\gamma ^2} + {\beta ^2} - 4){A^2}} / {[4{{(2 + \gamma )}^2}{{({\gamma ^2} + {\beta ^2} - 4)}^2}]}}. \end{aligned}$$

Because $\beta ,\gamma \in [0,1]$, we have $\Pi _{S2}^{*CC} \ge \Pi _{S1}^{*CC}$. Obviously, $\Pi _{S2}^{*CC} = \Pi _{S1}^{*CC}$ if and only if $\beta = 0$.

From Theorem 1, we have

$$\begin{aligned}&p_1^{*DD} - p_2^{*DD}\\&\quad =\displaystyle \frac{{(2 + \gamma )a + \gamma \beta \sqrt{e_2^{*DD}} + 2w_1^{*DD} + \gamma w_2^{*DD}}}{{4 - {\gamma ^2}}}\\&\qquad - \displaystyle \frac{{(2 + \gamma )a + 2\beta \sqrt{e_2^{*DD}} + 2w_2^{*DD} + \gamma w_1^{*DD}}}{{4 - {\gamma ^2}}}\\&\quad =\displaystyle \frac{{\mathrm{{(4}}{\gamma ^4}\mathrm{{ + 2}}{\gamma ^3} - \mathrm{{6}}\gamma - \mathrm{{20}}{\gamma ^2}\mathrm{{ + 24)}}\beta \sqrt{e_2^{*DD}} }}{{{\gamma ^2} - {{(2{\gamma ^2} - 4)}^2}}}. \end{aligned}$$

Because $\beta ,\gamma \in [0,1]$, we have $p_1^{*DD} \le p_2^{*DD}$. Obviously, $p_1^{*DD} = p_2^{*DD}$ if and only if $\beta = 0$.

Similarly, from Theorem 1, we have

$$\begin{aligned}&q_1^{*DD} - q_2^{*DD}\\&\quad = \displaystyle \frac{{\left( {2 - {\gamma ^2}} \right) \left[ {TA + 2({\gamma ^2} - 3) \gamma \beta \sqrt{e_2^{*DD}} } \right] }}{{({\gamma ^2} - 4)ST}}\\&\qquad - \displaystyle \frac{{\left( {2 - {\gamma ^2}} \right) \left[ {TA + (3{\gamma ^2} - 8)\beta \sqrt{e_2^{*DD}} } \right] }}{{({\gamma ^2} - 4)ST}}\\&\quad = \displaystyle \frac{{\left( {2 - {\gamma ^2}} \right) \beta \sqrt{e_2^{*DD}} }}{{({\gamma ^2} - 4)ST}}. \end{aligned}$$

Because $\beta ,\gamma \in [0,1]$ and $ST > 0$, we have $q_1^{*DD} \le q_2^{*DD}$. Obviously, $q_1^{*DD} = q_2^{*DD}$ if and only if $\beta = 0$.

Moreover, from Theorem 1, we have

$$\begin{aligned}&w_1^{*DD} - w_2^{*DD}\\&\quad = {{[2\gamma (3 - {\gamma ^2})\beta \sqrt{e_2^{*DD}} - TB]} / {(ST)}}\\&\qquad - {{[(8 - 3{\gamma ^2})\beta \sqrt{e_2^{*DD}} - TB]} / {(ST)}}\\&\quad =( - {\mathrm{2}}{\gamma ^3} + {\mathrm{3}}{\gamma ^2} + 6\gamma - 8 )\beta \sqrt{e_2^{*DD}} / {(ST)}. \end{aligned}$$

Because $\beta ,\gamma \in [0,1]$ and $ST > 0$, we have $w_1^{*DD} \le w_2^{*DD}$. Obviously, $w_1^{*DD} = w_2^{*DD}$ if and only if $\beta = 0$.

Meanwhile, we can prove $\Pi _{S1}^{*DD} \le \Pi _{S2}^{*DD}$ by the similar method.

(ii) From Theorem 1, we have

$$\begin{aligned}&p_1^{*CD} - p_2^{*CD}\\&\quad = {{[(2 + \gamma )a + 2c + \gamma \beta \sqrt{e_2^{*CD}} + \gamma w_2^{*CD}]} / {(4 - {\gamma ^2})}} \\&\qquad - {{[(2 + \gamma )a + \gamma c{ + 2\beta \sqrt{e_2^{*CD}} + 2w_2^{*CD}}]} / {(4 - {\gamma ^2})}}\\&\quad = {{[} - A + (2{\gamma ^2} - 6)\beta \sqrt{e_2^{*CD}} ]}/[2(2 - {\gamma ^2})(2 + \gamma )]. \end{aligned}$$

Because $\beta ,\gamma \in [0,1]$, we have $p_1^{*CD} \le p_2^{*CD}$. Obviously, $p_1^{*CD} = p_2^{*CD}$ if and only if $\beta = 0$.

Meanwhile, from Theorem 1, we have

$$\begin{aligned}&q_1^{*CD} - q_2^{*CD} \\&\quad = \displaystyle \frac{{{\varphi ^2}{A^2}}}{{{{(4 - {\gamma ^2})}^2}{{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}}}\\&\qquad - \displaystyle \frac{{[4(2 - {\gamma ^2}) - (4 - {\gamma ^2}){\beta ^2}]{A^2}}}{{{{(4 - {\gamma ^2})}^2}{{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}}}\\&\quad = \displaystyle \frac{{(2 - \gamma )[2(\gamma + 1) - {\beta ^2}]A}}{{{{(4 - {\gamma ^2})}^2}{{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}}}. \end{aligned}$$

Because $\beta ,\gamma \in [0,1]$ and $A > 0$, then we have $q_1^{*CD} \ge q_2^{*CD}$.

From Theorem 1, we have

$$\begin{aligned}&p_1^{*DC} - p_2^{*DC} \\&\quad = {{[(2 + \gamma )a + \gamma c + \gamma \beta \sqrt{e_2^{*DC}} + 2w_1^{*DC}]}/{(4 - {\gamma ^2})}}\\&\qquad - {{[(2 + \gamma )a + 2c + 2\beta \sqrt{e_2^{*DC}} + \gamma w_1^{*DC}]} /{(4 - {\gamma ^2})}}\\&\quad = {{(w_1^{*DC} - c - \beta \sqrt{e_2^{*DC}} )} /{(2 + \gamma )}}. \end{aligned}$$

It is easy to find that $p_1^{*DC} \ge p_2^{*DC}$ is satisfied if and only if $w_1^{*DC} - c \ge \beta \sqrt{e_2^{*DC}}$ is satisfied. Then, we have $w_1^{*DC} - c - \beta \sqrt{e_2^{*DC}} = {{[A + (\gamma - 4 + 2{\gamma ^2})\beta \sqrt{e_2^{*DC}} ]} /{(4 - 2{\gamma ^2})}}$.

Since $4 - 2{\gamma ^2} > 0$, we find $w_1^{*DC} - c \ge \beta \sqrt{e_2^{*DC}}$ is equivalent to $A + (\gamma - 4 + 2{\gamma ^2})\beta \sqrt{e_2^{*DC}} \ge 0$.

Then, we have $A + (\gamma - 4 + 2{\gamma ^2})\beta \sqrt{e_2^{*DC}} = [4(2 - {\gamma ^2})(4 - {\gamma ^2}) + (20{\gamma ^2} - 24 - 4{\gamma ^4}){{{\beta ^2}]A} /M}$.

Because $\beta ,\gamma \in [0,1]$, we have $p_1^{*DC} \ge p_2^{*DC}$.

Similarly, from Theorem 1, we can also show that $q_1^{*DC} \le q_2^{*DC}$.

Meanwhile, we can prove $\Pi _{S1}^{*CD} > \Pi _{S2}^{*CD}$ and $\Pi _{S1}^{*DC} < \Pi _{S2}^{*DC}$ by the similar method. $\square$

Proof of Proposition 1

(i) From Theorem 1, we have

$$\begin{aligned}&\Pi _{S2}^{*CC} - \Pi _{S2}^{*CD} \\&\quad = \displaystyle \frac{{{A^2}(4 - {\beta ^2})}}{{4{{({\gamma ^2} + {\beta ^2} - 4)}^2}}} - \frac{{[4(2 - {\gamma ^2}) - (4 - {\gamma ^2}){\beta ^2}]{A^2}}}{{(4 - {\gamma ^2}){{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}}}\\&={A^2}(4 - {\beta ^2})(4 - {\gamma ^2}){{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}\\&\quad \frac{{- 4{A^2}({\gamma ^2} + {\beta ^2} - 4)^2[4(2 - {\gamma ^2}) - (4 - {\gamma ^2}){\beta ^2}]}}{4(4 - {\gamma ^2})({\gamma ^2} + {\beta ^2} - 4)^2[4(2 - {\gamma ^2}) - 2{\beta ^2}]^2}. \end{aligned}$$

It is easy to find ${{{A^2}} /{\{ 4(4 - {\gamma ^2}){{({\gamma ^2} + {\beta ^2} - 4)}^2}}}{{{{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}\} }} > 0$. We assume

$$\begin{aligned}&{f_1}(\beta ,\gamma ) = 4(4 - {\gamma ^2})(2 - {\gamma ^2})(4 - 3{\gamma ^2})\\&\quad - (4 - {\gamma ^2})(16 - 16{\gamma ^2} + 3{\gamma ^4}){\beta ^2} + 2{(2 - {\gamma ^2})^2}{\beta ^4}. \end{aligned}$$

Then, there is $\Pi _{S2}^{*CC} - \Pi _{S2}^{*CD} \ge 0$ satisfied, if and only if the condition ${f_1}(\beta ,\gamma ) \ge 0$ is established. Here, we can regard ${f_1}(\beta ,\gamma )$ as a quadratic function of ${\beta ^2}$. Based on the properties of quadratic functions, we calculate

$$\begin{aligned} \Delta= & {} {(4 - {\gamma ^2})^2}{(16 - 16{\gamma ^2} + 3{\gamma ^4})^2}\\&\quad - 32{(2 - {\gamma ^2})^3}(4 - {\gamma ^2})(4 - 3{\gamma ^2}) < 0 \end{aligned}$$

Finally, we can obtain $\Pi _{S2}^{*CC} \ge \Pi _{S2}^{*CD}$.

Similarly, we can prove $\Pi _{S1}^{*CC} \ge \Pi _{S1}^{*DC}$ and omit the details here.

(ii) From Theorem 1, we have

$$\begin{aligned}&\Pi _{S1}^{*CD} - \Pi _{S1}^{*DD}\\&\quad = {{{\varphi ^2}{A^2}} /{{{\{ (4 - {\gamma ^2})[4(2 - {\gamma ^2}) - 2{\beta ^2}]\} }^2}}}\\&\qquad {{ - (2 - {\gamma ^2}){\varphi ^2}{A^2}} /{\{ (4 - {\gamma ^2}){{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]}^2}\} }}. \end{aligned}$$

Let ${Z_1}({\beta _1}\mid \gamma ) = 4{(4 - {\gamma ^2})^{1/2}}{(2 - {\gamma ^2})^{3/2}} - 2(4{\gamma ^4} - 17{\gamma ^2} + 16) + [8 - 3{\gamma ^2} - 2{(4 - {\gamma ^2})^{1/2}}{(2 - {\gamma ^2})^{1/2}}]\beta _1^2$.

Then, we find that $\Pi _{S1}^{*CD} - \Pi _{S1}^{*DD} \ge 0$ is equivalent to the condition of ${Z_1}({\beta _1}\mid \gamma ) \ge 0$.

• (i) When ${\beta _1} = 0$, we have ${{\partial {Z_1}(0\mid \gamma )}/ {\partial \gamma }} > 0$, ${Z_1}(0\mid 0) < 0$ and ${Z_1}(0\mid 1) > 0$;

• (ii) When ${\beta _1} = 1$, we have ${\partial {Z_1}(1\mid \gamma )} / {\partial \gamma } > 0$, ${Z_1}(1\mid 0) < 0$ and ${Z_1}(1\mid 1) > 0$. Finally, we can obtain

  • if $(\gamma ,\beta ) \in \{ 0 \le \gamma \le {\gamma _1}, 0 \le \beta \le 1\}$

    $\cup \{ {\gamma _1} \le \gamma \le {\gamma _2}, 0 \le \beta \le {\beta _1}\}$, $\Pi _{S1}^{*DD} \le \Pi _{S1}^{*CD}$;

  • if $(\gamma ,\beta ) \in \{ {\gamma _1} \le \gamma \le {\gamma _2}, {\beta _1} < \beta \le 1\}$

    $\cup \{ {\gamma _2} < \gamma \le 1, 0 \le \beta \le 1\}$, $\Pi _{S1}^{*DD} > \Pi _{S1}^{*CD}$, where the thresholds $\gamma _{1}$ ($\approx 0.8062$) and $\gamma _{2}$ ($\approx 0.9309$) are uniquely determined by the equations as ${Z_1}(0\mid {\gamma _1}) = 0$ and ${Z_1}(1\mid {\gamma _2}) = 0$.

• (iii) Similar to the proof of part (ii), we can obtain the results in part (iii).

$\square$

Proof of Corollary 2

(i) From Theorem 1, we have ${{\partial \Pi _{S1}^{*CC}} /{\partial {\beta ^2}}} = {{A\gamma } / {[2{{(4 - {\gamma ^2} - {\beta ^2})}^2}]}}$. Because $\beta ,\gamma \in [0,1]$, we have ${{\partial \Pi _{S1}^{*CC}} / {\partial {\beta ^2}}} \ge 0$.

Similarly, we can derive

$$\begin{aligned}&{{\partial \Pi _{S2}^{*CC}} / {\partial {\beta ^2}}}\\&\quad = {{A(4 + {\gamma ^2} - {\beta ^2})} / {[4{{(4 - {\gamma ^2} - {\beta ^2})}^3}]}} \ge 0. \end{aligned}$$

(ii) From Theorem 1, we have

$$\begin{aligned}&{{\partial \Pi _{S1}^{*DD}} / {\partial {\beta ^2}}}\\&\quad =\displaystyle \frac{{2(2 - {\gamma ^2})(\gamma - 2){A^2}\varphi }}{{(4 - {\gamma ^2}) {{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]}^2}}}\\&\qquad - \displaystyle \frac{{2(2 - {\gamma ^2})(3{\gamma ^2} - 8){A^2}{\varphi ^2}}}{{(4 - {\gamma ^2}){{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]}^2}}}. \end{aligned}$$

It is easy to find ${{2(2 - {\gamma ^2}){A^2}\varphi } /{\{ (4 - {\gamma ^2})}}{[2ST - (8 - 3{\gamma ^2})}{{\beta ^2}]^2}\} > 0$. Let ${f_2}(\beta ,\gamma ) = 2(2{\gamma ^2} - 4 - \gamma ) + (2 - \gamma ){\beta ^2}$.

Then, we find that ${{\partial \Pi _{S1}^{*DD}} / {\partial {\beta ^2}}} > 0$ is equivalent to the condition of ${f_2}(\beta ,\gamma ) < 0$. We can regard ${f_2}(\beta ,\gamma )$ as a quadratic function of $\beta$.

• (i) If $\beta =0$, ${f_2}(0,\gamma ) = 2(2{\gamma ^2} - 4 - \gamma ) < 0$;

• (ii) If $\beta =1$, ${f_2}(1,\gamma ) = 2(2{\gamma ^2} - 3 - \gamma ) - \gamma < 0$.

Therefore we have ${f_2}(\beta ,\gamma ) < 0,\ \ \forall \beta ,\gamma \in [0,1]$, namely ${{\partial \Pi _{S1}^{*DD}} / {\partial {\beta ^2}}} > 0$.

From Theorem 1, we have

$$\begin{aligned}&{{\partial \Pi _{S2}^{*DD}} / {\partial {\beta ^2}}}\\&\quad =\displaystyle \frac{{2({\gamma ^2} - 4){A^2}{T^2}}}{{(4 - {\gamma ^2}){{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]}^2}}}\\&\qquad - \displaystyle \frac{{2(3{\gamma ^2} - 8)[4(2 - {\gamma ^2}) - (4 - {\gamma ^2}){\beta ^2}] {A^2}{\varphi ^2}}}{{(4 - {\gamma ^2}){{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]}^2}}}. \end{aligned}$$

It is easy to find ${{2{A^2}} / {\{ (4 - {\gamma ^2})}}{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]^2}\} > 0$. Let ${Z_3}({\beta _3}\mid \gamma ) = 2({\gamma ^2} - 4)ST + 8(8 - 3{\gamma ^2})(2 - {\gamma ^2}) - (4 - {\gamma ^2})(8 - 3{\gamma ^2})\beta _3^2$, and ${{\partial \Pi _{S2}^{*DD}} / {\partial {\beta ^2}}} > 0$ is equivalent to the condition of ${Z_3}({\beta _3}\mid \gamma ) > 0$. Based on the properties of quadratic functions, we have

• (i) If $\beta _{3}=0$, ${Z_3}(0\mid \gamma ) = 2{\gamma ^2}(4{\gamma ^4} - 21{\gamma ^2} + 28) \ge 0$;

• (ii) If $\beta _{3}=1$, ${Z_3}(1\mid \gamma ) = 8{\gamma ^6} - 45{\gamma ^4} + 76{\gamma ^2} - 32$.

Next, we discuss ${Z_3}(1\mid \gamma )$. We have ${{\partial {Z_3}(1\mid \gamma )} / \partial }({\gamma ^2}) = 24{({\gamma ^2})^2} - 90{\gamma ^2} + 76 > 0$. Given ${Z_3}(1\mid 0) = - 32 < 0$ and ${Z_3}(1\mid 1) = 7 > 0$, $\exists {\gamma _5} \in [0,1],\ \ s.t.{Z_3}(1\mid \gamma ) = 0$.

Finally, we have $\left\{ \begin{array}{l} {Z_3}(1\mid \gamma ) \le 0\ \ \ \ 0 \le \gamma \le {\gamma _5} \\ {Z_3}(1\mid \gamma ) > 0\ \ \ \ {\gamma _5} < \gamma \le 1 \\ \end{array} \right.$ and $\left\{ \begin{array}{l} {Z_3}({\beta _3}\mid \gamma ) \ge 0\ \ \ \ if\ 0 \le \gamma \le {\gamma _5}, 0 \le {\beta ^2} \le \beta _3^2 \\ {Z_3}({\beta _3}\mid \gamma ) \le 0\ \ \ \ if\ 0 \le \gamma \le {\gamma _5}, \beta _3^2< {\beta ^2} \le 1 \\ {Z_3}({\beta _3}\mid \gamma ) \ge 0\ \ \ \ if\ {\gamma _5} < \gamma \le 1, \forall {\beta ^2} \in [0,1] \\ \end{array} \right.$, where $\gamma _{5}$ ($\approx 0.7933$) is determined by ${Z_3}(1\mid 0) = 0$ and $\beta _{3}$ is determined by ${Z_3}({\beta _3}\mid \gamma ) = 0$. $\square$

Proof of Theorem 2

Because the proof method of Theorem 2 is similar to that of Theorem 1, we omit the details here. $\square$

Proof of Corollary 3

Similar to the proof method of Corollary 3. $\square$

Proof of Proposition 2

Similar to the proof method of Proposition 2. $\square$

Proof of Corollary 4

Similar to the proof method of Corollary 4. $\square$

Proof of Proposition 3

Similar to the proof method of Proposition 3. $\square$

Proof of Proposition 4

(i). From Theorems 1 and 2, we have

$$\begin{aligned} {({\tilde{e}}_2^{*CC})^{1/2}} - {(e_2^{*CC}\mathrm{{)}}^{1/2}}= & {} {{A\beta } / {[(2 + \gamma )(4 - {\beta ^2} - 2\gamma )]}} \\&- {{A\beta } /{[2(4 - {\beta ^2} - {\gamma ^2})]}}. \end{aligned}$$

We find that the inequality of ${({\tilde{e}}_2^{*CC})^{1/2}} \ge {(e_2^{*CC}\mathrm{{)}}^{1/2}}$ is satisfied if and only if $2(4 - {\beta ^2} - {\gamma ^2}) - (2 + \gamma )(4 - {\beta ^2} - 2\gamma ) \ge 0$ is satisfied. Assuming ${f_6}(\beta ,\gamma ) = 2(4 - {\beta ^2} - {\gamma ^2}) - (2 + \gamma )(4 - {\beta ^2} - 2\gamma )$, we find that ${f_6}(\beta ,\gamma ) \ge 0$ due to $\gamma ,\beta \in [0,1]$, we have ${\tilde{e}}_2^{*CC} \ge e_2^{*CC}$.

Similarly, we can show other results. In the following, we give the comparisons on profits.

(ii) From Theorems 1 and 2, we have

$$\begin{aligned}&\tilde{\Pi }_{S2}^{*CC} - \Pi _{S2}^{*CC}\\&\quad = {{(4 - {\beta ^2}){A^2}} / {[{{(2 + \gamma )}^2}{{(4 - {\beta ^2} - 2\gamma )}^2}]}}\\&\qquad - {{{A^2}(4 - {\beta ^2})} / {[4{{({\gamma ^2} + {\beta ^2} - 4)}^2}]}}. \end{aligned}$$

It is easy to find that the inequality of $\tilde{\Pi }_{S2}^{*CC} \ge \Pi _{S2}^{*CC}$ is equivalent to $16 - 4{\gamma ^2} - 4{\beta ^2} - \gamma {\beta ^2} \ge 0$, which is always satisfied. Therefore, we obtain $\tilde{\Pi }_{S2}^{*CC} \ge \Pi _{S2}^{*CC}$. Meanwhile, $\tilde{\Pi }_{S2}^{*CC} = \Pi _{S2}^{*CC}$ if and only if $\beta$ or $\gamma$ is 0.

From Theorems 1 and 2, we find that the inequality of $\tilde{\Pi }_{S2}^{*CD} \ge \Pi _{S2}^{*CD}$ is equivalent to the condition of $8 - 4{\gamma ^2} + 2\gamma + (\gamma - 2){\beta ^2} \ge 0$. Let ${f_7}(\beta ,\gamma ) = 8 - 4{\gamma ^2} + 2\gamma + (\gamma - 2){\beta ^2}$. Obviously, the function ${f_7}(\beta ,\gamma )$ is a quadratic function of $\beta$.

Because ${f_7}(0,\gamma ) > 0$ and ${f_7}(1,\gamma ) > 0$, we can conclude that ${f_7}(\beta ,\gamma ) > 0$, $\forall \gamma ,\beta \in [0,1]$. Therefore, we have $\tilde{\Pi }_{S2}^{*CD} \ge \Pi _{S2}^{*CD}$ and $\tilde{\Pi }_{S2}^{*CD} = \Pi _{S2}^{*CD}$ if and only if $\beta$ or $\gamma$ is 0. The proof of $\tilde{\Pi }_{S2}^{*DC} \ge \Pi _{S2}^{*DC}$ is similar to that of $\tilde{\Pi }_{S2}^{*CD} \ge \Pi _{S2}^{*CD}$, and we omit the details here.

From Theorems 1 and 2, we find that the inequality of $\tilde{\Pi }_{S2}^{*DD} \ge \Pi _{S2}^{*DD}$ is satisfied if and only if the condition of $2(2{\gamma ^2} - 4 + \gamma )(2{\gamma ^2} - 4 - \gamma )(3 - {\gamma ^2}) + (\gamma - 3)( - {\gamma ^3} - 3{\gamma ^2} + 3\gamma + 8){\beta ^2} \ge 0$ is satisfied. We assume

$$\begin{aligned}&{Z_8}({\beta _8}\mid \gamma ) = 2(2{\gamma ^2} - 4 + \gamma )(2{\gamma ^2} - 4 - \gamma )(3 - {\gamma ^2}) \\&+ (\gamma - 3)( -{\gamma ^3} - 3{\gamma ^2} + 3\gamma + 8){\beta _8}^2. \end{aligned}$$

• (i) When $\beta =0$, we have ${Z_8}(0\mid \gamma ) > 0$;

• (ii) When $\beta =1$, we have ${Z_8}(1\mid \gamma ) = - 8{\gamma ^6} + 57{\gamma ^4} - 122{\gamma ^2} - \gamma + 72$.

Then, we use MATLAB to calculate the solution of equation $- 8{\gamma ^6} + 57{\gamma ^4} - 122{\gamma ^2} - \gamma + 72 = 0$ in $\gamma \in [0,1]$. The result is given by $\left\{ \begin{array}{l} {Z_8}(1\mid \gamma ) \ge 0\ \ \ 0 \le \gamma \le {\gamma _{12}} \\ {Z_8}(1\mid \gamma )< 0\ \ \ {\gamma _{12}} < \gamma \le 1 \\ \end{array} \right.$, where ${\gamma _{12}} \approx 0.9705$. Finally, we obtain $\left\{ \begin{array}{l} {Z_8}({\beta _8}\mid \gamma ) \ge 0\ \ 0 \le \gamma \le {\gamma _{12}}, \forall \beta \in \mathrm{{[0,1]}} \\ {Z_8}({\beta _8}\mid \gamma ) \ge 0\ \ {\gamma _{12}}< \gamma \le 1, 0 \le \beta \le {\beta _8} \\ {Z_8}({\beta _8}\mid \gamma )< 0\ \ {\gamma _{12}}< \gamma \le 1, {\beta _8} < \beta \le 1 \\ \end{array} \right.$, where the threshold $\beta _{8}$ is the unique solution determined by the equation as ${Z_8}({\beta _8}\mid \gamma ) = 0$. $\square$

Proof of Proposition 5

From Propositions 1 and 2, we can obtain the main results of Proposition 5. $\square$

Proof of Proposition 6

Here, comparisons of decision variables are obvious, and we omit them here. We only give the comparisons of profits.

From Theorem 1, we find that $\Pi _{S2}^{*CC} \ge \Pi _{S2}^{*DC}$ is satisfied if and only if the condition of $- 2(2 + \gamma ) + {\beta ^2} \ge 0$ is satisfied. Because $\beta ,\gamma \in [0,1]$, it is easy to prove $- 2(2 + \gamma ) + {\beta ^2} < 0$ . Therefore, we have $\Pi _{S2}^{*CC} \le \Pi _{S2}^{*DC}$ . Similarly, we can prove that $\Pi _{S2}^{*DD} \ge \Pi _{S2}^{*CD}$, $\tilde{\Pi }_{S2}^{*CC} \le \tilde{\Pi }_{S2}^{*DC}$ and $\tilde{\Pi }_{S2}^{*DD} \ge \tilde{\Pi }_{S2}^{*CD}$. $\square$

Proof of Proposition 7

(i) Here, the comparisons of decision variables will not be described in detail. The following discussions are mainly about comparisons of profits.

From Theorems 1 and 2, we find that $\tilde{\Pi }_{S1}^{*DC} \ge \Pi _{S1}^{*CC}$ is satisfied if and only if the following condition is satisfied

$$\begin{aligned}&{Z_9}({\beta _9}\mid \gamma ) = 2(4 - {\gamma ^2})[4(2 - {\gamma ^2}) - (4 - {\gamma ^2})\beta _9^2]\\&\quad {[}(4 - {\gamma ^2}) - \beta _9^2] - [8(2 - {\gamma ^2})(4 - {\gamma ^2}) - 2(8 - 3{\gamma ^2})\beta _9^2\\&\qquad - (4 - \beta _9^2)(4 - {\gamma ^2})\beta _9^2]^2 \ge 0. \end{aligned}$$

Because ${{{\partial ^2}{Z_9}({\beta _9}\mid \gamma )} / {{{(\beta _9^2)}^2}}} < 0$ and ${{\partial {Z_9}({\beta _9}\mid \gamma )} / {(\beta _9^2)}} > 0$, we find that ${{\partial {Z_9}({\beta _9}\mid \gamma )} / {(\beta _9^2)}}$ is monotonically decrease of $\beta _9^2$ and ${Z_9}({\beta _9}\mid \gamma )$ is monotonically increasing of $\beta _9^2$. In addition, since ${Z_9}(0\mid \gamma ) < 0$ and ${Z_9}(1\mid \gamma ) > 0$, we have $\left\{ \begin{array}{ll} {\tilde{\Pi }_{S1}^{*DC} \le \Pi _{S1}^{*CC}} &{} {0 \le \beta \le {\beta _9},\ \forall \gamma \in [0,1]} \\ {\tilde{\Pi }_{S1}^{*DC} > \Pi _{S1}^{*CC}} &{} {{\beta _9} < \beta \le 1,\ \forall \gamma \in [0,1]} \\ \end{array} \right.$, where $\beta _{9}$ is the solution of equation as ${Z_9}({\beta _9}\mid \gamma ) = 0$.

Based on Proposition 3 (i) and Proposition 5 (ii) (a), we can finally obtain Proposition 7 (i) (a).

In addition, we have $\tilde{\Pi }_{S1}^{*CC} \ge \Pi _{S1}^{*CC}$ and $\Pi _{S1}^{*DC} < \Pi _{S1}^{*CC}$ according to Propositions 3 (i) and Proposition 5 (i) (a). Therefore, we have $\Pi _{S1}^{*DC} \le \tilde{\Pi }_{S1}^{*CC}$.

(ii) Similar to the proofs of part (i), we can obtain the results of part (ii). $\square$

Proof of Proposition 8

Similar to the proof method of Proposition 7. $\square$