Shared manufacturing in a differentiated duopoly with capacity constraints

Abstract

This study proposes a differentiated duopoly model considering capacity constraints and shared manufacturing, investigates the effects of product differentiation and capacity constraints in three scenarios, and compares the equilibrium outcomes in the three cases under Cournot and Stackelberg competitions. We find that capacity constraints affect the relationships between product differentiation and equilibrium results, even the market share of enterprises. Shared manufacturing impacts the degree of excess capacity, profits, consumer surplus, and social welfare. However, shared manufacturing may sometimes play a negative role in alleviating excess capacity. Moreover, the Cournot competition is a better choice for enterprises with capacity constraints than the Stackelberg competition.

Appendices

Appendix A: Proof of Proposition 1

The effects of $r$:

$$ \begin{aligned} & \frac{{\partial x_{i}^{{{\text{CSS}}}} }}{\partial r} = \frac{{16\left( {a - b} \right)\left( {3r^{2} + 8r - 16} \right)}}{{\left( {32 + 16r - 4r^{2} - r^{3} } \right)^{2} }} < 0;\;\frac{{\partial q_{i}^{{{\text{CSS}}}} }}{\partial r} = \frac{{\left( {64r + 32r^{2} - 256 - r^{4} } \right)\left( {a - b} \right)}}{{\left( {32 + 16r - 4r^{2} - r^{3} } \right)^{2} }} < 0; \\ & \frac{{\partial \left( {x_{i}^{{{\text{CSS}}}} - q_{i}^{{{\text{CSS}}}} } \right)}}{\partial r} = \frac{{r\left( {r^{3} + 16r + 64} \right)\left( {a - b} \right)}}{{\left( {32 + 16r - 4r^{2} - r^{3} } \right)^{2} }} > 0;\;\frac{{\partial \pi_{i}^{{{\text{CSS}}}} }}{\partial r} = \frac{{4\left[ {\left( {r^{6} - 48r^{4} - 64r^{3} + 384r^{2} - 2048} \right)\left( {a - b} \right)^{2} } \right]}}{{\left( {32 + 16r - 4r^{2} - r^{3} } \right)^{3} }} < 0. \\ & \frac{{\partial {\text{CS}}^{{{\text{CSS}}}} }}{\partial r} = \frac{{\left( {r^{6} - 2r^{5} - 48r^{4} - 64r^{3} + 640r^{2} + 1536r - 2048} \right)r\left( {a - b} \right)^{2} }}{{\left( {32 + 16r - 4r^{2} - r^{3} } \right)^{3} }},\;{\text{if}}\;0 < r \le 0.99, \\ & {\text{then}}\;\frac{{\partial {\text{CS}}^{{{\text{CSS}}}} }}{\partial r} < 0;\;{\text{if}}\;0.99 < r < 1,\;{\text{then}}\;\frac{{\partial {\text{CS}}^{{{\text{CSS}}}} }}{\partial r} > 0. \\ & \frac{{\partial {\text{SW}}^{{{\text{CSS}}}} }}{\partial r} = \frac{{\left( {r^{7} + 6r^{6} - 48r^{5} - 448r^{4} + 128r^{3} + 4608r^{2} - 2048r - 16384} \right)\left( {a - b} \right)^{2} }}{{\left( {32 + 16r - 4r^{2} - r^{3} } \right)^{3} }} < 0. \\ \end{aligned} $$

Appendix B: Proof of Proposition 2

• (1) The effects of $r$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{CSI}}}} }}{\partial r} = \frac{{16\left[ {\left( {3r^{4} - 16r^{3} - 32r^{2} + 256r - 128} \right)\left( {a - b} \right) + 2\left( {3r^{4} - 32r^{2} - 128} \right)k} \right]}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0,\;\frac{{\partial x_{2}^{{{\text{CSI}}}} }}{\partial r} = 0; \\ & \frac{{\partial q_{1}^{{{\text{CSI}}}} }}{\partial r} = - \frac{{\left( {r^{6} - 8r^{5} - 16r^{4} + 256r^{3} + 128r^{2} - 3072r + 2048} \right)\left( {a - b} \right) + 2\left( {r^{6} - 16r^{4} + 128r^{2} + 2048} \right)k}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0; \\ & \frac{{\partial q_{2}^{{{\text{CSI}}}} }}{\partial r} = - \frac{{\left( {r^{6} - 8r^{5} - 16r^{4} + 128r^{3} + 128r^{2} - 1024r + 2048} \right)\left( {a - b} \right) - 16\left( {r^{5} - 16r^{3} + 128r} \right)k}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{1}^{{{\text{CSI}}}} }}{\partial r} = - \frac{{4\left[ {\left( {r^{5} - 12r^{4} + 32r^{3} + 128r^{2} - 640r + 512} \right)\left( {a - b} \right)^{2} + 4\left( {r^{5} - 6r^{4} + 64r^{2} - 128r + 256} \right)\left( {a - b} \right)k + 4\left( {r^{5} - 128r} \right)k^{2} } \right]}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{CSI}}}} }}{\partial r} = \frac{{4\left[ {\begin{array}{*{20}c} {\left( {r^{9} - 12r^{8} + 352r^{6} - 384r^{5} - 4096r^{4} + 8192r^{3} + 12288r^{2} - 65536r + 65536} \right)\left( {a - b} \right)^{2} } \\ { - 8\left( {3r^{8} - 16r^{7} - 88r^{6} + 384r^{5} + 1024r^{4} - 4096r^{3} - 3072r^{2} + 16384r - 16384} \right)\left( {a - b} \right)kx} \\ { + 128\left( {r^{7} - 24r^{5} + 256r^{3} - 1024r} \right)k^{2} } \\ \end{array} } \right]}}{{ - \left( {r^{4} - 32r^{2} + 128} \right)^{3} }} < 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{CSI}}}} - q_{1}^{{{\text{CSI}}}} } \right)}}{\partial r} = \frac{{r\left[ {\left( {r^{5} - 8r^{4} + 32r^{3} - 384r + 1024} \right)\left( {a - b} \right) + 2rk\left( {r^{4} + 32r^{2} - 384} \right)} \right]}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} > 0; \\ & \frac{{\partial \left( {x_{2}^{{{\text{CSI}}}} - q_{2}^{{{\text{CSI}}}} } \right)}}{\partial r} = \frac{{\left( {r^{6} - 8r^{5} - 16r^{4} + 128r^{3} + 128r^{2} - 1024r + 2048} \right)\left( {a - b} \right) - 16\left( {r^{5} - 16r^{3} + 128r} \right)k}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{CSI}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {\left( {r^{10} - 14r^{9} + 24r^{8} + 384r^{7} - 1184r^{6} - 4608r^{5} + 15872r^{4} + 20480r^{3} - 86016r^{2} + 98304r - 65536} \right)\left( {a - b} \right)^{2} } \hfill \\ {\quad + 2\left( {r^{10} - 14r^{9} + 24r^{8} + 256r^{7} - 1184r^{6} - 768r^{5} + 15872r^{4} - 16384r^{3} - 86016r^{2} + 131072r - 65536} \right)\left( {a - b} \right)k} \hfill \\ {\quad - 4\left( {7r^{9} - 128r^{7} + 384r^{5} + 8192r^{3} - 65536r} \right)k^{2} } \hfill \\ \end{array} }}{{ - \left( {r^{4} - 32r^{2} + 128} \right)^{3} }} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{CSI}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {\left( {r^{10} - 6r^{9} - 72r^{8} + 384r^{7} - 2272r^{6} - 12288r^{5} - 20992r^{4} + 151552r^{3} - 36864r^{2} - 491520r + 458752} \right)\left( {a - b} \right)^{2} } \hfill \\ {\quad + 2\left( {r^{10} - 6r^{9} - 72r^{8} + 25r^{7} - 2272r^{6} - 6912r^{5} - 20992r^{4} + 81920r^{3} - 36864r^{2} - 262144r + 458752} \right)\left( {a - b} \right)k} \hfill \\ {\quad - 4\left( {3r^{9} - 128r^{7} + 3456r^{5} - 40960r^{3} + 131072r} \right)k^{2} } \hfill \\ \end{array} }}{{ - \left( {r^{4} - 32r^{2} + 128} \right)^{3} }} < 0. \\ \end{aligned} $$

• (2) The effects of $k$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{CSI}}}} }}{\partial k} = \frac{ - 32r}{{r^{4} - 32r^{2} + 128}} < 0,\;\frac{{\partial x_{2}^{C2} }}{\partial k} = 1 > 0;\;\frac{{\partial q_{1}^{{{\text{CSI}}}} }}{\partial k} = \frac{{2r\left( {r^{2} - 16} \right)}}{{r^{4} - 32r^{2} + 128}} < 0, \\ & \frac{{\partial q_{2}^{C2} }}{\partial k} = \frac{{ - 8\left( {r^{2} - 8} \right)}}{{r^{4} - 32r^{2} + 128}} > 0;\;\frac{{\partial \pi_{1}^{{{\text{CSI}}}} }}{\partial k} = \frac{{8r\left[ {\left( {a - b)(r - 4} \right) + 2kr} \right]}}{{r^{4} - 32r^{2} + 128}} < 0, \\ & \frac{{\partial \pi_{2}^{C2} }}{\partial k} = \frac{{ - 2\left[ {\begin{array}{*{20}c} {16\left( {r^{5} - 4r^{4} - 24r^{3} + 64r^{2} + 128r - 256} \right)\left( {a - b} \right) + } \\ {\left( {r^{8} - 64r^{6} + 1152r^{4} - 6144r^{2} + 8192} \right)k} \\ \end{array} } \right]}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} > 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{CSI}}}} - q_{1}^{{{\text{CSI}}}} } \right)}}{\partial k} = - \frac{{2r^{3} }}{{r^{4} - 32r^{2} + 128}} < 0, \frac{{\partial \left( {x_{2}^{C2} - q_{2}^{C2} } \right)}}{\partial k} = \frac{{r^{4} - 24r^{2} + 64}}{{r^{4} - 32r^{2} + 128}} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{CSI}}}} }}{\partial k} = \frac{{2\left[ {\left( {r^{7} - 7r^{6} - 24r^{5} + 176r^{4} + 96r^{3} - 1024r^{2} + 512r + 1024} \right)\left( {a - b} \right) - 2\left( {7r^{6} - 176r^{4} + 1024r^{2} - 1024} \right)k} \right]}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{CSI}}}} }}{\partial k} = \frac{{2\left[ {\begin{array}{*{20}c} {\left( {r^{7} - 3r^{6} - 56r^{5} + 112r^{4} + 992r^{3} - 1536r^{2} - 3584r + 5120} \right)\left( {a - b} \right)} \\ { - \left( {r^{8} - 58r^{6} + 1056r^{4} - 5120r^{2} + 6144} \right)k} \\ \end{array} } \right]}}{{\left( {r^{4} - 32r^{2} + 128} \right)^{2} }} > 0. \\ \end{aligned} $$

Appendix C: Proof of Proposition 3

$$ \begin{aligned} & \pi_{2}^{{{\text{CSP}}}} - \pi_{2}^{{{\text{CSI}}}} \\ & \quad = \frac{{\begin{array}{*{20}l} {\left\{ {1024\left( {r^{8} - 8r^{7} - 24r^{6} + 256r^{5} + 128r^{4} - 2560r^{3} + 1024r^{2} + 8192r - 8192} \right)A^{2} } \right.} \hfill \\ {\quad + 64\left( {r^{11} - 4r^{10} - 80r^{9} + 288r^{8} + 2304r^{7} - 7168r^{6} - 28673r^{5} + 73728r^{4} + 147456r^{3} - 327680r^{2} - 262144r + 524288} \right)AB} \hfill \\ {\quad + 128\left( {r^{11} - 4r^{10} - 80r^{9} + 288r^{8} + 2176r^{7} - 6656r^{6} - 24576r^{5} + 61440r^{4} + 106496r^{3} - 229376r^{2} - 131072r + 262144} \right)Ak} \hfill \\ {\quad \left. { + \left( {r^{14} - 120r^{12} + 5632r^{10} - 131072r^{8} + 1589248r^{6} - 9830400r^{4} + 29360128r^{2} - 33554432} \right)(B + 2k)^{2} } \right\}} \hfill \\ \end{array} }}{{512\left( {128 - 32r^{2} + r^{4} } \right)^{2} \left( {8 - r^{2} } \right)}}. \\ \end{aligned} $$

From $\pi_{2}^{{{\text{CSP}}}} - \pi_{2}^{{{\text{CSI}}}} = 0$, we can obtain:

$$ \begin{aligned} & k_{1}^{{{\text{CSP}}}} = \frac{{\left( { - 32r^{3} + 128r^{2} + 512{\text{r}} - 1024} \right)A + \left( { - r^{6} + 48r^{4} - 640r^{2} + 2048} \right)B}}{{2\left( {r^{6} - 48r^{4} + 512r^{2} - 1024} \right)}}; \\ & k_{2}^{{{\text{CSP}}}} = \frac{{\left( { - 32r^{5} + 128r^{4} + 768r^{3} - 2048r^{2} - 4096r + 8192} \right)A + \left( { - r^{8} + 72r^{6} - 1536r^{4} + 9216r^{2} - 16384} \right)B}}{{2\left( {r^{8} - 72r^{6} + 1408r^{4} - 7168r^{2} + 8192} \right)}}. \\ \end{aligned} $$

From $0 < X^{{{\text{CSP}}}} < \frac{1}{3}e - Q$, we derive $k_{1}^{{{\text{CSP}}}} < k < k_{2}^{{{\text{CSP}}}}$, where $k_{3}^{{{\text{CSP}}}} = - \frac{{96\left( {r^{3} - 4r^{2} - 16r + 32} \right)A + (3r^{6} - 144r^{4} + 1664r^{2} - 4096)B + 512\left( {r^{2} - 8} \right)Q}}{{6\left( {r^{6} - 48r^{4} + 512r^{2} - 1024} \right)}}$. Thus, $k_{1}^{{{\text{CSP}}}} < k < k_{3}^{{{\text{CSP}}}}$.

• (1) The effects of $r$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{CSP}}}} }}{\partial r} = \frac{{64Ar - \left( {B + 2k} \right)\left( {r^{4} - 8r^{2} + 128} \right)}}{{8\left( {r^{2} - 8} \right)^{2} }} > 0; \\ & \frac{{\partial x_{2}^{{{\text{CSP}}}} }}{\partial r} = \frac{{\left( {1024 - 64r^{2} + 8r^{4} } \right)A + \left( {B + 2k} \right)\left( {r^{7} - 36r^{5} + 384r^{3} - 1536r} \right)}}{{64\left( {r^{2} - 8} \right)^{2} }} > 0; \\ & \frac{{\partial q_{1}^{{{\text{CSP}}}} }}{\partial r} = \frac{{512Ar + \left( {B + 2k} \right)(3r^{6} - 72r^{4} + 512r^{2} - 2048)}}{{128\left( {r^{2} - 8} \right)^{2} }} < 0,\;\frac{{\partial q_{2}^{CSP} }}{\partial r} = - \frac{{\left( {B + 2k} \right)r}}{16} < 0; \\ & \frac{{\partial \pi_{1}^{{{\text{CSP}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {163843A^{2} r^{3} + 32\left( {B + 2k} \right)\left( {3r^{8} - 104r^{6} + 1280r^{4} - 9216r^{2} + 16384} \right)} \hfill \\ {\quad + \left( {B + 2k} \right)^{2} \left( {3r^{11} - 168r^{9} + 3456r^{7} - 33792r^{5} + 163840r^{3} - 262144r} \right)} \hfill \\ \end{array} }}{{4096\left( {r^{2} - 8} \right)^{3} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{CSP}}}} }}{\partial r} = - \frac{{16A\left( {B + 2k} \right)\left( {r^{4} - 128r^{2} + 2048} \right) + \left( {B + 2k} \right)^{2} \left( {r^{7} - 40r^{5} + 448r^{3} - 2048} \right)}}{{128\left( {r^{2} - 8} \right)^{2} }} < 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{CSP}}}} - q_{1}^{{{\text{CSP}}}} } \right)}}{\partial r} = \frac{{r\left[ {512A - \left( {B + 2k} \right)\left( {3r^{5} - 56r^{3} + 384r} \right)} \right]}}{{128\left( {r^{2} - 8} \right)^{2} }} > 0; \\ & \frac{{\partial \left( {x_{2}^{{{\text{CSP}}}} - q_{2}^{{{\text{CSP}}}} } \right)}}{\partial r} = \frac{{\left( {1024 - 64r^{2} + 8r^{4} } \right)A - \left( {B + 2k} \right)\left( {r^{7} - 32r^{5} + 320r^{3} - 1280r} \right)}}{{64\left( {r^{2} - 8} \right)^{2} }} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{CSP}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {16384A^{2} r\left( {r^{2} - 16} \right) - 32\left( {9r^{8} - 280r^{6} + 3200r^{4} - 18432r^{2} + 32768} \right)A\left( {B + 2k} \right)} \hfill \\ {\quad - (21r^{10} - 1080r^{8} + 21248r^{6} - 202752r^{4} + 96656r^{2} - 1835008)\left( {B + 2k} \right)^{2} r} \hfill \\ \end{array} }}{{16384\left( {r^{2} - 8} \right)^{3} }} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{CSP}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {16384\left( {5r^{2} - 16} \right)A^{2} r + 32\left( {3r^{8} - 200r^{6} + 2944r^{4} - 30720r^{2} + 98304} \right)A\left( {B + 2k} \right)} \hfill \\ {\quad - (9r^{10} - 280r^{8} + 1280r^{6} + 30720r^{4} - 409600r^{2} + 1310720)\left( {B + 2k} \right)^{2} r} \hfill \\ \end{array} }}{{16384\left( {r^{2} - 8} \right)^{3} }} < 0. \\ \end{aligned} $$

• (2) The effects of $k$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{CSP}}}} }}{\partial k} = - \frac{{r\left( {r^{2} - 16} \right)}}{{4\left( {r^{2} - 8} \right)}} < 0,\;\frac{{\partial x_{3}^{{{\text{CSP}}}} }}{\partial k} = \frac{{r^{6} - 48r^{4} + 640r^{2} - 2048}}{{128\left( {r^{2} - 8} \right)}} > 0; \\ & \frac{{\partial q_{1}^{{{\text{CSP}}}} }}{\partial k} = \frac{{r\left( {r^{4} - 32r^{2} + 256} \right)}}{{64\left( {r^{2} - 8} \right)}} < 0,\; \frac{{\partial q_{2}^{{{\text{CSP}}}} }}{\partial k} = - \frac{1}{16}r^{2} + 1 > 0; \\ & \frac{{\partial \pi_{1}^{{{\text{CSP}}}} }}{\partial k} = \frac{{r\left[ {32\left( {r^{6} - 48r^{4} + 640r^{2} - 2048} \right)A + \left( {B + 2k} \right)\left( {r^{9} - 64r^{7} + 1408r^{5} - 12288r^{3} + 32768r} \right)} \right]}}{{2048\left( {r^{2} - 8} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{CSP}}}} }}{\partial k} = \frac{{32A\left( {r^{3} - 4r^{2} - 16r + 32} \right) + \left( {B + 2k} \right)\left( {r^{6} - 56r^{4} + 640r^{2} - 1024} \right)}}{{128\left( {8 - r^{2} } \right)}} > 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{CSP}}}} - q_{1}^{{{\text{CSP}}}} } \right)}}{\partial k} = - \frac{{r^{3} \left( {r^{2} - 16} \right)}}{{64\left( {r^{2} - 8} \right)}} < 0,\;\frac{{\partial \left( {x_{2}^{{{\text{CSP}}}} - q_{2}^{{{\text{CSP}}}} } \right)}}{\partial k} = \frac{{r^{6} - 40r^{4} + 448r^{2} - 1024}}{{128\left( {r^{2} - 8} \right)}} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{CSP}}}} }}{\partial k} = \frac{{32Ar\left( {4096 - 1280r^{2} + 112r^{4} - 3r^{6} } \right) - \left( {B + 2k} \right)\left( {7r^{10} - 400r^{8} + 8448r^{6} - 78848r^{4} + 294912r^{2} - 262144} \right)}}{{8192\left( {r^{2} - 8} \right)^{2} }} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{CSP}}}} }}{\partial k} = \frac{{\begin{array}{*{20}l} {32A\left( {r^{7} - 144r^{5} + 256r^{4} + 2816r^{3} - 4096r^{2} - 12288r + 16384} \right)} \hfill \\ {\quad - \left( {B + 2k} \right)\left( {3r^{10} - 80r^{8} - 1280r^{6} + 39936r^{4} - 229376r^{2} + 262144} \right)} \hfill \\ \end{array} }}{{8192\left( {r^{2} - 8} \right)^{2} }} > 0. \\ \end{aligned} $$

Appendix D: Proof of Proposition 4

The effects of $r$:

$$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{SSS}}}} }}{\partial r} = - \frac{{4\left[ {\left( {7r^{6} - 42r^{5} - 96r^{4} + 448r^{3} + 192r^{2} - 1536r + 1024} \right)\left( {a - b} \right)} \right]}}{{\left( {7r^{4} - 72r^{2} + 128} \right)^{2} }},\;{\text{if}}\;0 < r \le 0.99, \\ & {\text{then}}\;\frac{{\partial x_{1}^{{{\text{SSS}}}} }}{\partial r} \le 0;\;{\text{if}}\;0.99 < r < 1,\;{\text{then}}\;\frac{{\partial x_{1}^{{{\text{SSS}}}} }}{\partial r} > 0; \\ & \frac{{\partial x_{2}^{{{\text{SSS}}}} }}{\partial r} = - \frac{{2\left[ {\left( {7r^{6} - 48r^{5} - 264r^{4} + 768r^{3} + 768r^{2} - 3072r + 2048} \right)\left( {a - b} \right)} \right]}}{{\left( {7r^{4} - 72r^{2} + 128} \right)^{2} }} < 0; \\ & \frac{{\partial q_{1}^{{{\text{SSS}}}} }}{\partial r} = - \frac{{4\left[ {\left( {7r^{6} - 42r^{5} - 96r^{4} + 448r^{3} + 192r^{2} - 1536r + 1024} \right)\left( {a - b} \right)} \right]}}{{(r^{2} - 8)^{2} \left( {7r^{2} - 16} \right)^{2} }}, \\ & {\text{if}}\;0 < r \le 0.99,\;{\text{then}}\;\frac{{\partial q_{1}^{S1} }}{\partial r} \le 0;\;{\text{if}}\;0.99 < r < 1,\;{\text{then}}\;\frac{{\partial q_{1}^{S1} }}{\partial r} > 0. \\ & \frac{{\partial q_{2}^{{{\text{SSS}}}} }}{\partial r} = - \frac{{4\left[ {(7r^{2} - 20r + 16} \right)\left( {a - b} \right)]}}{{\left( {7r^{2} - 16} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{1}^{{{\text{SSS}}}} }}{\partial r} = \frac{{8\left[ {\left( {21r^{10} - 67r^{9} - 444r^{8} + 1360r^{7} + 3648r^{6} - 12288r^{5} - 12544r^{4} + 55296r^{3} - 98304r + 65536} \right)\left( {a - b} \right)^{2} } \right]}}{{(r^{2} - 8)\left( {16 - 7r^{2} } \right)\left( {7r^{4} - 72r^{2} + 128} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{SSS}}}} }}{\partial r} = \frac{{ - 2\left[ {\left( {\begin{array}{*{20}c} {49r^{10} + 84r^{9} - 1680r^{8} - 160r^{7} + 18624r^{6} - 23040r^{5} - 76800r^{4} + } \\ {180224r^{3} + 49152r^{2} - 39316r + 262144} \\ \end{array} } \right)\left( {a - b} \right)^{2} } \right]}}{{(r^{2} - 8)^{3} \left( {7r^{2} - 16} \right)^{3} }} < 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{SSS}}}} - q_{1}^{{{\text{SSS}}}} } \right)}}{\partial r} = 0,\;\frac{{\partial \left( {x_{2}^{{{\text{SSS}}}} - q_{2}^{{{\text{SSS}}}} } \right)}}{\partial r} = \frac{{2r\left[ {\left( {7r^{5} + 8r^{4} + 72r^{3} - 128r^{2} - 384r + 512} \right)\left( {a - b} \right)} \right]}}{{(r^{2} - 8)^{2} \left( {7r^{2} - 16} \right)^{2} }} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{SSS}}}} }}{\partial r} = \frac{{ - 8\left[ {\left( {7r^{10} - 90r^{9} - 162r^{8} + 1788r^{7} + 608r^{6} - 11712r^{5} + 5760r^{4} + 25600r^{3} - 30720r^{2} + 8192r} \right)\left( {a - b} \right)^{2} } \right]}}{{(r^{2} - 8)^{3} \left( {7r^{2} - 16} \right)^{3} }}, \\ & {\text{if}}\;0 < r \le 0.408,\;{\text{then}}\; \frac{{\partial {\text{CS}}^{S1} }}{\partial r} \le 0;\;{\text{if}}\;0.408 < r < 1,\;{\text{then}}\; \frac{{\partial {\text{CS}}^{S1} }}{\partial r} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{SSS}}}} }}{\partial r} = \frac{{ - 2\left[ {\left( {\begin{array}{*{20}c} {161r^{10} - 544r^{9} - 4104r^{8} + 12432r^{7} + 35648r^{6} - 119040r^{5} - 103936r^{4} + 503808r^{3} } \\ { - 73728r^{2} - 753664r + 524288} \\ \end{array} } \right)\left( {a - b} \right)^{2} } \right]}}{{(r^{2} - 8)^{3} \left( {7r^{2} - 16} \right)^{3} }} < 0. \\ \end{aligned} $$

Appendix E: Proof of Proposition 5

• (1) The effects of $r$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{SSI}}}} }}{\partial r} = - \frac{{\left( {r^{2} - 8r + 4} \right)\left( {a - b} \right) + 2k(r^{2} + 4)}}{{2(r^{2} - 4)^{2} }}, \\ & {\text{if}}\;0 < k < \frac{{\left( {a - b} \right)\left( {4 + 8r - r^{2} } \right)}}{{2(r^{2} + 4)}}\;{\text{and}}\;0 < r < 4 - 2\sqrt 3 ,{\text{then}}\;\frac{{\partial x_{1}^{{{\text{SSI}}}} }}{\partial r} < 0; \\ & {\text{if}}\;0 < k < \frac{{\left( {a - b} \right)\left( {4 + 8r - r^{2} } \right)}}{{2(r^{2} + 4)}}\;{\text{and}}\;4 - 2\sqrt 3 < r < 1,{\text{then}}\;\frac{{\partial x_{1}^{{{\text{SSI}}}} }}{\partial r} > 0; \\ & {\text{if}}\;\frac{{\left( {a - b} \right)\left( {4 + 8r - r^{2} } \right)}}{{2(r^{2} + 4)}} < k < \frac{{\left( {a - b} \right)\left( {r^{4} + 4r^{3} - 24r^{2} - 64r + 128} \right)}}{{2\left( {7r^{4} - 72r^{2} + 128} \right)}}\;{\text{and}}\;4 - 2\sqrt 3 < r < 1, \\ & {\text{then}}\;\frac{{\partial x_{1}^{{{\text{SSI}}}} }}{\partial r} < 0. \\ \end{aligned} $$
  $$ \begin{aligned} & \frac{{\partial x_{2}^{{{\text{SSI}}}} }}{\partial r} = 0,\;\frac{{\partial q_{1}^{{{\text{SSI}}}} }}{\partial r} = - \frac{{\left( {r^{2} - 8r + 4} \right)\left( {a - b} \right) + 2k(r^{2} + 4)}}{{2(r^{2} - 4)^{2} }},\; \\ & {\text{if}}\;0 < k < \frac{{\left( {a - b} \right)\left( {r^{4} + 4r^{3} - 24r^{2} - 64r + 128} \right)}}{{2\left( {7r^{4} - 72r^{2} + 128} \right)}}\;{\text{and}}\;0 < r < 4 - 2\sqrt 3 , \\ & {\text{then}}\;\frac{{\partial q_{1}^{{{\text{SSI}}}} }}{\partial r} < 0;\;{\text{if}}\;0 < k < \frac{{\left( {a - b} \right)\left( {4 + 8r - r^{2} } \right)}}{{2(r^{2} + 4)}}\;{\text{and}}\;4 - 2\sqrt 3 < r < 1,{\text{then}}\;\frac{{\partial q_{1}^{{{\text{SSI}}}} }}{\partial r} > 0; \\ & {\text{if}}\;\frac{{\left( {a - b} \right)\left( {4 + 8r - r^{2} } \right)}}{{2(r^{2} + 4)}} < k < \frac{{\left( {a - b} \right)\left( {r^{4} + 4r^{3} - 24r^{2} - 64r + 128} \right)}}{{2\left( {7r^{4} - 72r^{2} + 128} \right)}}\;{\text{and}}\;4 - 2\sqrt 3 < r < 1, \\ & {\text{then}}\;\frac{{\partial q_{1}^{{{\text{SSI}}}} }}{\partial r} < 0. \\ \end{aligned} $$
  $$ \begin{aligned} & \frac{{\partial q_{2}^{{{\text{SSI}}}} }}{\partial r} = - \frac{{\left( {r^{2} - 2r + 4} \right)\left( {a - b} \right) - 4kr}}{{2(r^{2} - 4)^{2} }} < 0; \\ & \frac{{\partial \pi_{1}^{{{\text{SSI}}}} }}{\partial r} = - \frac{{\left[ {\left( {a - b} \right)^{2} + 2k\left( {a - b} \right)} \right]r^{2} - \left[ {5\left( {a - b} \right)^{2} + 4kr\left( {a - b + k} \right)} \right] + 4\left( {a - b} \right)^{2} + 8k\left( {a - b} \right)}}{{2(r^{2} - 4)^{2} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{SSI}}}} }}{\partial r} = - \frac{{\left( {a - b} \right)^{2} \left( {r^{4} + 2r^{3} - 12r^{2} + 32r - 32} \right) + 2k\left[ {\left( {r^{4} - 4r^{3} - 12r^{2} + 32r - 32} \right)\left( {a - b} \right)} \right] + 64k^{2} r}}{{4(r^{2} - 4)^{3} }} < 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{SSI}}}} - q_{1}^{{{\text{SSI}}}} } \right)}}{\partial r} = 0,\;\frac{{\partial \left( {x_{2}^{{{\text{SSI}}}} - q_{2}^{{{\text{SSI}}}} } \right)}}{\partial r} = \frac{{\left( {r^{2} - 2r + 4} \right)\left( {a - b} \right) - 4kr}}{{2(r^{2} - 4)^{2} }} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{SSI}}}} }}{\partial r} = \frac{{\left( {a - b} \right)^{2} \left( {r^{4} - 26r^{3} + 36r^{2} - 16r + 32} \right) + 2k\left[ {\left( {r^{4} + 4r^{3} + 36r^{2} - 64r + 32} \right)\left( {a - b} \right)} \right] + 8k^{2} r\left( {r^{2} - 16} \right)}}{{ - 16(r^{2} - 4)^{3} }} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{SSI}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {\left( {a - b} \right)^{2} \left( {13r^{4} - 58r^{3} - 12r^{2} + 272r - 224} \right)} \hfill \\ {\quad + 2k\left[ {\left( {13r^{4} - 28r^{3} - 12r^{2} + 128r - 224} \right)\left( {a - b} \right)} \right] - 8k^{2} r\left( {7r^{2} - 32} \right)} \hfill \\ \end{array} }}{{ - 16(r^{2} - 4)^{3} }} < 0. \\ \end{aligned} $$

• (2) The effects of $k$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{SSI}}}} }}{\partial k} = \frac{r}{{r^{2} - 4}} < 0,\;\frac{{\partial x_{2}^{{{\text{SSI}}}} }}{\partial k} = 1 > 0;\; \\ & \frac{{\partial q_{1}^{{{\text{SSI}}}} }}{\partial k} = \frac{r}{{r^{2} - 4}} < 0,\;\frac{{\partial q_{2}^{{{\text{SSI}}}} }}{\partial k} = \frac{{r^{2} - 8}}{{4(r^{2} - 4)}} > 0; \\ & \frac{{\partial \pi_{1}^{{{\text{SSI}}}} }}{\partial k} = \frac{{r\left[ {\left( {a - b} \right)\left( {r - 4} \right) + 2kr} \right]}}{{4\left( {4 - r^{2} } \right)}} < 0, \\ & \frac{{\partial \pi_{2}^{{{\text{SSI}}}} }}{\partial k} = - \frac{{\left( {a - b} \right)\left( {r^{4} + 4r^{3} - 16r^{2} - 32r + 64} \right) - 2k\left( {7r^{4} - 48r^{2} + 64} \right)}}{{8(r^{2} - 4)^{2} }} > 0; \\ \user2{ } & \frac{{\partial \left( {x_{1}^{{{\text{SSI}}}} - q_{1}^{{{\text{SSI}}}} } \right)}}{\partial k} = 0,\;\frac{{\partial \left( {x_{2}^{{{\text{SSI}}}} - q_{2}^{{{\text{SSI}}}} } \right)}}{\partial k} = \frac{{8 - 3r^{2} }}{{4\left( {4 - r^{2} } \right)}} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{SSI}}}} }}{\partial k} = \frac{{\left( {a - b} \right)\left( {9r^{4} + 4r^{3} - 64r^{2} + 32r + 64} \right) + 2k\left( {9r^{4} - 64r^{2} + 64} \right)}}{{32(r^{2} - 4)^{2} }} > 0, \\ & \frac{{\partial {\text{SW}}^{{{\text{SSI}}}} }}{\partial k} = \frac{{\left( {a - b} \right)\left( {5r^{4} + 52r^{3} - 96r^{2} - 224r + 320} \right) - 2k\left( {27r^{4} - 160r^{2} + 192} \right)}}{{32(r^{2} - 4)^{2} }} > 0. \\ \end{aligned} $$

Appendix F: Proof of Proposition 6

$$ \begin{aligned} & \pi_{2}^{{{\text{SSP}}}} - \pi_{2}^{{{\text{SSI}}}} \\ & \quad = - \frac{{\begin{array}{*{20}l} {\left( {r^{10} + 8r^{9} - 40r^{8} - 384r^{7} + 576r^{6} + 6656r^{5} - 4608r^{4} - 49152r^{3} + 32768r^{2} + 131072r + 131072} \right)A^{2} } \hfill \\ {\quad - 16\left( {r^{10} + 4r^{9} - 48r^{8} - 160r^{7} + 848r^{6} + 2112r^{5} - 6784r^{4} - 10240r^{3} + 24576r^{2} + 16384r - 32768} \right)AB} \hfill \\ {\quad - \left( {28r^{10} + 112r^{9} - 1312r^{8} - 4352r^{7} + 22272r^{6} + 54272r^{5} - 165888r^{4} - 229376r^{3} + 524288r^{2} + 262144r - 524288} \right)Ak} \hfill \\ {\quad + 64\left( {r^{10} - r^{8} - 64r^{6} + 1152r^{4} - 6144r^{2} + 8192} \right)Bk + 4\left( {49r^{10} - 1848r^{8} + 21440r^{6} - 100864r^{4} + 196608r^{2} - 131072} \right)k^{2} } \hfill \\ \end{array} }}{{32(r^{2} - 8)^{2} \left( {r^{2} - 16} \right)^{2} \left( {r^{2} - 4} \right)^{2} }}. \\ \end{aligned} $$

From $\pi_{2}^{{{\text{SSP}}}} - \pi_{2}^{{{\text{SSI}}}} = 0$, we can obtain:

$$ \begin{gathered} k_{1}^{SSP} = \frac{{\left( {r^{4} + 4r^{3} - 24r^{2} - 64r + 128} \right)A - 8\left( {r^{4} - 12r^{2} + 32} \right)B}}{{2\left( {7r^{4} - 72r^{2} + 128} \right)}}; \hfill \\ k_{2}^{SSP} = \frac{{\left( {r^{6} + 4r^{5} - 32r^{4} - 96r^{3} + 320r^{2} + 512r - 1024} \right)A - 8\left( {r^{6} - 28r^{4} + 160r^{2} - 256} \right)B}}{{2\left( {7r^{6} - 192r^{4} + 960r^{2} - 1024} \right)}}. \hfill \\ \end{gathered} $$

If $\pi_{2}^{{{\text{SSP}}}} - \pi_{2}^{{{\text{SSI}}}} > 0$, then $k_{1}^{SSP} < k < k_{2}^{SSP} .$

From $0 < X < \frac{1}{3}e - Q$, we can derive $k < x_{2}^{{{\text{SSP}}}}$ and $k < x_{2}^{{{\text{SSS}}}}$, thus $k_{1}^{{{\text{SSP}}}} < k < k_{3}^{{{\text{SSP}}}}$.

• (1) The effects of $r$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{SSP}}}} }}{\partial r} = \frac{{4\left[ {2\left( {r^{5} - 32r^{3} + 256r} \right)A - \left( {8r^{6} - 768r^{2} + 4096} \right)\left( {B + 2k} \right)} \right]}}{{\left( {r^{4} - 24r^{2} + 128} \right)^{2} }} < 0; \\ & \frac{{\partial x_{2}^{{{\text{SSP}}}} }}{\partial r} = \frac{{2\left[ {\left( {r^{6} - 24r^{4} + 2048} \right)A - 48\left( {r^{5} - 16r^{3} + 64} \right)\left( {B + 2k} \right)} \right]}}{{\left( {r^{4} - 24r^{2} + 128} \right)^{2} }} > 0; \\ & \frac{{\partial q_{1}^{{{\text{SSP}}}} }}{\partial r} = \frac{{4\left[ {2\left( {r^{5} - 32r^{3} + 256r} \right)A - \left( {r^{6} - 192r^{2} + 1024} \right)\left( {B + 2k} \right)} \right]}}{{(r^{2} - 8)^{2} \left( {r^{2} - 16} \right)^{2} }} < 0, \\ & \frac{{\partial q_{2}^{{{\text{SSP}}}} }}{\partial r} = - \frac{{16r\left( {B + 2k} \right)}}{{\left( {r^{2} - 16} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{1}^{{{\text{SSP}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} {8\left\{ {\left( {r^{9} - 48r^{7} + 788r^{5} - 4096r^{3} } \right)A^{2} - \left( {r^{10} - 12r^{8} - 448r^{6} - 8960r^{4} + 49152r^{2} - 65536} \right)A\left( {B + 2k} \right)} \right.} \hfill \\ {\quad + 4\left( {7r^{9} - 184r^{7} + 1728r^{5} - 6656r^{3} + 8768r} \right)(B + 2k)^{2} \} } \hfill \\ \end{array} }}{{(r^{2} - 8)^{3} \left( {r^{2} - 16} \right)^{3} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{SSP}}}} }}{\partial r} = \frac{{\begin{array}{*{20}l} { - 2\left( {r^{8} - 40r^{6} + 384r^{4} + 2048r^{2} - 32768} \right)AB - 4\left( {r^{8} - 40r^{6} + 384r^{4} + 2048r^{2} - 32768} \right)kA} \hfill \\ {\quad + 32\left( {r^{7} - 48r^{5} + 576r^{3} - 2048r} \right)B^{2} + 128\left( {r^{7} - 48r^{5} + 576r^{3} - 2048r} \right)k\left( {B + k} \right)} \hfill \\ \end{array} }}{{(r^{2} - 8)^{2} \left( {r^{2} - 16} \right)^{3} }} < 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{SSP}}}} - q_{1}^{{{\text{SSP}}}} } \right)}}{\partial r} = 0,\;\frac{{\partial \left( {x_{2}^{{{\text{SSP}}}} - q_{2}^{{{\text{SSP}}}} } \right)}}{\partial r} = \frac{{2\left( {r^{6} - 24r^{4} + 1024} \right)A - 80\left( {r^{5} - 16r^{3} + 64r} \right)\left( {B + 2k} \right)}}{{(r^{2} - 8)^{2} \left( {r^{2} - 16} \right)^{2} }} < 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{SSP}}}} }}{\partial r} = - \frac{{4\left\{ {\begin{array}{*{20}c} {8\left( {r^{7} - 48r^{5} + 768r^{3} - 4096r} \right)A^{2} - \left( {r^{10} - 12r^{8} - 448r^{6} - 8960r^{4} + 49152r^{2} - 65536} \right)A\left( {B + 2k} \right)} \\ { + 56\left( {r^{9} - 28r^{7} + 288r^{5} - 1280r^{3} + 2048r} \right)\left( {B + 2k} \right)^{2} } \\ \end{array} } \right\}}}{{(r^{2} - 8)^{3} \left( {r^{2} - 16} \right)^{3} }} > 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{SSP}}}} }}{\partial r} = \frac{{2\left\{ {\begin{array}{*{20}c} {4\left( {r^{9} - 52r^{7} + 960r^{5} - 7168r^{3} + 16384r} \right)A^{2} } \\ { - 3\left( {r^{10} - 24r^{8} - 64r^{6} + 5632r^{4} - 49152r^{2} + 131072} \right)A\left( {B + 2k} \right)} \\ { + 16\left( {r^{9} - 44r^{7} + 672r^{5} - 4352r^{3} + 10240r} \right)(B + 2k)^{2} } \\ \end{array} } \right\}}}{{(r^{2} - 8)^{3} \left( {r^{2} - 16} \right)^{3} }} < 0. \\ \end{aligned} $$

• (2) The effects of $k$:

  $$ \begin{aligned} & \frac{{\partial x_{1}^{{{\text{SSP}}}} }}{\partial k} = \frac{8r}{{r^{2} - 16}} < 0,\;\frac{{\partial x_{3}^{{{\text{SSP}}}} }}{\partial k} = \frac{{8(r^{2} - 4)}}{{r^{2} - 16}} > 0; \\ & \frac{{\partial q_{1}^{{{\text{SSP}}}} }}{\partial k} = \frac{8r}{{r^{2} - 16}} < 0,\;\frac{{\partial q_{2}^{{{\text{SSP}}}} }}{\partial k} = \frac{{2r^{2} - 16}}{{r^{2} - 16}} > 0; \\ & \frac{{\partial \pi_{1}^{{{\text{SSP}}}} }}{\partial k} = \frac{{16\left\{ {r\left( {r^{4} - 20r^{2} + 64} \right)A - r^{2} \left( {r^{4} - 12r^{2} + 32} \right)\left( {B + 2k} \right)} \right\}}}{{(r^{2} - 8)\left( {r^{2} - 16} \right)^{2} }} < 0; \\ & \frac{{\partial \pi_{2}^{{{\text{SSP}}}} }}{\partial k} = \frac{{\left\{ {\left( {r^{6} + 4r^{5} - 40r^{4} - 128r^{3} + 512r^{2} + 1024r - 2048} \right)A - \left( {7r^{6} - 216r^{4} + 1536r^{2} - 2048} \right)\left( {B + 2k} \right)} \right\}}}{{(r^{2} - 8)\left( {r^{2} - 16} \right)^{2} }} > 0; \\ & \frac{{\partial \left( {x_{1}^{{{\text{SSP}}}} - q_{1}^{{{\text{SSP}}}} } \right)}}{\partial k} = 0,\;\frac{{\partial \left( {x_{3}^{{{\text{SSP}}}} - q_{2}^{{{\text{SSP}}}} } \right)}}{\partial k} = \frac{{2\left( {3r^{2} - 8} \right)}}{{r^{2} - 16}} > 0; \\ & \frac{{\partial {\text{CS}}^{{{\text{SSP}}}} }}{\partial k} = \frac{{2\left\{ {4r\left( {r^{4} - 20r^{2} + 64} \right)A - \left( {9r^{6} - 136r^{4} + 576r^{2} - 512} \right)\left( {B + 2k} \right)} \right\}}}{{ - (r^{2} - 8)\left( {r^{2} - 16} \right)^{2} }} < 0; \\ & \frac{{\partial {\text{SW}}^{{{\text{SSP}}}} }}{\partial k} = \frac{{\left\{ {\left( {r^{4} + 12r^{3} - 32r^{2} - 192r + 256} \right)A - \left( {5r^{4} - 96r^{2} + 128} \right)\left( {B + 2k} \right)} \right\}}}{{\left( {r^{2} - 16} \right)^{2} }} > 0. \\ \end{aligned} $$

Appendix G: Proof of Corollaries 1, 2 and 3

• (1) Comparison of different cases under the Cournot competition:

  $$ \left( {x_{1}^{{{\text{CSP}}}} - q_{1}^{{{\text{CSP}}}} } \right) - \left( {x_{1}^{{{\text{CSS}}}} - q_{1}^{{{\text{CSS}}}} } \right) = \frac{{r^{3} \left[ {32\left( {16 - r^{2} } \right)A - \left( {r^{5} + r^{4} - 32r^{3} - 96r^{2} + 256r + 512} \right)\left( {B + 2k} \right)} \right]}}{{128(r^{2} - 8)\left( {r^{3} + 4r^{2} - 16r - 32} \right)}} > 0; $$
  $$ \left( {x_{1}^{{{\text{CSP}}}} - q_{1}^{{{\text{CSP}}}} } \right) - \left( {x_{1}^{{{\text{CSI}}}} - q_{1}^{{{\text{CSI}}}} } \right) = - \frac{{r^{3} \left[ {32\left( {r^{3} - 4r^{2} - 16r + 32} \right)A + \left( {r^{6} - 48r^{4} + 640r^{2} - 2048} \right)\left( {B + 2k} \right)} \right]}}{{128(r^{2} - 8)\left( {r^{4} - 32r^{2} + 128} \right)}} < 0; $$
  $$ \left( {x_{1}^{{{\text{CSI}}}} - q_{1}^{{{\text{CSI}}}} } \right) - \left( {x_{1}^{{{\text{CSS}}}} - q_{1}^{{{\text{CSS}}}} } \right) = - \frac{{2r^{3} \left[ {16A + \left( {r^{3} + 4r^{2} - 16r - 32} \right)k} \right]}}{{\left( {r^{3} + 4r^{2} - 16r - 32} \right)\left( {r^{4} - 32r^{2} + 128} \right)}} > 0; $$
  $$ \left( {x_{2}^{{{\text{CSP}}}} - q_{2}^{{{\text{CSP}}}} } \right) - \left( {x_{2}^{{{\text{CSS}}}} - q_{2}^{{{\text{CSS}}}} } \right) = \frac{{\begin{array}{*{20}l} {\left\{ {32\left( {r^{6} - 40r^{4} + 448r^{2} - 1024} \right)A} \right.} \hfill \\ {\quad + \left. {\left( {\begin{array}{*{20}c} {r^{9} + 4r^{8} - 56r^{7} - 192r^{6} + 1088r^{5} + 3072r^{4} - 8192r^{3} } \\ { - 18432r^{2} + 16384r + 32768} \\ \end{array} } \right)\left( {B + 2k} \right)} \right\}} \hfill \\ \end{array} }}{{256(r^{2} - 8)\left( {r^{3} + 4r^{2} - 16r - 32} \right)}} < 0; $$
  $$ \left( {x_{2}^{{{\text{CSP}}}} - q_{2}^{{{\text{CSP}}}} } \right) - \left( {x_{2}^{{{\text{CSI}}}} - q_{2}^{{{\text{CSI}}}} } \right) = \frac{{\begin{array}{*{20}l} {\left\{ {32\left( {r^{7} - 4r^{6} - 40r^{5} + 128r^{4} + 448r^{3} - 1024r^{2} - 1024r + 2048} \right)A} \right.} \hfill \\ {\quad \left. { + \left( {r^{10} - 72r^{8} + 1856r^{6} - 20480r^{4} + 90112r^{2} - 131072} \right)\left( {B + 2k} \right)} \right\}} \hfill \\ \end{array} }}{{256(r^{2} - 8)\left( {r^{4} - 32r^{2} + 128} \right)}} > 0; $$
  $$ \left( {x_{2}^{{{\text{CSI}}}} - q_{2}^{{{\text{CSI}}}} } \right) - \left( {x_{2}^{{{\text{CSI}}}} - q_{2}^{{{\text{CSI}}}} } \right) = \frac{{\begin{array}{*{20}l} {\left\{ {16\left( {r^{4} - 24r^{2} + 64} \right)A} \right.} \hfill \\ {\quad \left. { - \left( {r^{7} + 4r^{6} - 40r^{5} - 128r^{4} + 448r^{3} + 1024r^{2} - 1024r - 2048} \right)k} \right\}} \hfill \\ \end{array} }}{{\left( {r^{3} + 4r^{2} - 16r - 32} \right)\left( {r^{4} - 32r^{2} + 128} \right)}} < 0; $$
  $$ \pi_{1}^{{{\text{CSP}}}} - \pi_{1}^{{{\text{CSP}}}} = \frac{{\left\{ {\begin{array}{*{20}c} {1024A^{2} \left( {r^{9} + 8r^{8} - 64r^{7} - 448r^{6} + 1408r^{5} + 8192r^{4} - 12288r^{3} - 57344r^{2} + 32768r + 131072} \right)} \\ { + 64A\left( {B + 2k} \right)\left( {\begin{array}{*{20}c} {r^{12} + 8r^{11} - 64r^{10} - 576r^{9} + 1408r^{8} + 15360r^{7} - 11264r^{6} - 188416r^{5} - } \\ {16384r^{4} + 1048576r^{3} + 655360r^{2} - 2097152r - 2097152} \\ \end{array} } \right)} \\ { + \left( {\begin{array}{*{20}c} {r^{15} + 8r^{14} - 80r^{13} - 704r^{12} + 2432r^{11} + 24576r^{10} - 33792r^{9} - 434176r^{8} + 163840r^{7} } \\ { + 4063232r^{6} + 917504r^{5} - 18874368r^{4} - 12582912r^{3} + 33554432r^{2} + 33554432r} \\ \end{array} } \right)\left( {B + 2k} \right)^{2} } \\ \end{array} } \right\}}}{{8192(r^{2} - 8)^{2} \left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} }} > 0; $$
  $$ \pi_{1}^{{{\text{CSP}}}} - \pi_{1}^{{{\text{CSI}}}} = \frac{{\left\{ {\begin{array}{*{20}c} {1024A^{2} \left( {r^{7} - 80r^{5} + 128r^{4} + 1280r^{3} - 2048r^{2} - 5120r + 8192} \right)} \\ { + 64AB\left( {r^{10} - 80r^{8} + 2304r^{6} - 28672r^{4} + 147456r^{2} - 262144} \right)} \\ { + 128Ak\left( {r^{10} - 80r^{8} + 2304r^{6} - 512r^{5} - 26624r^{4} + 8192r^{3} + 114688r^{2} - 32768r - 131072} \right)} \\ { + \left( {r^{13} - 96r^{11} + 3584r^{9} - 65536r^{7} + 606208r^{5} - 2621440r^{3} + 4194304r} \right)\left( {B + 2k} \right)^{2} } \\ \end{array} } \right\}}}{{8192(r^{2} - 8)^{2} \left( {r^{4} - 32r^{2} + 128} \right)}} < 0; $$
  $$ \pi_{1}^{{{\text{CSI}}}} - \pi_{1}^{{{\text{CSS}}}} = \frac{{8r\left\{ {\begin{array}{*{20}c} {16A^{2} \left( {r^{4} - 32r^{2} + 16r + 128} \right) + Ak\left( {r^{7} + 4r^{6} - 48r^{5} - 128r^{4} + 768r^{3} + 1024r^{2} - 3072r - 4096} \right)} \\ { + \left( {r^{7} + 8r^{6} - 16r^{5} - 192r^{4} + 1024r^{2} + 1024r} \right)k^{2} } \\ \end{array} } \right\}}}{{\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} \left( {r^{4} - 32r^{2} + 128} \right)}} > 0 $$
  $$ \pi_{2}^{{{\text{CSP}}}} - \pi_{2}^{{{\text{CSS}}}} = \frac{{\begin{array}{*{20}l} {1024A^{2} \left( {r^{6} - 40r^{4} + 384r^{2} - 1024} \right) + 64A\left( {B + 2k} \right)\left( {\begin{array}{*{20}c} {r^{9} + 4r^{8} - 64r^{7} - 224r^{6} + 1280r^{5} } \\ { + 3584r^{4} - 9216r^{3} - 20480r^{2} + 16384r + 32768} \\ \end{array} } \right)} \hfill \\ {\quad + B^{2} \left( {\begin{array}{*{20}c} {r^{12} + 8r^{11} - 72r^{10} - 640r^{9} + 1664r^{8} + 17920r^{7} - 13312r^{6} - 221184r^{5} } \\ { - 24576r^{4} + 1179648r^{3} + 786432r^{2} - 2097152r - 2097152} \\ \end{array} } \right)} \hfill \\ {\quad + 4k\left( {B + k} \right)\left( {\begin{array}{*{20}c} {r^{12} + 8r^{11} - 72r^{10} - 640r^{9} + 1536r^{8} + 16896r^{7} - 10240r^{6} - 188416r^{5} } \\ { - 40960r^{4} + 851968r^{3} + 655360r^{2} - 1048576r - 1048576} \\ \end{array} } \right)} \hfill \\ \end{array} }}{{ - 512\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} \left( {r^{2} - 8} \right)}} < 0; $$
  $$ \pi_{2}^{{{\text{CSP}}}} - \pi_{2}^{{{\text{CSI}}}} = \frac{{\begin{array}{*{20}l} {1024\left( {r^{8} - 8r^{7} - 24r^{6} + 256r^{5} + 128r^{4} - 2560r^{3} + 1024r^{2} + 8192r - 8192} \right)A^{2} } \hfill \\ {\quad + 64\left( {\begin{array}{*{20}c} {r^{11} - 4r^{10} - 80r^{9} + 288r^{8} + 2304r^{7} - 7168r^{6} - 28673r^{5} + 73728r^{4} + 147456r^{3} } \\ { - 327680r^{2} - 262144r + 524288} \\ \end{array} } \right)A\left( {B + 2k} \right)} \hfill \\ {\quad + \left( {\begin{array}{*{20}c} {r^{14} - 120r^{12} + 5632r^{10} + 131072r^{8} + 1589248r^{6} - 9830400r^{4} } \\ { + 29360128r^{2} - 33554432} \\ \end{array} } \right)(B + 2k)^{2} } \hfill \\ \end{array} }}{{512(128 - 32r^{2} + r^{4} )^{2} \left( {8 - r^{2} } \right)}} > 0; $$
  $$ \pi_{2}^{{{\text{CSI}}}} - \pi_{2}^{{{\text{CSS}}}} = \frac{{\begin{array}{*{20}l} {256A^{2} \left( {r^{8} - 48r^{6} + 640r^{4} - 4096r^{2} + 8192} \right)} \hfill \\ {\quad + 32Ak\left( {\begin{array}{*{20}c} {r^{11} + 4r^{10} - 72r^{9} - 256r^{8} + 1792r^{7} + 5376r^{6} - 19456r^{5} - 49152r^{4} } \\ { + 90112r^{3} + 196608r^{2} - 131072r - 262144} \\ \end{array} } \right)} \hfill \\ { + \left( {\begin{array}{*{20}c} {r^{14} + 8r^{13} - 80r^{12} - 704r^{11} + 2176r^{10} + 22528r^{9} - 23552r^{8} - 335872r^{7} } \\ { + 40960r^{6} + 2424832r^{5} + 1048576r^{4} - 6291456r^{2} + 8388608} \\ \end{array} } \right)k^{2} } \hfill \\ \end{array} }}{{\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} \left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0; $$
  $$ \begin{aligned} & {\text{CS}}^{{{\text{CSP}}}} - {\text{CS}}^{{{\text{CSS}}}} \\ & \quad = \frac{{\begin{array}{*{20}l} {1024A^{2} \left( {\begin{array}{*{20}c} {r^{10} - 24r^{9} - 80r^{8} + 1088r^{7} + 2304r^{6} - 17408r^{5} - 29696r^{4} } \\ { + 114688r^{3} + 163840r^{2} - 262144r - 262144} \\ \end{array} } \right)} \hfill \\ {\quad - 64A\left( {B + 2k} \right)\left( {\begin{array}{*{20}c} {3r^{13} + 24r^{12} - 160r^{11} - 1472r^{10} + 3072r^{9} + 34816r^{8} - 21504r^{7} } \\ { - 393216r^{6} - 49152r^{5} + 2097152r^{4} + 1310720r^{3} - 4294304r^{2} - 4294304r} \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {7r^{16} + 56r^{15} - 512r^{14} - 4544r^{13} + 14848r^{12} + 151552r^{11} - 206848r^{10} - 2662400r^{9} } \\ { + 1146880r^{8} + 26148864r^{7} + 3670016r^{6} - 139460608r^{5} } \\ { - 76546048r^{4} + 352321536r^{3} + 301989888r^{2} - 268435456r - 268435456} \\ \end{array} } \right)\left( {B + 2k} \right)^{2} } \hfill \\ \end{array} }}{{32768\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} (r^{2} - 8)^{2} }} < 0; \\ \end{aligned} $$
  $$ {\text{CS}}^{{{\text{CSP}}}} - {\text{CS}}^{{{\text{CSI}}}} = \frac{{\begin{array}{*{20}l} {1024A^{2} \left( {\begin{array}{*{20}c} {r^{12} - 32r^{11} + 128r^{10} + 1280r^{9} - 6656r^{8} - 17408r^{7} + 108544r^{6} + 81920r^{5} } \\ { - 753664r^{4} + 65536r^{3} + 2097152r^{2} - 1048576r - 1048576} \\ \end{array} } \right)} \hfill \\ {\quad - 64AB\left( {\begin{array}{*{20}c} {3r^{15} - 304r^{13} + 12288r^{11} - 253952r^{9} + 2867200r^{7} - } \\ {17563648r^{5} + 54525952r^{3} - 67108864r} \\ \end{array} } \right)} \hfill \\ {\quad - 128Ak\left( {\begin{array}{*{20}c} {3r^{15} - 304r^{13} + 12288r^{11} - 3584r^{10} - 274432r^{9} + 147456r^{8} + 3145728r^{7} - 2195456r^{6} } \\ { - 18874368r^{5} + 1468006r^{4} + 534777376r^{3} - 41943040r^{2} - 5033164r + 33554432} \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {7r^{18} - 848r^{16} + 43008r^{14} - 1188864r^{12} + 19546112r^{10} - 195821568r^{8} } \\ { + 1178599424r^{6} - 4043309056r^{4} + 6979321856r^{2} - 4294967296} \\ \end{array} } \right)\left( {B + 2k} \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {7r^{18} - 848r^{16} + 43008r^{14} - 1188864r^{12} + 19431424r^{10} - 191102976r^{8} } \\ { + 1108344832r^{6} - 3573547008r^{4} + 5637144576r^{2} - 3221225472} \\ \end{array} } \right)k^{2} } \hfill \\ \end{array} }}{{32768\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} \left( {r^{2} - 8} \right)^{2} }} < 0; $$
  $$ {\text{CS}}^{{{\text{CSI}}}} - {\text{CS}}^{{{\text{CSS}}}} = \frac{{\begin{array}{*{20}l} {16A^{2} \left( {\begin{array}{*{20}c} {r^{10} - 3r^{9} - 68r^{8} + 160r^{7} + 1520r^{6} - 2688r^{5} - 13568r^{4} } \\ { + 16384r^{3} + 45056r^{2} - 32768r - 49152} \\ \end{array} } \right)} \hfill \\ {\quad - Ak\left( {\begin{array}{*{20}c} {r^{13} + r^{12} - 96r^{11} - 96r^{10} + 3232r^{9} + 2560r^{8} - 49152r^{7} - 28672r^{6} + 352256r^{5} } \\ { + 163840r^{4} - 1146880r^{3} - 524288r^{2} + 1572864r + 1048576} \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {7r^{12} + 56r^{11} - 288r^{10} - 2752r^{9} + 3840r^{8} + 49152r^{7} - 10240r^{6} } \\ { - 385024r^{5} - 163840r^{4} + 1245184r^{3} + 1048576r^{2} - 1048576r - 1048576} \\ \end{array} } \right)k^{2} } \hfill \\ \end{array} }}{{\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} \left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0; $$
  $$ \begin{aligned} & {\text{SW}}^{{{\text{CSP}}}} - {\text{SW}}^{{{\text{CSS}}}} \\ & \quad = \frac{{\begin{array}{*{20}l} {1024A^{2} \left( {\begin{array}{*{20}c} {5r^{10} + 8r^{9} + 400r^{8} - 704r^{7} + 11008r^{6} + 15360r^{5} - 123904r^{4} } \\ { - 114688r^{3} + 57056r^{2} + 262144r - 786432} \\ \end{array} } \right)} \hfill \\ {\quad + 64A\left( {B + 2k} \right)\left( {\begin{array}{*{20}c} {r^{13} + 8r^{12} - 160r^{11} - 1088r^{10} + 7168r^{9} + 43008r^{8} - 138240r^{7} - 704512r^{6} } \\ { + 1228800r^{5} + 5242880r^{4} - 4456448r^{3} - 16777216r^{2} + 4194304r + 16777216} \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {3r^{16} + 24r^{15} - 128r^{14} - 1216r^{13} + 8192r^{11} + 71680r^{10} + 548864r^{9} - 1212416r^{8} - 13434880r^{7} + 5242880r^{6} } \\ { + 124780544r^{5} + 36700160r^{4} - 520093696r^{3} - 369098752r^{2} + 805306368r + 805306368} \\ \end{array} } \right)\left( {B + 2k} \right)^{2} } \hfill \\ \end{array} }}{{32768\left( {r^{4} - 32r^{2} + 128} \right)^{2} (r^{2} - 8)^{2} }} < 0; \\ \end{aligned} $$
  $$ \begin{aligned} & {\text{SW}}^{{{\text{CSP}}}} - {\text{SW}}^{{{\text{CSI}}}} \\ & \quad = \frac{{\begin{array}{*{20}l} {1024A^{2} \left( {\begin{array}{*{20}c} {5r^{12} - 32r^{11} - 384r^{10} + 2304r^{9} + 11264r^{8} - 62464r^{7} - 137216r^{6} + 737280r^{5} + 557056r^{4} } \\ { - 3866624r^{3} + 524288r^{2} + 7340032r - 5242880} \\ \end{array} } \right)} \hfill \\ {\quad + 64AB\left( {\begin{array}{*{20}c} {r^{15} - 208r^{13} + 256r^{12} + 13312r^{11} - 20480r^{10} - 385024r^{9} + 606208r^{8} + 5586944r^{7} - 8388608r^{6} } \\ { - 41156608r^{5} + 58720256r^{4} + 146800640r^{3} - 201326592r^{2} + 201326592r + 268435456} \\ \end{array} } \right)} \hfill \\ {\quad + 128Ak\left( {\begin{array}{*{20}c} {r^{15} - 208r^{13} + 256r^{12} + 12800r^{11} - 18944r^{10} - 348160r^{9} + 524288r^{8} + 4587520r^{7} - 6586368r^{6} } \\ { - 29360128r^{5} + 39845888r^{4} + 84934656r^{3} - 109051904r^{2} - 83886080r + 100663296} \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {3r^{18} - 272r^{16} + 7680r^{14} + 3072r^{12} - 434528r^{10} + 94633984r^{8} - 926941184r^{6} } \\ { + 4747952128r^{4} - 12348030976r^{2} + 12884901888} \\ \end{array} } \right)B^{2} } \hfill \\ {\quad - 4Bk\left( {\begin{array}{*{20}c} {3r^{18} - 272r^{16} + 7680r^{14} - 5120r^{12} - 3719168r^{10} + 75235328r^{8} - 658505728r^{6} } \\ { + 2868903936r^{4} - 5905580032r^{2} + 4294967296} \\ \end{array} } \right)} \hfill \\ {\quad - 4k^{2} \left( {\begin{array}{*{20}c} {3r^{18} - 272r^{16} + 7680r^{14} - 13312r^{12} - 3112960r^{10} + 58458112r^{8} - 447741952r^{6} } \\ { + 1593835520r^{4} - 2415919104r^{2} + 1073741824} \\ \end{array} } \right)} \hfill \\ \end{array} }}{{32768\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} (r^{2} - 8)^{2} }} > 0; \\ \end{aligned} $$
  $$ {\text{SW}}^{{{\text{CSI}}}} - {\text{SW}}^{{{\text{CSS}}}} = \frac{{\begin{array}{*{20}l} {32A^{2} \left( {\begin{array}{*{20}c} {r^{10} + r^{9} - 76r^{8} - 96r^{7} + 1968r^{6} + 2432r^{5} - 20736r^{4} } \\ { - 16384r^{3} + 86016r^{2} + 32768r - 114688} \\ \end{array} } \right)} \hfill \\ {\quad + 2Ak\left( {\begin{array}{*{20}c} {r^{13} + 5r^{12} - 96r^{11} - 480r^{10} + 3360r^{9} + 16384r^{8} - 55296r^{7} - 249856r^{6} } \\ { + 450560r^{5} + 1736704r^{4} - 1540096r^{3} - 5242880r^{2} + 1572864r + 5242880} \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {r^{14} + 8r^{13} - 74r^{11} - 656r^{10} + 1984r^{9} + 20608r^{8} - 303104r^{7} + 28672r^{6} } \\ { + 2113536r^{5} + 983040r^{4} - 6422528r^{3} - 5242880r^{2} + 6291456r + 6291456} \\ \end{array} } \right)k^{2} } \hfill \\ \end{array} }}{{\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} \left( {r^{4} - 32r^{2} + 128} \right)^{2} }} < 0. $$

• (2) Comparison of different cases under the Stackelberg competition:

  $$ \left( {x_{2}^{{{\text{SSP}}}} - q_{2}^{{{\text{SSP}}}} } \right) - \left( {x_{2}^{{{\text{SSS}}}} - q_{2}^{{{\text{SSS}}}} } \right) = \frac{{\begin{array}{*{20}l} {\left( {3r^{6} + 12r^{5} - 80r^{4} - 224r^{3} + 576r^{2} + 512r - 1024} \right)A} \hfill \\ {\quad - \left( {21r^{6} - 272r^{4} + 960r^{2} - 1024} \right)\left( {B + 2k} \right)} \hfill \\ \end{array} }}{{ - (r^{2} - 8)(r^{2} - 16)(7r^{2} - 16)}} < 0; $$
  $$ \left( {x_{2}^{{{\text{SSP}}}} - q_{2}^{{{\text{SSP}}}} } \right) - \left( {x_{2}^{{{\text{SSI}}}} - q_{2}^{{{\text{SSI}}}} } \right) = \frac{{\begin{array}{*{20}l} {\left( {3r^{6} + 12r^{5} - 80r^{4} - 224r^{3} + 576r^{2} + 512r - 1024} \right)A} \hfill \\ {\quad - 8\left( {3r^{6} - 44r^{4} + 192r^{2} - 256} \right)B - 2\left( {21r^{6} - 272r^{4} + 960r^{2} - 1024} \right)k} \hfill \\ \end{array} }}{{ - 8(r^{2} - 8)(r^{2} - 16)(r^{2} - 4)}} > 0; $$
  $$ \left( {x_{2}^{{{\text{SSI}}}} - q_{2}^{{{\text{SSI}}}} } \right) - \left( {x_{2}^{{{\text{SSS}}}} - q_{2}^{{{\text{SSS}}}} } \right) = \frac{{\begin{array}{*{20}l} {\left( {3r^{6} + 12r^{5} - 80r^{4} - 224r^{3} + 576r^{2} + 512r - 1024} \right)A} \hfill \\ {\quad - 2\left( {21r^{6} - 272r^{4} + 960r^{2} - 1024} \right)k} \hfill \\ \end{array} }}{{ - 8(r^{2} - 8)(r^{2} - 16)(r^{2} - 4)}} < 0; $$
  $$ \pi_{1}^{{{\text{SSP}}}} - \pi_{1}^{{{\text{SSS}}}} = \frac{{4\left[ {\begin{array}{*{20}c} {A^{2} \left( {\begin{array}{*{20}c} {r^{11} - 6r^{10} - 92r^{9} + 296r^{8} + 2592r^{7} - 5440r^{6} - 29440r^{5} + 46080r^{4} + 131072r^{3} } \\ { - 180224r^{2} - 196608r + 262144} \\ \end{array} } \right)} \\ { + 2A\left( {B + 2k} \right)\left( {49r^{10} - 1596r^{8} + 17504r^{6} - 82432r^{4} + 172032r^{2} - 131072} \right)} \\ { - \left( {49r^{11} - 1024r^{9} + 11008r^{7} - 46336r^{5} + 90112r^{3} - 65536} \right)\left( {B + 2k} \right)^{2} } \\ \end{array} } \right]}}{{\left( {7r^{4} - 72r^{2} + 128} \right)(7r^{2} - 16)(r^{2} - 16)(r^{4} - 24r^{2} + 128)}} > 0; $$
  $$ \pi_{1}^{{{\text{SSP}}}} - \pi_{1}^{{{\text{SSI}}}} = \frac{{r\left\{ {\begin{array}{*{20}c} {A^{2} \left( {r^{9} - 8r^{8} - 96r^{7} + 384r^{6} + 2624r^{5} - 6656r^{4} - 26624r^{3} + 49152r^{2} + 81920r - 131072} \right)} \\ { + 128AB\left( {r^{8} - 32r^{6} + 336r^{4} - 1408r^{2} + 2048} \right)} \\ { + 4Ak\left( {r^{9} + 60r^{8} - 48r^{7} - 1856r^{6} + 832r^{5} + 18176r^{4} - 6144r^{3} - 655362r^{2} + 16384r + 65536} \right)} \\ { - 64\left( {r^{9} - 24r^{7} + 208r^{5} - 768r^{3} + 1024r} \right)\left( {B + 2k} \right)^{2} } \\ \end{array} } \right\}}}{{16\left( {r^{2} - 4} \right)(r^{2} - 8)(r^{2} - 16)(r^{4} - 24r^{2} + 128)}} < 0; $$
  $$ \pi_{1}^{{{\text{SSI}}}} - \pi_{1}^{{{\text{SSS}}}} = \frac{{r\left\{ {\begin{array}{*{20}c} {A^{2} \left( {\begin{array}{*{20}c} {15r^{9} + 8r^{8} - 736r^{7} + 128r^{6} + 11712r^{5} - 7680r^{4} - 67584r^{3} } \\ { + 65536r^{2} + 114688r - 131072} \\ \end{array} } \right)} \\ { - 4Ak\left( {\begin{array}{*{20}c} {49r^{9} - 196r^{8} - 1008r^{7} + 4032r^{6} + 6976r^{5} - 27904r^{4} - 18432r^{3} } \\ { + 73728r^{2} + 16384r - 65536} \\ \end{array} } \right)} \\ { - 4\left( {49r^{9} - 1008r^{7} + 6976r^{5} - 18432r^{3} + 16384r} \right)k^{2} } \\ \end{array} } \right\}}}{{16\left( {r^{2} - 4} \right)(r^{2} - 8)(7r^{2} - 16)(r^{4} - 72r^{2} + 128)}} > 0; $$
  $$ \pi_{2}^{{{\text{SSP}}}} - \pi_{1}^{{{\text{SSS}}}} = \frac{{\begin{array}{*{20}l} {A^{2} \left( {\begin{array}{*{20}c} {7r^{12} + 56r^{11} - 320r^{10} - 3008r^{9} + 5568r^{8} + 61440r^{7} - 53248r^{6} - 589824r^{5} } \\ { + 393216r^{4} + 2621440r^{3} - 2097152r^{2} - 4194304r + 4194304} \\ \end{array} } \right)} \hfill \\ {\quad - 2A\left( {B + 2k} \right)\left( {\begin{array}{*{20}c} {49r^{12} + 196r^{11} - 2576r^{10} - 8736r^{9} + 51776r^{8} + 137216r^{7} - 499712r^{6} } \\ { - 901120r^{5} + 2392064r^{4} + 2359296r^{3} - 5242880r^{2} - 2097152r + 4194304} \\ \end{array} } \right)} \hfill \\ {\quad + 8\left( {49r^{12} - 2184r^{10} + 34304r^{8} + 34304r^{6} - 250368r^{4} + 905216r^{2} + 1048576} \right)\left( {B + 2k} \right)^{2} } \hfill \\ \end{array} }}{{ - 4(r^{2} - 8)^{2} (r^{2} - 16)^{2} \left( {7r^{2} - 16} \right)^{2} }} < 0; $$
  $$ \pi_{2}^{{{\text{SSP}}}} - \pi_{2}^{{{\text{SSI}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {r^{10} + 8r^{9} - 40r^{8} - 384r^{7} + 576r^{6} + 6656r^{5} - 4608r^{4} - 49152r^{3} } \\ { + 32768r^{2} + 131072r + 131072} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 16\left( {\begin{array}{*{20}c} {r^{10} + 4r^{9} - 48r^{8} - 160r^{7} + 848r^{6} + 2112r^{5} - 6784r^{4} - 10240r^{3} } \\ { + 24576r^{2} + 16384r - 32768} \\ \end{array} } \right)AB} \hfill \\ {\quad - \left( {\begin{array}{*{20}c} {28r^{10} + 112r^{9} - 1312r^{8} - 4352r^{7} + 22272r^{6} + 54272r^{5} - 165888r^{4} } \\ { - 229376r^{3} + 524288r^{2} + 262144r - 524288} \\ \end{array} } \right)Ak} \hfill \\ {\quad + 64\left( {r^{10} - r^{8} - 64r^{6} + 1152r^{4} - 6144r^{2} + 8192} \right)Bk} \hfill \\ {\quad + 4\left( {49r^{10} - 1848r^{8} + 21440r^{6} - 100864r^{4} + 196608r^{2} - 131072} \right)k^{2} } \hfill \\ \end{array} }}{{ - 32(r^{2} - 8)\left( {r^{2} - 16} \right)^{2} \left( {r^{2} - 4} \right)^{2} }} > 0; $$
  $$ \pi_{2}^{{{\text{SSI}}}} - \pi_{2}^{{{\text{SSS}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {7r^{12} + 56r^{11} - 208r^{10} - 2112r^{9} + 2560r^{8} + 30208r^{7} - 23552r^{6} - 217088r^{5} + } \\ {151552r^{4} + 786432r^{3} - 655360r^{2} - 1048576r + 1048576} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 4\left( {\begin{array}{*{20}c} {49r^{12} + 196r^{11} - 1792r^{10} - 5600r^{9} + 26240r^{8} + 60160r^{7} - 194560r^{6} - } \\ {296960r^{5} + 757760r^{4} + 655360r^{3} - 1441792r^{2} - 524288r + 1048576} \\ \end{array} } \right)Ak} \hfill \\ {\quad + 4\left( {49r^{10} - 1848r^{8} + 21440r^{6} - 100864r^{4} + 196608r^{2} - 131072} \right)k^{2} } \hfill \\ \end{array} }}{{ - 32(r^{2} - 8)\left( {7r^{2} - 16} \right)^{2} \left( {r^{2} - 4} \right)^{2} }} < 0; $$
  $$ {\text{CS}}^{{{\text{SSP}}}} - {\text{CS}}^{{{\text{SSS}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {9r^{12} + 16r^{11} - 576r^{10} - 800r^{9} + 13568r^{8} + 14592r^{7} - 145408r^{6} - 114688r^{5} } \\ { + 724992r^{4} + 327680r^{3} - 1703936r^{2} + 1048576} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 8\left( {49r^{11} - 1596r^{9} + 17504r^{7} - 82432r^{5} + 172032r^{3} - 131072r} \right)A\left( {B + 2k} \right)} \hfill \\ {\quad + \left( {\begin{array}{*{20}c} {441r^{12} - 12208r^{10} + 130432r^{8} - 676864r^{6} + 1773568r^{4} } \\ { - 2228224r^{2} + 1048576} \\ \end{array} } \right)\left( {B + 2k} \right)^{2} } \hfill \\ \end{array} }}{{ - 2(r^{2} - 8)^{2} \left( {7r^{2} - 16} \right)^{2} \left( {r^{2} - 16} \right)^{2} }} < 0; $$
  $$ {\text{CS}}^{{{\text{SSP}}}} - {\text{CS}}^{{{\text{SSI}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {9r^{12} + 8r^{11} - 608r^{10} - 320r^{9} + 15744r^{8} + 3584r^{7} - 196608r^{6} + 4096r^{5} + 1224704r^{4} } \\ { - 262144r^{3} - 3538944r^{2} + 1048576r + 3145728} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 256\left( {r^{11} - 36r^{9} + 464r^{7} - 2752r^{5} + 7680r^{3} - 8192r} \right)AB} \hfill \\ {\quad + 4\left( {\begin{array}{*{20}c} {9r^{12} + 132r^{11} - 496r^{10} - 4768r^{9} + 10624r^{8} + 61184r^{7} - 111616r^{6} - 350208r^{5} } \\ { + 593920r^{4} + 851968r^{3} - 1441792r^{2} - 5524288r + 1048576} \\ \end{array} } \right)Ak} \hfill \\ {\quad - 32\left( {9r^{12} - 280r^{10} + 3472r^{8} - 21760r^{6} + 71680r^{4} - 114688r^{2} + 65536} \right)B\left( {B + 4k} \right)} \hfill \\ {\quad - 4\left( {279r^{12} - 8464r^{10} + 100480r^{8} - 584704r^{6} + 1699840r^{4} - 2228224r^{2} + 1048576} \right)k^{2} } \hfill \\ \end{array} }}{{ - 64(r^{2} - 8)^{2} \left( {r^{2} - 4} \right)^{2} \left( {r^{2} - 16} \right)^{2} }} < 0; $$
  $$ {\text{CS}}^{{{\text{SSI}}}} - {\text{CS}}^{{{\text{SSS}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {153r^{12} - 120r^{11} - 6176r^{10} + 8384r^{9} + 86912r^{8} - 131584r^{7} - 532480r^{6} } \\ { + 774144r^{5} + 1421312r^{4} - 1703936r^{3} - 1703936r^{2} + 1048576r + 1048576} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 8\left( {49r^{11} - 1596r^{9} + 17504r^{7} - 82432r^{5} + 172032r^{3} - 131072r} \right)A\left( {B + 2k} \right)} \hfill \\ {\quad + \left( {\begin{array}{*{20}c} {441r^{12} - 12208r^{10} + 130432r^{8} - 676864r^{6} + 1773568r^{4} } \\ { - 2228224r^{2} + 1048576} \\ \end{array} } \right)\left( {B + 2k} \right)^{2} } \hfill \\ \end{array} }}{{64(r^{2} - 8)^{2} \left( {7r^{2} - 16} \right)^{2} \left( {r^{2} - 4} \right)^{2} }} > 0; $$
  $$ {\text{SW}}^{{{\text{SSP}}}} - {\text{SW}}^{{{\text{SSS}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {9r^{12} + 184r^{11} - 9344r^{9} - 8768r^{8} + 177664r^{7} + 126976r^{6} - 1556480r^{5} - 253952r^{4} } \\ { + 6160384r^{3} - 2359296r^{2} - 8388608r + 6291456} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 2\left( {\begin{array}{*{20}c} {49r^{12} + 588r^{11} - 2576r^{10} - 21504r^{9} + 51776r^{8} + 277248r^{7} - 499712r^{6} } \\ { - 1560576r^{5} + 2392064r^{4} + 3735552r^{3} - 5242880r^{2} - 3145728r + 4194304} \\ \end{array} } \right)A\left( {B + 2k} \right)} \hfill \\ {\quad - 2\left( {147r^{12} - 6160r^{10} + 94848r^{8} - 695296r^{6} + 2568192r^{4} - 4587520r^{2} + 3145728} \right)B^{2} } \hfill \\ {\quad - 4(245r^{12} - 9744r^{10} + 137920r^{8} - 890880r^{6} + 2744320r^{4} } \hfill \\ {\quad - 3932160r^{2} + 2097152)\left( {B + k} \right)k} \hfill \\ \end{array} }}{{4(r^{2} - 8)^{2} \left( {7r^{2} - 16} \right)^{2} \left( {r^{2} - 16} \right)^{2} }} < 0; $$
  $$ {\text{SW}}^{{{\text{SSP}}}} - {\text{SW}}^{{{\text{SSI}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {7r^{12} + 56r^{11} - 304r^{10} - 2880r^{9} + 5504r^{8} + 55808r^{7} - 66560r^{6} - 503808r^{5} } \\ { + 610304r^{4} + 2097152r^{3} - 3014656r^{2} - 3145728r + 5242880} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad - 32\left( {\begin{array}{*{20}c} {r^{12} + 12r^{11} - 56r^{10} - 480r^{9} + 1232r^{8} + 7104r^{7} - 13568r^{6} - 49152r^{5} + 78848r^{4} } \\ { + 159744r^{3} - 229376r^{2} - 196608r + 262144} \\ \end{array} } \right)AB} \hfill \\ {\quad - 4\left( {\begin{array}{*{20}c} {9r^{12} + 164r^{11} - 480r^{10} - 6240r^{9} + 9856r^{8} + 85760r^{7} - 98304r^{6} - 534528r^{5} } \\ { + 495616r^{4} + 1507328r^{3} - 1179648r^{2} - 1572864r + 1048576} \\ \end{array} } \right)} \hfill \\ {\quad + 32\left( {3r^{12} - 136r^{10} + 2352r^{8} - 20224r^{6} + 92160r^{4} - 212992r^{2} + 196608} \right)B^{2} } \hfill \\ {\quad + 64\left( {5r^{12} - 216r^{10} + 3472r^{8} - 26880r^{6} + 105472r^{4} - 196608r^{2} + 131072} \right)Bk} \hfill \\ {\quad + 4\left( {71r^{12} - 2976r^{10} + 45696r^{8} - 331776r^{6} + 1191936r^{4} - 196608r^{2} + 1048576} \right)k^{2} } \hfill \\ \end{array} }}{{ - 64(r^{2} - 8)^{2} \left( {r^{2} - 16} \right)^{2} \left( {r^{2} - 4} \right)^{2} }} < 0; $$
  $$ {\text{SW}}^{{{\text{SSI}}}} - {\text{SW}}^{{{\text{SSS}}}} = \frac{{\begin{array}{*{20}l} {\left( {\begin{array}{*{20}c} {199r^{12} - 200r^{11} - 8944r^{10} + 12992r^{9} + 140416r^{8} - 224768r^{7} - 943104r^{6} + 1593344r^{5} } \\ { + 2658304r^{4} - 4849664r^{3} - 2228224r^{2} + 5242880r - 1048576} \\ \end{array} } \right)A^{2} } \hfill \\ {\quad + 4\left( {\begin{array}{*{20}c} {343r^{12} + 1372r^{11} - 10976r^{10} - 32928r^{9} + 138880r^{8} + 292096r^{7} - 880640r^{6} } \\ { - 1185792r^{5} + 2928640r^{4} + 2228224r^{3} - 4849664r^{2} - 1572864r + 3145728} \\ \end{array} } \right)Ak} \hfill \\ {\quad - 4\left( {441r^{12} - 11424r^{10} + 114304r^{8} - 565248r^{6} + 1478656r^{4} - 196608r^{2} + 1048576} \right)k^{2} } \hfill \\ \end{array} }}{{64(r^{2} - 8)^{2} \left( {7r^{2} - 16} \right)^{2} \left( {r^{2} - 4} \right)^{2} }} > 0; $$
  $$ \left( {x_{1}^{{{\text{CSS}}}} - q_{1}^{{{\text{CSS}}}} } \right) - \left( {x_{1}^{{{\text{SSS}}}} - q_{1}^{{{\text{SSS}}}} } \right) = - \frac{{r^{2} A}}{{r^{3} + 4r^{2} - 16r - 32}} > 0; $$
  $$ \left( {x_{2}^{{{\text{CSS}}}} - q_{2}^{{{\text{CSS}}}} } \right) - \left( {x_{2}^{{{\text{SSS}}}} - q_{2}^{{{\text{SSS}}}} } \right) = \frac{{r^{4} A\left( {r^{3} - 6r^{2} - 8r + 16} \right)}}{{2(r^{2} - 8)(7r^{2} - 16)\left( {r^{3} + 4r^{2} - 16r - 32} \right)}} < 0; $$
  $$ \pi_{1}^{{{\text{CSS}}}} - \pi_{1}^{{{\text{SSS}}}} = \frac{{2r^{4} A^{2} \left( {\begin{array}{*{20}c} {2r^{10} + 4r^{9} - 101r^{8} - 160r^{7} + 1736r^{6} + 1728r^{5} } \\ { - 12992r^{4} - 6656r^{3} + 43008r^{2} + 8192r - 49152} \\ \end{array} } \right)}}{{(r^{2} - 8)\left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} (7r^{2} - 16)\left( {7r^{4} - 72r^{2} + 128} \right)}} < 0; $$
  $$ \pi_{2}^{{{\text{CSS}}}} - \pi_{2}^{{{\text{SSS}}}} = - \frac{{r^{3} \left\{ {A^{2} \left( {\begin{array}{*{20}c} {7r^{11} + 112r^{10} - 152r^{9} - 4224r^{8} + 2368r^{7} + 58880r^{6} } \\ { - 17920r^{5} - 372736r^{4} + 32768r^{3} + 1048576r^{2} - 1048576} \\ \end{array} } \right)} \right\}}}{{4(r^{2} - 8)^{2} (7r^{2} - 16)^{2} \left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} }} > 0; $$
  $$ {\text{CS}}^{{{\text{CSS}}}} - {\text{CS}}^{{{\text{SSS}}}} = - \frac{{r^{2} \left\{ {A^{2} \left( {\begin{array}{*{20}c} {9r^{12} - 10r^{11} - 402r^{10} + 576r^{9} + 7216r^{8} - 10496r^{7} - 67456r^{6} } \\ { + 78848r^{5} + 327680r^{4} - 245760r^{3} - 704512r^{2} + 262144r + 524288} \\ \end{array} } \right)} \right\}}}{{2(r^{2} - 8)^{2} (7r^{2} - 16)^{2} \left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} }} < 0; $$
  $$ {\text{SW}}^{{{\text{CSS}}}} - {\text{SW}}^{{{\text{SSS}}}} = - \frac{{r^{2} \left\{ {A^{2} \left( {\begin{array}{*{20}c} {9r^{12} + 60r^{11} - 148r^{10} - 1792r^{9} + 2912r^{8} + 24064r^{7} - 48896r^{6} } \\ { - 161792r^{5} + 344064r^{4} + 491520r^{3} - 1015808r^{2} - 524288r + 1048576} \\ \end{array} } \right)} \right\}}}{{4(r^{2} - 8)^{2} (7r^{2} - 16)^{2} \left( {r^{3} + 4r^{2} - 16r - 32} \right)^{2} }} < 0. $$

• (3) Comparison of different competition modes:

  $$ \left( {x_{1}^{{{\text{CSI}}}} - q_{1}^{{{\text{CSI}}}} } \right) > \left( {x_{1}^{{{\text{SSI}}}} - q_{1}^{{{\text{SSI}}}} } \right) = \frac{{r^{2} \left[ {\left( {4 - r} \right)A - 2kr} \right]}}{{r^{4} - 32r^{2} + 128}} > 0; $$
  $$ \left( {x_{2}^{{{\text{CSI}}}} - q_{2}^{{{\text{CSI}}}} } \right) < \left( {x_{2}^{{{\text{SSI}}}} - q_{2}^{{{\text{SSI}}}} } \right) = \frac{{r^{3} \left[ {\left( {r^{3} - 4r^{2} - 8r + 32} \right)A + 2kr\left( {r^{2} - 16} \right)} \right]}}{{8(r^{2} - 4)(r^{4} - 32r^{2} + 128)}} < 0; $$
  $$ \pi_{1}^{{{\text{CSI}}}} - \pi_{1}^{{{\text{SSI}}}} = \frac{{\left( {\left( {r - 4} \right)A + 2kr} \right)^{2} }}{{16\left( {r^{2} - 4} \right)\left( {r^{4} - 32r^{2} + 128} \right)}} < 0; $$
  $$ \pi_{2}^{{{\text{CSI}}}} - \pi_{2}^{{{\text{SSI}}}} = - \frac{{\begin{array}{*{20}l} {r^{3} \left\{ {A^{2} \left( {r^{9} + 8r^{8} - 128r^{7} - 64r^{6} + 2880r^{5} - 2048r^{4} - 21504r^{3} + 24576r^{2} + 49152r - 65536} \right)} \right.} \hfill \\ {\quad + 4Ak\left( {r^{9} + 4r^{8} - 90r^{7} - 32r^{6} + 1344r^{5} - 1024r^{4} - 8192r^{3} + 12288r^{2} + 16384r - 32768} \right)} \hfill \\ {\quad \left. { + 4k^{2} \left( {r^{9} - 90r^{7} + 1344r^{5} - 8192r^{3} + 16384r} \right)} \right\}} \hfill \\ \end{array} }}{{32(r^{4} - 32r^{2} + 128)^{2} (r^{2} - 4)^{2} }} > 0; $$
  $$ {\text{CS}}^{{{\text{CSI}}}} - {\text{CS}}^{{{\text{SSI}}}} = \frac{{\begin{array}{*{20}l} {A^{2} \left( {\begin{array}{*{20}c} {9r^{12} - 56r^{11} - 304r^{10} + 1600r^{9} + 6208r^{8} - 13312r^{7} - 99328r^{6} + 57344r^{5} } \\ { + 770048r^{4} - 229376r^{3} - 2490368r^{2} + 52428r + 2621440} \\ \end{array} } \right)} \hfill \\ {\quad + 4Ak\left( {\begin{array}{*{20}c} {9r^{12} - 28r^{11} - 416r^{10} + 800r^{9} + 8256r^{8} - 6656r^{7} - 78336r^{6} } \\ { + 28672r^{5} + 368640r^{4} - 114688r^{3} - 786432r^{2} + 262144r + 524288} \\ \end{array} } \right)} \hfill \\ {\quad + 4k^{2} \left( {9r^{12} - 416r^{10} + 8256r^{8} - 78336r^{6} + 368640r^{4} - 786432r^{2} + 524288} \right)} \hfill \\ \end{array} }}{{ - 64(r^{4} - 32r^{2} + 128)^{2} (r^{2} - 4)^{2} }} < 0; $$
  $$ {\text{SW}}^{{{\text{CSI}}}} - {\text{SW}}^{{{\text{SSI}}}} = - \frac{{\begin{array}{*{20}l} {A^{2} \left( {\begin{array}{*{20}c} {7r^{12} - 8r^{11} - 480r^{10} + 320r^{9} + 13248r^{8} - 9216r^{7} - 156672r^{6} + 90112r^{5} } \\ { - 901120r^{4} - 360448r^{3} - 2490368r^{2} + 52428r + 2621440} \\ \end{array} } \right)} \hfill \\ {\quad + 4Ak\left( {\begin{array}{*{20}c} {7r^{12} - 4r^{11} - 432r^{10} + 160r^{9} + 9920r^{8} - 4608r^{7} - 92672r^{6} + 45056r^{5} } \\ { + 401408r^{4} - 180224r^{3} - 786432r^{2} + 262144r + 524288} \\ \end{array} } \right)} \hfill \\ {\quad + 4k^{2} \left( {7r^{12} - 432r^{10} + 9920r^{8} - 92672r^{6} + 401408r^{4} - 786432r^{2} + 524288} \right)} \hfill \\ \end{array} }}{{64(r^{4} - 32r^{2} + 128)^{2} (r^{2} - 4)^{2} }} < 0; $$
  $$ \left( {x_{1}^{{{\text{SSP}}}} - q_{1}^{{{\text{SSP}}}} } \right) - \left( {x_{1}^{{{\text{CSP}}}} - q_{1}^{{{\text{CSP}}}} } \right) = - \frac{{r^{2} \left[ {32A + \left( {r^{2} - 16} \right)r\left( {B + 2k} \right)} \right]}}{{128(r^{2} - 8)}} > 0; $$
  $$ \left( {x_{2}^{{{\text{SSP}}}} - q_{2}^{{{\text{SSP}}}} } \right) - \left( {x_{2}^{{{\text{CSP}}}} - q_{2}^{{{\text{CSP}}}} } \right) = \frac{{r^{3} \left[ {32\left( {r^{2} - 16} \right)A + \left( {r^{4} - 56r^{2} + 320} \right)r\left( {B + 2k} \right)} \right]}}{{256(r^{2} - 8)(r^{2} - 16)}} < 0; $$
  $$ \pi_{1}^{{{\text{CSP}}}} - \pi_{1}^{{{\text{SSP}}}} = \frac{{r^{4} \left\{ {\begin{array}{*{20}c} {1024A^{2} \left( {r - 16} \right)^{2} + \left( {r^{6} - 80r^{4} + 1408r^{2} - 6144} \right)A\left( {B + 2k} \right)r} \\ { + \left( {r^{8} - 96r^{6} + 3712r^{4} - 40960r^{2} + 131072} \right)\left( {B + 2k} \right)^{2} r^{2} } \\ \end{array} } \right\}}}{{8192(r^{2} - 8)(r^{2} - 16)\left( {r^{4} - 24r^{2} + 128} \right)}} > 0; $$
  $$ \pi_{2}^{{{\text{CSP}}}} - \pi_{2}^{{{\text{SSP}}}} = - \frac{{r^{3} \left\{ {64\left( {r - 16} \right)^{2} A\left( {B + 2k} \right) + \left( {r^{6} - 88r^{4} + 1792r^{2} - 8192} \right)\left( {B + 2k} \right)^{2} r} \right\}}}{{512(r^{2} - 8)(r^{2} - 16)^{2} }} > 0; $$
  $$ {\text{CS}}^{{{\text{CSP}}}} - {\text{CS}}^{{{\text{SSP}}}} = \frac{{r^{2} \left\{ {\begin{array}{*{20}c} {1024A^{2} \left( {r^{6} - 64r^{4} + 1280r^{2} - 8192} \right) + 64r^{3} \left( {3r^{6} - 208r^{4} + 3584r^{2} - 16384} \right)A\left( {B + 2k} \right)r} \\ { - \left( {7r^{10} - 624r^{8} + 23040r^{6} - 304128r^{4} + 1572864r^{2} - 2621440} \right)\left( {B + 2k} \right)^{2} r^{2} } \\ \end{array} } \right\}}}{{32768\left( {r^{2} - 16} \right)^{2} \left( {r^{2} - 8} \right)^{2} }} < 0; $$
  $$ {\text{SW}}^{{{\text{CSP}}}} - {\text{SW}}^{{{\text{SSP}}}} = \frac{{r^{2} \left\{ {\begin{array}{*{20}c} {1024A^{2} \left( {5r^{6} - 192r^{4} + 2304r^{2} - 8192} \right) + 64\left( {r^{8} - 176r^{6} + 4608r^{4} - 40960r^{2} + 131072} \right)A\left( {B + 2k} \right)r} \\ { - \left( {3r^{10} - 176r^{8} + 2048r^{6} + 19456r^{4} - 393216r^{2} + 1572864} \right)\left( {B + 2k} \right)^{2} r^{2} } \\ \end{array} } \right\}}}{{32768\left( {r^{2} - 16} \right)^{2} \left( {r^{2} - 8} \right)^{2} }} < 0. $$