A game theoretic approach for pricing and advertising of an integrated product family in a duopoly

Abstract

Product family concept predicates to a products’ group or services that derive from a common base and they have usually the same production process and physical features, in different industries. In this paper, we assume two manufacturers that produce some homogeneous product families that they want to decide upon their wholesale price and national advertising expenditure, as a new problem in the multi-agent environment. According to this problem, we propose the non-cooperative, cooperative and two-stage game models to maximize manufacturers' profits and then, based on the optimization approaches i.e., Nash and Stackelberg methods, we obtain the appropriate equilibrium strategies in a special example. Also, we propose a new profit-sharing approach for the Stackelberg model of the defined problem and discuss the solutions as some propositions and marginal points, based on the numerical studies. The results show that the centralized model leads to better profit than the non-centralized and Stackelberg models.

Appendices

Appendix A

$$ B = \gamma^{4} {\text{sr}}_{ij}^{2} - \left( {4\beta_{j} - \gamma^{2} } \right)\left( {4\beta_{i} - \gamma^{2} } \right) + 4\lambda_{i} \lambda_{j} + 2\gamma^{2} {\text{sr}}_{ij} \left( {\lambda_{i} + \lambda_{j} } \right) $$

$$ B_{i} = \gamma^{4} {\text{sr}}_{ij}^{2} - \left( {6\beta_{i} - \gamma^{2} } \right)\left( {4\beta_{j} - \gamma^{2} } \right) + 2\gamma^{2} {\text{sr}}_{ij} \lambda_{j} $$

$$ B_{2} = \gamma^{2} {\text{sr}}_{ij} \beta_{j} + \left( {6\beta_{j} - \gamma^{2} } \right)\lambda_{j} $$

$$ B_{3} = \alpha_{i} \left( {4\beta_{j} - \gamma^{2} } \right) - \gamma^{2} sr_{ij} \alpha_{j} - 2\alpha_{j} \lambda_{j} $$

$$ B_{4} = 2\gamma^{2} sr_{ij} \beta_{i} + 2\left( {6\beta_{i} - \gamma^{2} } \right)\lambda_{i} $$

$$ B_{5} = \gamma^{4} sr_{ij}^{2} - \left( {6\beta_{j} - \gamma^{2} } \right)\left( {4\beta_{i} - \gamma^{2} } \right) + 2\gamma^{2} sr_{ij} \lambda_{i} $$

$$ B_{6} = \alpha_{j} \left( {4\beta_{i} - \gamma^{2} } \right) - \gamma^{2} sr_{ij} \alpha_{i} - 2\alpha_{i} \lambda_{i} $$

$$ B_{7} = \beta_{i} \left( {4\beta_{j} - \gamma^{2} } \right) + \gamma^{2} sr_{ij} \lambda_{i} + 2\lambda_{i} \lambda_{j} $$

$$ B_{8} = \gamma^{3} sr_{ij} + 2\gamma^{2} \lambda_{j} \alpha_{i} \left( {4\beta_{j} - \gamma^{2} } \right) $$

$$ B_{9} = \lambda_{i} \gamma^{2} + \gamma^{2} \beta_{i} sr_{ij} - 2\beta_{i} \lambda_{i} $$

$$ B_{10} = \beta_{j} \left( {4\beta_{i} - \gamma^{2} } \right) + \gamma^{2} sr_{ij} \lambda_{j} + 2\lambda_{i} \lambda_{ij} $$

$$ B_{11} = \gamma^{2} sr_{ij} \beta_{j} + \left( {6\beta_{j} - \gamma^{2} } \right)\lambda_{j} $$

$$ E = \left( { - \gamma^{2} \left( {1 + sr_{ij} } \right)^{2} + 2\beta_{i} + 2\beta_{j} - 2\lambda_{i} - 2\lambda_{j} } \right) $$

$$ E_{1} = 2\left( {\beta_{i} - \lambda_{j} } \right) - 2\gamma^{2} \left( {1 + sr_{ij} } \right)^{2} $$

$$ E_{2} = 4\left( {\beta_{j} - \lambda_{i} } \right) - 2\gamma^{2} \left( {1 + sr_{ij} } \right)^{2} $$

$$ E_{3} = \gamma \left( {1 + sr_{ij} } \right)\left( {\lambda_{i} - \beta_{j} } \right) $$

$$ E_{4} = \gamma^{2} \left( {1 + sr_{ij} } \right)^{2} \left( {\lambda_{j} - \beta_{i} } \right) - 2\left( {\beta_{j} - \lambda_{i} } \right)\left( {E - 2\beta_{i} + 2\lambda_{j} } \right) $$

$$ E_{5} = \alpha_{i} \left( {E + \beta_{j} - \lambda_{i} } \right) - 2c_{j} \left( {\beta_{j} - \lambda_{i} } \right)\left( {\beta_{i} - \lambda_{j} } \right) - \alpha_{j} \left( {E - 2\beta_{j} + 2\lambda_{i} } \right) $$

$$ E_{6} = \beta_{i} \left( {\gamma^{2} \left( {1 + sr_{ij} } \right)^{2} + 2\lambda_{j} } \right) - \beta_{i}^{2} - \gamma^{2} \left( {1 + sr_{ij} } \right)^{2} \left( {\beta_{j} - \lambda_{i} } \right) - \gamma^{2} \left( {1 + sr_{ij} } \right)^{2} \lambda_{j} - \left( {\lambda_{j} } \right)^{2} $$

$$ E_{7} = \alpha_{i} \left( {E - \beta_{i} + \lambda_{j} } \right) + \alpha_{j} \left( {E + 2\beta_{i} - 2\lambda_{j} } \right) $$

$$ \begin{aligned} M & = \gamma^{6} {\text{sr}}_{ij}^{4} + 4\gamma^{4} {\text{sr}}_{ij}^{3} \lambda_{j} + 4\gamma^{2} {\text{sr}}_{ij} \left( {4\beta_{j} - \gamma^{2} } \right)\left( {\lambda_{i} + \lambda_{j} } \right) - \left( {4\beta_{j} - \gamma^{2} } \right) \\ & \quad \left( {\left( {4\beta_{j} - \gamma^{2} } \right)\left( {4\beta_{i} - \gamma^{2} } \right) - 8\lambda_{i} \lambda_{j} } \right) + {\text{sr}}_{ij}^{2} \left( { - 2\gamma^{6} + 8\gamma^{4} \beta_{j} + 4\gamma^{2} \left( {\lambda_{j} } \right)^{2} } \right) \\ \end{aligned} $$

$$ M_{1} = M + 2\gamma^{2} sr_{ij} \lambda_{i} - M_{2} \left( {4\beta_{i} - \gamma^{2} } \right) + 4\lambda_{i} \lambda_{j} $$

$$ M_{2} = 4\beta_{j} - \gamma^{2} $$

$$ \begin{aligned} M_{3} & = \gamma^{6} sr_{ij}^{4} - 2sr_{ij}^{2} \left( {\gamma^{6} - 3\gamma^{4} \beta_{j} } \right) + 2\gamma^{4} sr_{ij}^{3} \lambda_{j} - 2\gamma^{2} sr_{ij} \\ & \quad \left( {2\left( {M_{2} + \beta_{j} } \right)\lambda_{i} + \left( {M_{2} + 2\beta_{j} } \right)\lambda_{j} } \right) + \left( {M_{2} + 2\beta_{j} } \right)\left( {M_{2} \left( {4\beta_{i} - \gamma^{2} } \right) - 4\lambda_{i} \lambda_{j} } \right) \\ \end{aligned} $$

$$ \begin{aligned} M_{4} & = \gamma^{4} {\text{ sr}}_{ij}^{3} \left( {\beta_{i} c_{i} - \alpha_{i} } \right) + M_{2} \left( {\left( {c_{i} \left( {2\beta_{i} - \left( {4\beta_{i} - \gamma^{2} } \right)} \right) - 2\alpha_{i} } \right)\lambda_{i} - \alpha_{j} \left( {4\beta_{i} - \gamma^{2} } \right)} \right) \\ & \quad + 4\lambda_{i} \left( {\alpha_{j} + c_{i} \lambda_{i} } \right)\lambda_{j} + \gamma^{2} {\text{sr}}_{ij}^{2} \left( {\gamma^{2} \left( {\alpha_{j} + c_{i} \lambda_{i} } \right) - 2\left( {\alpha_{i} - c_{i} \beta_{i} } \right)\lambda_{j} } \right) \\ & \quad + \gamma^{2} {\text{sr}}_{ij} ( - \alpha_{i} M_{2} + 2\alpha_{j} \left( {\lambda_{i} + \lambda_{j} } \right) + c_{i} \left( {\beta_{i} M_{2} + 2\lambda_{i} \left( {\lambda_{i} + \lambda_{j} } \right)} \right) \\ \end{aligned} $$

$$ M_{5} = \gamma^{2} sr_{ij}^{2} - \gamma^{2} + 4\beta_{j} + 2sr_{ij} \lambda_{j} $$

$$ M_{6} = c_{i} \beta_{i} M_{2} - \alpha_{i} M_{2} - \gamma^{2} {\text{sr}}_{ij} \left( {\alpha_{j} - c_{j} \beta_{j} + c_{i} \lambda_{i} } \right) + \left( {c_{j} \left( { - M_{2} + 2\beta_{j} } \right) - 2\left( {\alpha_{j} + c_{i} \lambda_{i} } \right)} \right)\lambda_{j} $$

$$ \begin{aligned} M_{7} & = \left( {2{\upalpha }_{i} - {\text{c}}_{i} { }\left( {2{\upbeta }_{i} - \left( {4\beta_{i} - \gamma^{2} { }} \right)} \right)} \right){\uplambda }_{i} ) - \left( {4\beta_{i} - \gamma^{2} } \right)\left( {c_{j} \beta_{j} - \alpha_{j} } \right) \\ & \quad + 2{\uplambda }_{{\text{i}}} { }\left( {c_{j} { }\left( { - M_{2} - 2\beta_{j} } \right) + 2\left( {\alpha_{j} + {\text{c}}_{i} {\uplambda }_{i} } \right)} \right)\lambda_{j} - {\upgamma }^{4} {\text{sr}}_{{{\text{ij}}}}^{3} \left( {{\upalpha }_{i} - {\text{c}}_{i} {\upbeta }_{{\text{i}}} + c_{j} \lambda_{j} } \right) + {\upgamma }^{2} {\text{sr}}_{ij}^{2} { }({\upgamma }^{2} { }(\alpha_{j} \\ & \quad - c_{j} \beta_{j} + {\text{c}}_{i} {\uplambda }_{i} ) - 2\left( {{\upalpha }_{i} - {\text{c}}_{i} {\upbeta }_{i} } \right)\lambda_{j} - 2c_{j} \left( {\lambda_{j} } \right)^{2} ) + {\upgamma }^{2} {\text{sr}}_{ij} { }(2\left( {\alpha_{j} - c_{j} \beta_{j} } \right){\uplambda }_{i} - {\upalpha }_{i} M_{2} + (2\alpha_{j} \\ & \quad + c_{j} { }\left( {6\beta_{j} - M_{2} } \right))\lambda_{j} + c_{i} \left( {\beta_{i} M_{2} + 2\lambda_{i} \left( {\lambda_{i} + \lambda_{j} } \right)} \right) \\ \end{aligned} $$

Appendix B

2.1 Proof of Proposition 1

To prove that the obtained equilibrium strategies for the non-centralized game will maximize the total profit of manufacturers, we calculate the first and second partial derivatives of Eqs. (14) and (15), respectively, and solve them simultaneously, therefore, for the simple proposed example we have:

– Product $i$ :

$$ \frac{{\partial \pi_{1} \left( {w_{i} ,A_{i} } \right)}}{{\partial w_{i} }} = \gamma \left( {A_{i} + A_{j} {\text{sr}}_{ij} } \right) + \alpha_{i} - w_{i} \beta_{i} - \left( {w_{i} - c_{i} } \right)\beta_{i} + w_{j} \lambda_{j} = 0 $$

We must have a negative definite matrix for maximizing the profit function of the first manufacturer:

$$ \left| {\frac{{\partial^{2} \pi_{1} }}{{\partial w_{i}^{2} }}} \right| = \left| { - 2\beta_{i} } \right| = - 2\beta_{i} < 0 $$

– Product $j$ :

$$ \frac{{\partial \pi_{1} \left( {w_{j} ,A_{j} } \right)}}{{\partial w_{j} }} = \gamma \left( {A_{j} + A_{i} {\text{sr}}_{ij} } \right) + \alpha_{j} - w_{j} \beta_{j} - \left( {w_{j} - c_{j} } \right)\beta_{j} + p_{i} \lambda_{i} = 0 $$

Similar to the product $i$, If we must have a negative definite matrix for maximizing the profit function of the second manufacturer:

$$ \left| {\frac{{\partial^{2} \pi_{1} }}{{\partial \left( {w_{j} } \right)^{2} }}} \right| = \left| { - 2\beta_{j} } \right| = - 2\beta_{j} < 0. $$

Now, considering the partial derivatives and solving the system, we have:

$$ \begin{aligned} w_{i} & = \frac{{c_{i} \left( {\gamma^{4} {\text{sr}}_{ij}^{2} - \left( {\gamma^{2} - 2\beta_{i} } \right)\left( {\gamma^{2} - 4\beta_{j} } \right) + 2\gamma^{2} {\text{sr}}_{ij} \lambda_{j} } \right) + 2\left( {\alpha_{i} \left( {\gamma^{2} - 4\beta_{j} } \right) + \gamma^{2} {\text{sr}}_{ij} \left( { - \alpha_{j} + c_{j} \beta_{j} } \right) + \left( { - 2\alpha_{j} + c_{j} \left( {\gamma^{2} - 2\beta_{j} } \right)} \right)\lambda_{j} } \right)}}{{\gamma^{4} {\text{sr}}_{ij}^{2} - \left( {\gamma^{2} - 4\beta_{i} } \right)\left( {\gamma^{2} - 4\beta_{j} } \right) + 4\lambda_{i} \lambda_{j} + 2\gamma^{2} {\text{sr}}_{ij} \left( {\lambda_{i} + \lambda_{j} } \right)}} \\ w_{j} & = \frac{{c_{j} \left( {\gamma^{4} {\text{sr}}_{ij}^{2} - \left( {\gamma^{2} - 4\beta_{i} } \right)\left( {\gamma^{2} - 2\beta_{j} } \right) + 2\gamma^{2} {\text{sr}}_{ij} \lambda_{i} } \right) + 2\left( {\alpha_{j} \left( {\gamma^{2} - 4\beta_{i} } \right) + \gamma^{2} {\text{sr}}_{ij} \left( { - \alpha_{i} + c_{i} \beta_{i} } \right) + \left( { - 2\alpha_{i} + c_{i} \left( {\gamma^{2} - 2\beta_{i} } \right)} \right)\lambda_{i} } \right)}}{{\gamma^{4} {\text{sr}}_{ij}^{2} - \left( {\gamma^{2} - 4\beta_{i} } \right)\left( {\gamma^{2} - 4\beta_{j} } \right) + 4\lambda_{i} \lambda_{j} + 2\gamma^{2} {\text{sr}}_{ij} \left( {\lambda_{i} + \lambda_{j} } \right)}} \\ \end{aligned} $$

□

2.2 Proof of Proposition 2

Similar to the proposition 1, we calculate the first and second partial derivatives of Eqs. (14) and (15), respectively, and solve them simultaneously, therefore, for the simple proposed example we have:

– Product $i$ :

$$ \frac{{\partial \pi_{1} \left( {w_{i} ,A_{i} } \right)}}{{\partial A_{i} }} = \gamma \left( {w_{i} - c_{i} } \right) - 2A_{i} = 0 $$

Moreover, for hessian matrix we have: $\left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{1} }}{{\partial w_{i}^{2} }}} & {\frac{{\partial^{2} \pi_{1} }}{{\partial A_{i} \partial w_{i} }}} \\ {\frac{{\partial^{2} \pi_{1} }}{{\partial w_{i} \partial A_{i} }}} & {\frac{{\partial^{2} \pi_{1} }}{{\partial A_{i}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2\beta_{i} } & \gamma \\ \gamma & { - 2} \\ \end{array} } \right]$.

Now, if we must have a negative definite matrix to maximize the profit function of the first manufacturer:

$$ \left| {\frac{{\partial^{2} \pi_{1} }}{{\partial A_{i}^{2} }}} \right| = \left| { - 2} \right| = - 2 < 0 $$

$\left| {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{1} }}{{\partial w_{i}^{2} }}} & {\frac{{\partial^{2} \pi_{1} }}{{\partial A_{i} \partial w_{i} }}} \\ {\frac{{\partial^{2} \pi_{1} }}{{\partial w_{i} \partial A_{i} }}} & {\frac{{\partial^{2} \pi_{1} }}{{\partial A_{i}^{2} }}} \\ \end{array} } \right| = \left| {\begin{array}{*{20}c} { - 2\beta_{i} } & \gamma \\ \gamma & { - 2} \\ \end{array} } \right| = 4\beta_{i} - \gamma^{2} > 0 \Rightarrow 0 < {\upgamma } < 2\sqrt {\beta_{i} }$.

– Product $j$ :

$$ \frac{{\partial \pi_{2} \left( {w_{j} ,A_{i} } \right)}}{{\partial A_{j} }} = \gamma \left( {w_{j} - c_{j} } \right) - 2A_{j} = 0 $$

For hessian matrix we have: $\left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{2} }}{{\partial \left( {w_{j} } \right)^{2} }}} & {\frac{{\partial^{2} \pi_{2} }}{{\partial A_{j} \partial w_{j} }}} \\ {\frac{{\partial^{2} \pi_{2} }}{{\partial w_{j} \partial A_{j} }}} & {\frac{{\partial^{2} \pi_{2} }}{{\partial \left( {A_{j} } \right)^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2\beta_{j} } & \gamma \\ \gamma & { - 2} \\ \end{array} } \right]$.

Similar to the product $i$, If we must have a negative definite matrix for maximizing the profit function of the second manufacturer:

$$ \left| {\frac{{\partial^{2} \pi }}{{\partial \left( {A_{j} } \right)^{2} }}} \right| = \left| { - 2} \right| = - 2 < 0 $$

$$ \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{2} }}{{\partial \left( {w_{j} } \right)^{2} }}} & {\frac{{\partial^{2} \pi_{2} }}{{\partial A_{j} \partial w_{j} }}} \\ {\frac{{\partial^{2} \pi_{2} }}{{\partial w_{j} \partial A_{j} }}} & {\frac{{\partial^{2} \pi_{2} }}{{\partial \left( {A_{j} } \right)^{2} }}} \\ \end{array} } \right| = \left| {\begin{array}{*{20}c} { - 2\beta_{j} } & \gamma \\ \gamma & { - 2} \\ \end{array} } \right| = 4\beta_{j} - \gamma^{2} > 0 \Rightarrow 0 < {\upgamma } < 2\sqrt {\beta_{j} } . $$

With respect the first obtained relation ($0 < {{\upgamma }} < 2\sqrt {\beta _{i} }$) and present relation, we have:

$$ 0 < {\upgamma } < 2\sqrt {\min \left( {\beta_{i} ,\beta_{j} } \right)} $$

Now, with considering the partial derivatives Eqs. and solving the system of Eqs., we have:

$$ A_{i} = \frac{{\gamma \left( {\alpha_{i} \left( {\gamma^{2} - 4\beta_{j} } \right) - c_{i} \beta_{i} \left( {\gamma^{2} - 4\beta_{j} } \right) - \gamma^{2} {\text{sr}}_{ij} \left( {\alpha_{j} - c_{j} \beta_{j} + c_{i} \lambda_{i} } \right) + \left( {c_{j} \left( {\gamma^{2} - 2\beta_{j} } \right) - 2\left( {\alpha_{j} + c_{i} \lambda_{i} } \right)} \right)\lambda_{j} } \right)}}{{\gamma^{4} {\text{sr}}_{ij}^{2} - \left( {\gamma^{2} - 4\beta_{i} } \right)\left( {\gamma^{2} - 4\beta_{j} } \right) + 4\lambda_{i} \lambda_{j} + 2\gamma^{2} {\text{sr}}_{ij} \left( {\lambda_{i} + \lambda_{j} } \right)}} $$

$$ A_{j} = \frac{{ - \gamma \left( { - \alpha_{j} \left( {\gamma^{2} - 4\beta_{i} } \right) + \gamma^{2} c_{j} \beta_{j} - 4c_{j} \beta_{i} \beta_{j} - \gamma^{2} c_{i} \lambda_{i} + 2\alpha_{i} \lambda_{i} + 2c_{i} \beta_{i} \lambda_{i} + 2c_{j} \lambda_{i} \lambda_{j} + \gamma^{2} {\text{sr}}_{ij} \left( {\alpha_{i} - c_{i} \beta_{i} + c_{j} \lambda_{j} } \right)} \right)}}{{\gamma^{4} {\text{sr}}_{ij}^{2} - \left( {\gamma^{2} - 4\beta_{i} } \right)\left( {\gamma^{2} - 4\beta_{j} } \right) + 4\lambda_{i} \lambda_{j} + 2\gamma^{2} {\text{sr}}_{ij} \left( {\lambda_{i} + \lambda_{j} } \right)}} $$

□

2.3 Proof of Proposition 3

We have the same argument for the centralized model and therefore, we calculate the first and second partial derivatives of Eq. (18) and solve them simultaneously as follow:

$$ \begin{aligned} \frac{{\partial \pi_{c} \left( {w_{ij}^{c} ,A_{ij}^{c} } \right)}}{{\partial w_{ij}^{c} }} & = 2\gamma A_{ij}^{c} \left( {1 + {\text{sr}}_{ij} } \right) + \alpha_{i} + \alpha_{j} + w_{ij}^{c} \left( {\lambda_{i} - \beta_{j} } \right) + \left( {w_{ij}^{c} - c_{j} } \right)\left( {\lambda_{i} - \beta_{j} } \right) \\ & \quad + w_{ij}^{c} \left( {\lambda_{j} - \beta_{i} } \right) + \left( {w_{ij}^{c} - c_{i} } \right)\left( {\lambda_{j} - \beta_{i} } \right) = 0 \\ \end{aligned} $$

Similar to the previous section, if we must have a negative definite matrix to maximize the total profit of the manufacturers:

Proposition 1. $\left| {\frac{{\partial^{2} \pi_{c} }}{{\partial \left( {w_{ij}^{c} } \right)^{2} }}} \right| = \left| { - 2\beta_{i} - 2\beta_{j} + 2\lambda_{i} + 2\lambda_{j} } \right| = 2\lambda_{i} + 2\lambda_{j} - 2\beta_{i} - 2\beta_{j} < 0$

Proposition 2. $\Rightarrow \lambda_{i} + \lambda_{j} < \beta_{i} + \beta_{j}$.

Now, with considering the partial derivatives Eqs. and solving the system of Eqs., we have:

Proposition 3. $w_{ij}^{c} = - \frac{{2\gamma^{2} \left( {c_{i} + c_{j} } \right)\left( {1 + {\text{sr}}_{ij} } \right)^{2} - 4\left( {\alpha_{i} + \alpha_{j} + c_{j} \left( {\beta_{j} - \lambda_{i} } \right) + c_{i} \left( {\beta_{i} - \lambda_{j} } \right)} \right)}}{{4\left( { - \gamma^{2} \left( {1 + {\text{sr}}_{ij} } \right)^{2} + 2\beta_{i} + 2\beta_{j} - 2\lambda_{i} - 2\lambda_{j} } \right)}}$ □

2.4 Proof of Proposition 4

We have the same argument for the centralized model and therefore, we calculate the first and second partial derivatives of Eq. (18) and solve them simultaneously as follow:

$$ \frac{{\partial \pi_{c} \left( {w_{ij}^{c} ,A_{ij}^{c} } \right)}}{{\partial A_{ij}^{c} }} = - 4A_{ij}^{c} + \gamma \left( {w_{ij}^{c} - c_{i} } \right)\left( {1 + {\text{sr}}_{ij} } \right) + \gamma \left( { - c_{j} + w_{ij}^{c} } \right)\left( {1 + {\text{sr}}_{ij} } \right) = 0 $$

Hessian matrix: $\left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{c} }}{{\partial \left( {w_{ij}^{c} } \right)^{2} }}} & {\frac{{\partial^{2} \pi_{c} }}{{\partial A_{ij}^{c} \partial w_{ij}^{c} }}} \\ {\frac{{\partial^{2} \pi_{c} }}{{\partial w_{ij}^{c} \partial A_{ij}^{c} }}} & {\frac{{\partial^{2} \pi_{c} }}{{\partial \left( {A_{ij}^{c} } \right)^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2\beta_{i} - 2\beta_{j} + 2\lambda_{i} + 2\lambda_{j} } & {2\gamma \left( {1 + {\text{sr}}_{ij} } \right)} \\ {2\gamma \left( {1 + {\text{sr}}_{ij} } \right)} & { - 4} \\ \end{array} } \right]$.

Similar to the previous section, if we must have a negative definite matrix to maximize the total profit of the manufacturers:

Proposition 4. $\left| {\frac{{\partial^{2} \pi_{c} }}{{\partial \left( {A_{ij}^{c} } \right)^{2} }}} \right| = \left| { - 4} \right| = - 4 < 0$

Proposition 5. $\left| {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{c} }}{{\partial \left( {w_{ij}^{c} } \right)^{2} }}} & {\frac{{\partial^{2} \pi_{c} }}{{\partial A_{ij}^{c} \partial w_{ij}^{c} }}} \\ {\frac{{\partial^{2} \pi_{c} }}{{\partial w_{ij}^{c} \partial A_{ij}^{c} }}} & {\frac{{\partial^{2} \pi_{c} }}{{\partial \left( {A_{ij}^{c} } \right)^{2} }}} \\ \end{array} } \right| = \left| {\begin{array}{*{20}c} {2\lambda_{i} + 2\lambda_{j} - 2\beta_{i} - 2\beta_{j} } & {2\gamma \left( {1 + {\text{sr}}_{ij} } \right)} \\ {2\gamma \left( {1 + {\text{sr}}_{ij} } \right)} & { - 4} \\ \end{array} } \right| = 8\beta_{i} + 8\beta_{j} - 4\gamma^{2} - 8\gamma^{2} {\text{sr}}_{ij} - 4\gamma^{2} {\text{sr}}_{ij}^{2} - 8\lambda_{i} - 8\lambda_{j} > 0$$\Rightarrow 2\beta_{i} + 2\beta_{j} > \gamma^{2} + 2\gamma^{2} {\text{sr}}_{ij} + \gamma^{2} {\text{sr}}_{ij}^{2} + 2\lambda_{i} + 2\lambda_{j}$.

Now, with considering the partial derivatives Eqs. and solving the system of Eqs., we have:

$$ A_{ij}^{c} = - \frac{{\gamma \left( {1 + {\text{sr}}_{ij} } \right)\left( {\alpha_{i} + \alpha_{j} + c_{i} \left( { - \beta_{j} + \lambda_{i} } \right) + c_{j} \left( { - \beta_{i} + \lambda_{j} } \right)} \right)}}{{2\left( {\gamma^{2} + \gamma^{2} {\text{sr}}_{ij} \left( {2 + {\text{sr}}_{ij} } \right) - 2\beta_{i} - 2\beta_{j} + 2\lambda_{i} + 2\lambda_{j} } \right)}}. $$

□