Differential game of product–service supply chain considering consumers’ reference effect and supply chain members’ reciprocity altruism in the online-to-offline mode

Abstract

Supply chain members’ reciprocal altruism and consumers’ quality and service reference effects are important behavioral factors that affect the decision-making of supply chain members. This article incorporates these factors into a product–service supply chain consisting of a manufacturer and a retailer in the online-to-offline (O2O) environment. Based on the inherent dynamics of the model, we construct a differential game model between the manufacturer and the retailer. Based on the Bellman continuous dynamic programming theory, this study analyzes the quality strategy of the manufacturer, the service level strategy of the retailer, and the performance of the supply chain system under three decision-making patterns (decentralization, centralization, and reciprocal altruism) within the O2O framework. The results show that compared with the decentralized decision-making model, reciprocal altruism helps members develop higher quality and service levels, improve brand goodwill, and obtain greater utility. The results are verified by numerical examples, and sensitivity analysis of consumer quality and the service reference effect, channel preference, and members’ reciprocal altruism behavior on the supply chain performance is carried out. The results show: (1) Consumers’ reference effects cause an “anchoring mentality” among consumers, which leads the manufacturer to lower the quality level and the retailer to lower the service level. This hurts the performance of the product–service supply chain. Consumers’ channel preference has an important impact on supply chain members’ strategies and performance. (2) Retailers should encourage consumers to purchase products online and use offline channel services as sales assistance measures to satisfy consumers’ experience utility. (3) As a positive social preference, the supply chain performance under the members’ reciprocal altruism decision-making model is Pareto-improved and receives additional social benefits. (4) Only when the manufacturer and the retailer have pure altruistic preference, that is, minimum return and maximum altruism, the total profit of the supply chain can reach that of the centralized decision-making scenario.

Appendix

1.1 Proof of Proposition 1

For simplicity, time argument $t$ is omitted when there is no controversy. According to Bellman continuous dynamic programming theory, for any state $G^{C} \ge 0$, there exists a continuously differentiable value function $V^{C}$ satisfying Hamilton-Jacobi-Bellman (HJB)function.

$$ rV^{C} = \mathop {\max }\limits_{{q^{C} ,s^{C} }} \left\{ {\left( {\rho_{1} + \rho_{2} } \right)\left[ {\chi \left( {\alpha \left( {q^{C} - \xi_{1} G^{C} } \right) + \theta G^{C} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s^{C} - \xi_{2} G^{C} } \right) + \theta G^{C} } \right)} \right] - \frac{1}{2}k_{1} \left( {q^{C} } \right)^{2} - \frac{1}{2}k_{2} \left( {s^{C} } \right)^{2} + V^{C^{\prime}} \left( {\gamma_{1} q^{C} + \gamma_{2}^{C} s - \delta G^{C} } \right)} \right\} $$

(11)

where $V^{C}$ is the value function of the supply chain system which represents the total profit during the planning horizon. $V^{C\prime } = \tfrac{{\partial V^{C} }}{{\partial G^{C} }}$ represents the first order derivative of the value function $V^{C}$. According to optimal first order condition, the feedback quality strategy of manufacturer and feedback price strategy of the retailer are as follows:

$$ q^{C} (G^{C} ) = \frac{1}{{k_{1} }}\left[ {\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} V^{C\prime } } \right] $$

(12)

$$ s^{C} (G^{C} ) = \frac{1}{{k_{2} }}\left[ {\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V^{C\prime } } \right] $$

(13)

To further identify the explicit form of value function, we substitute (12) and (13) into (11) to get HJB function.

$$ \begin{aligned} rV^{C} (G^{C} ) = & \left( {\rho_{1} + \rho_{2} } \right)\left[ \begin{gathered} \chi \left( {\alpha \left( {\frac{1}{{k_{1} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} V^{C} } \right) - \xi_{1} G^{C} } \right) + \theta G^{C} } \right) \hfill \\ + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{1}{{k_{2} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V^{C} } \right) - \xi_{2} G^{C} } \right) + \theta G^{C} } \right) \hfill \\ \end{gathered} \right] \\ & - \frac{1}{{2k_{1} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} V^{C} } \right)^{2} - \frac{1}{{2k_{2} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V^{C} } \right)^{2} \\ & + \frac{{\partial V^{C} }}{{\partial G^{C} }}\left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} V^{C} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V^{C} } \right) - \delta G^{C} } \right] \\ \end{aligned} $$

(14)

Based on the structure above, we assume the value function to be $V^{C} = l_{1} G^{C} + l_{2}$, where $l_{1}$ and $l_{2}$ are the undetermined coefficient. Substitute the value function and its first order derivative into (14) and we get the following equation sets of the undetermined the coefficients. $\left\{ \begin{gathered} rl_{1} = \left( {\rho_{1} + \rho_{2} } \right)\left[ {\chi \left( { - \alpha \xi_{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi_{2} + \theta } \right)} \right] - l_{1} \delta \hfill \\ rl_{2} = \left( {\rho_{1} + \rho_{2} } \right)\left[ {\chi \alpha \left( {\frac{{\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} l_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} l_{1} }}{{k_{2} }}} \right)} \right] \hfill \\ \, - \frac{1}{{2k_{1} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} l_{1} } \right)^{2} - \frac{1}{{2k_{2} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} l_{1} } \right)^{2} \hfill \\ { + }l_{1} \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha + \gamma_{1} l_{1} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} l_{1} } \right)} \right] \hfill \\ \end{gathered} \right.$

Solve the equation sets above we get the value of coefficient

$$ \left\{ \begin{gathered} l_{1} = \frac{{\left( {\rho_{1} + \rho_{2} } \right)M}}{r + \delta } \hfill \\ l_{2} = \left( {\frac{{\left( {\rho_{1} + \rho_{2} } \right)^{2} \gamma_{1}^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\left( {\rho_{1} + \rho_{2} } \right)^{2} \gamma_{2}^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{2} }}} \right)M^{2} { + }\left( {\frac{{\left( {\rho_{1} + \rho_{2} } \right)^{2} \gamma_{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\rho_{1} + \rho_{2} } \right)^{2} \gamma_{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right)M \hfill \\ \, + \frac{{\left( {\rho_{1} + \rho_{2} } \right)^{2} \left( {\chi \alpha } \right)^{2} }}{{2rk_{1} }} + \frac{{\left( {\rho_{1} + \rho_{2} } \right)^{2} \left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{2rk_{2} }} \hfill \\ \end{gathered} \right. $$

(15)

where ${\rm M} = \chi \left( { - \alpha \xi_{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi_{2} + \theta } \right)$.

Substitute (14) into (11)and (12) and we can get the optimal strategies of supply chain members.

$$ \begin{gathered} q^{C} = \frac{{\left( {\rho_{1} + \rho_{2} } \right)\chi \alpha \left( {r + \delta } \right) + \left( {\rho_{1} + \rho_{2} } \right)\gamma_{1} {\rm M}}}{{\left( {r + \delta } \right)k_{1} }} \hfill \\ s^{C} = \frac{{\left( {\rho_{1} + \rho_{2} } \right)\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \left( {\rho_{1} + \rho_{2} } \right)\gamma_{2} {\rm M}}}{{\left( {r + \delta } \right)k_{2} }} \hfill \\ \end{gathered} $$

(16)

Substitute (16) into the dynamic equation of goodwill and we have

$$ \dot{G}^{C} (t) = \frac{{\left( {\rho_{1} + \rho_{2} } \right)\gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\rho_{1} + \rho_{2} } \right)\gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }} - \delta G^{C} (t), \, G^{C} (0) = G_{0} $$

(17)

Solve the differential equation above, we can get time trajectory of goodwill under centralized patter. Substitute (15) into value function and we can get the total profit of supply chain system of this pattern. According to (2) and (3) we can get the time trajectory of quality and service reference level. Proposition 1 proved.

1.2 Proof of proposition 2

For any state $G^{N} \ge 0$, there exists a continuous differentiable value function $V_{i}^{N} ,i = M,R$ which satisfies HJB function.

$$ rV_{M}^{N} = \mathop {\max }\limits_{{q^{N} }} \left\{ {\rho_{1} \left[ {\chi \left( {\alpha \left( {q^{N} - \xi_{1} G^{N} } \right) + \theta G^{N} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s - \xi_{2} G^{N} } \right) + \theta G^{N} } \right)} \right] - \frac{{k_{1} }}{2}\left( {q^{N} } \right)^{2} + V_{M}^{N} \left[ {\gamma_{1} q^{N} + \gamma_{2} s^{N} - \delta G^{N} } \right]} \right\} $$

(18)

$$ rV_{R}^{N} = \mathop {\max }\limits_{{s^{N} }} \left\{ {\rho_{2} \left[ {\chi \left( {\alpha \left( {q^{N} - \xi_{1} G^{N} } \right) + \theta G^{N} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s - \xi_{2} G^{N} } \right) + \theta G^{N} } \right)} \right] - \frac{{k_{2} }}{2}\left( {s^{N} } \right)^{2} + V_{R}^{N^{\prime}} \left[ {\gamma_{1} q^{N} + \gamma_{2} s^{N} - \delta G^{N} } \right]} \right\} $$

(19)

In (18) and (19), $V_{M}^{N}$ and $V_{R}^{N}$ represent the value function of manufacturer and retailer respectively, and indicate the total profit during the planning horizon. $V_{M}^{N^{\prime}} = \tfrac{{\partial V_{M}^{N} }}{{\partial G^{N} }}$ and $V_{R}^{N} { = }\tfrac{{\partial V_{R}^{N} }}{{\partial G^{N} }}$ represent the derivatives of value functions of manufacturer and retailer respectively. According to optimal first order condition, we can get the feedback quality strategy of manufacturer and feedback service strategy of the retailer.

$$ q^{N} (G^{N} ) = \frac{1}{{k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} V_{M}^{N} } \right) $$

(20)

$$ s(G^{N} ) = \frac{1}{{k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{N^{\prime}} } \right) $$

(21)

Substitute (20) and (21) into (18) and (19), we can get HJB equation

$$ \begin{gathered} rV_{M}^{N} (G^{N} ) = \rho_{1} \left[ {\chi \left( {\alpha \left( {\frac{{\rho_{1} \chi \alpha + \gamma_{1} V_{M}^{N^{\prime}} }}{{k_{1} }} - \xi_{1} G^{N} } \right) + \theta G^{N} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{{\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{N^{\prime}} }}{{k_{2} }} - \xi_{2} G^{N} } \right) + \theta G^{N} } \right)} \right] \hfill \\ \, - \frac{1}{{2k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} V_{M}^{N^{\prime}} } \right)^{2} + V_{M}^{N^{\prime}} \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} V_{M}^{N^{\prime}} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{N^{\prime}} } \right) - \delta G^{N} } \right] \hfill \\ rV_{R}^{N} (G^{N} ) = \rho_{2} \left[ {\chi \left( {\alpha \left( {\frac{{\rho_{1} \chi \alpha + \gamma_{1} V_{M}^{N^{\prime}} }}{{k_{1} }} - \xi_{1} G^{N} } \right) + \theta G^{N} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{{\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} V_{M}^{N^{\prime}} }}{{k_{2} }} - \xi_{2} G^{N} } \right) + \theta G^{N} } \right)} \right] \hfill \\ \, - \frac{1}{{2k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{N^{\prime}} } \right)^{2} + V_{R}^{N^{\prime}} \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} V_{M}^{N^{\prime}} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{N^{\prime}} } \right) - \delta G^{N} } \right] \hfill \\ \end{gathered} $$

(22)

According to the structure of (22), the value functions of manufacturer and retailer can be expressed as $V_{M}^{N} = f_{1} G^{N} + f_{2}$, $V_{R}^{N} = g_{1} G^{N} + g_{2}$ where $f_{1} ,f_{2} ,g_{1} ,g_{2} > 0$ are undetermined coefficients of value functions. Substitute the value function and its first order derivative into (22) and we can get the non-linear equation sets of the undetermined coefficients.

$$ \left\{ \begin{gathered} rf_{1} = \rho_{1} \left[ {\chi \left( { - \alpha \xi_{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi_{2} + \theta } \right)} \right] - f_{1} \delta \hfill \\ rf_{2} = \rho_{1} \left[ {\chi \alpha \left( {\frac{{\rho_{1} \chi \alpha + \gamma_{1} f_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} g_{1} }}{{k_{2} }}} \right)} \right] - \frac{1}{{2k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} f_{1} } \right)^{2} + f_{1} \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} f_{1} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} g_{1} } \right)} \right] \hfill \\ rg_{1} = \rho_{2} \left[ {\chi \left( { - \alpha \xi_{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi_{2} + \theta } \right)} \right] - g_{1} \delta \hfill \\ rg_{2} = \rho_{2} \left[ {\chi \alpha \left( {\frac{{\rho_{1} \chi \alpha + \gamma_{1} f_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} g_{1} }}{{k_{2} }}} \right)} \right] - \frac{1}{{2k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} g_{1} } \right)^{2} + g_{1} \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\rho_{1} \chi \alpha + \gamma_{1} f_{1} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\rho_{2} \left( {1 - \chi } \right)\beta + \gamma_{2} g_{1} } \right)} \right] \, \hfill \\ \end{gathered} \right. $$

solve the equations above and we can get the explicit expression of coefficient

$$ \left\{ \begin{gathered} f_{1} = \frac{{\rho_{1} }}{r + \delta }{\rm M} \hfill \\ f_{2} = \left( {\frac{{\rho_{1}^{2} \gamma_{1}^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\rho_{1} \rho_{2} \gamma_{2}^{2} }}{{r\left( {r + \delta } \right)^{2} k_{2} }}} \right){\rm M}^{2} + \left( {\frac{{\rho_{1}^{2} \gamma_{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + \frac{{2\rho_{1} \rho_{2} \gamma_{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right){\rm M} + \frac{{\rho_{1}^{2} \left( {\chi \alpha } \right)^{2} }}{{2rk_{1} }} + \frac{{\rho_{1} \rho_{2} \left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{rk_{2} }} \hfill \\ g_{1} = \frac{{\rho_{2} }}{r + \delta }{\rm M} \hfill \\ g_{2} = \left( {\frac{{\rho_{1} \rho_{2} \gamma_{1}^{2} }}{{r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\rho_{2}^{2} \gamma_{2}^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{2} }}} \right){\rm M}^{2} + \left( {\frac{{2\rho_{1} \rho_{2} \gamma_{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + \frac{{\rho_{2}^{2} \gamma_{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right){\rm M} + \frac{{\rho_{1} \rho_{2} \left( {\chi \alpha } \right)^{2} }}{{rk_{1} }} + \frac{{\rho_{2}^{2} \left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{2rk_{2} }} \hfill \\ \end{gathered} \right. $$

(23)

Substitute (23) into (20) and (21) and we can get the optimal strategies of supply chain members.

$$ \begin{gathered} q^{N} = \frac{{\rho_{1} \chi \alpha \left( {r + \delta } \right) + \rho_{1} \gamma_{1} {\rm M}}}{{\left( {r + \delta } \right)k_{1} }} \hfill \\ s^{N} = \frac{{\rho_{2} \left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \rho_{2} \gamma_{2} {\rm M}}}{{\left( {r + \delta } \right)k_{2} }} \hfill \\ \end{gathered} $$

(24)

Substitute (24) into the dynamic equation of the objective functional and we can get the differential equation of goodwill

$$ \dot{G}^{N} (t) = \frac{{\rho_{1} \gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\rho_{2} \gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }} - \delta G^{N} (t), \, G^{N} (0) = G_{0} $$

(25)

Solve the differential equation above and we can get the time trajectory of goodwill under N patter. Substitute the goodwill and the coefficients (23) into value function and we can get the total profit of manufacturer and retailer of this pattern. According to (2) and (3), we can get the time trajectory of quality and service reference level. Proposition 2 proved.

1.3 Proof of proposition 3

By backward induction, we firstly determined the pricing strategy of retailer. According to Bellman continuous dynamic programming theory, for any state $G^{N} \ge 0$, there exists a continuous differentiable value function $V_{R}^{A}$ satisfying Hamilton–Jacobi-Bellman(HJB)function.

$$ rV_{R}^{A} = \mathop {\max }\limits_{{s^{A} }} \left\{ \begin{gathered} \rho_{2} \left[ {\chi \left( {\alpha \left( {q^{A} - \xi_{1} G} \right) + \theta G^{A} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s^{A} - \xi_{2} G^{A} } \right) + \theta G^{A} } \right)} \right] - \frac{{k_{2} }}{2}\left( {s^{A} } \right)^{2} \hfill \\ + \phi_{R} \left[ {\rho_{1} \left[ {\chi \left( {\alpha \left( {q - \xi_{1} G^{A} } \right) + \theta G^{A} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s^{A} - \xi_{2} G^{A} } \right) + \theta G^{A} } \right)} \right] - \frac{{k_{1} }}{2}\left( {q^{A} } \right)^{2} } \right] \hfill \\ + V_{R}^{A^{\prime}} \left[ {\gamma_{1} q^{A} + \gamma_{2} s^{A} - \delta G^{A} } \right] \hfill \\ \end{gathered} \right\} $$

(26)

where $V_{R}^{A}$ is the value function of retailer representing the total utility of retailer under altruistic and reciprocal pattern.$V_{R}^{A^{\prime}} = \tfrac{{\partial V_{R}^{A} }}{{\partial G^{A} }}$ is the first order derivative of $G^{A}$. According to first order optimal condition, we can get the optimal reaction function of retailer’s service

$$ s^{A} = \frac{1}{{k_{2} }}\left[ {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime}} } \right] $$

(27)

(27) excludes the quality strategy of the manufacturer, which means that the optimal reaction function of service also belongs to the manufacturer. Substitute (27) into the manufacturer’s objective functional and we can get the HJB equation of manufacturer.

$$ rV_{M}^{A} = \mathop {\max }\limits_{{q^{A} }} \left\{ \begin{gathered} \rho_{1} \left[ {\chi \left( {\alpha \left( {q^{A} - \xi_{1} G^{A} } \right) + \theta G^{A} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s^{A} - \xi_{2} G^{A} } \right) + \theta G^{A} } \right)} \right] - \frac{{k_{1} }}{2}\left( {q^{A} } \right)^{2} \hfill \\ + \phi_{M} \left[ {\rho_{2} \left[ {\chi \left( {\alpha \left( {q^{A} - \xi_{1} G^{A} } \right) + \theta G^{A} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {s - \xi_{2} G^{A} } \right) + \theta G^{A} } \right)} \right] - \frac{{k_{2} }}{2}\left( {s^{A} } \right)^{2} } \right] \hfill \\ + V_{M}^{A^{\prime}} \left[ {\gamma_{1} q^{A} + \gamma_{2} s^{A} - \delta G^{A} } \right] \hfill \\ \end{gathered} \right\} $$

(28)

where $V_{M}^{A}$ is the value function of a manufacturer and represents the total utility of manufacturer under altruistic and reciprocal pattern. According to the optimal first order condition, we can get the feedback quality strategy of the manufacturer.

$$ q^{A} = \frac{1}{{k_{1} }}\left[ {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime}} } \right] $$

(29)

To acquire the explicit expression of manufacturer’s and retailer’s value function, we substitute (27) and (29) into (26) and (28)

$$ \begin{gathered} rV_{M}^{A} = \rho_{1} \left[ {\chi \left( {\alpha \left( {\frac{1}{{k_{1} }}\left( {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime} } } \right) - \xi_{1} G} \right) + \theta G} \right) + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{1}{{k_{2} }}\left( {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right) - \xi_{2} G} \right) + \theta G} \right)} \right] \hfill \\ \, \quad \quad \; - \frac{1}{{2k_{1} }}\left[ {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime} } } \right]^{2} \hfill \\ \quad \quad \; + \phi_{M} \left\{ \begin{gathered} \rho_{2} \left[ {\chi \left( {\alpha \left( {\frac{1}{{k_{1} }}\left( {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime} } } \right) - \xi_{1} G} \right) + \theta G} \right) + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{1}{{k_{2} }}\left( {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right) - \xi_{2} G} \right) + \theta G} \right)} \right] \hfill \\ - \frac{1}{{2k_{2} }}\left[ {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right]^{2} \hfill \\ \end{gathered} \right\} \hfill \\ \quad \quad \; + V_{M}^{A^{\prime}} \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime}} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right) - \delta G^{A} } \right] \hfill \\ rV_{R}^{A} = \rho_{2} \left[ {\chi \left( {\alpha \left( {\frac{1}{{k_{1} }}\left( {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime}} } \right) - \xi_{1} G^{A} } \right) + \theta G^{A} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{1}{{k_{2} }}\left( {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right) - \xi_{2} G^{A} } \right) + \theta G^{A} } \right)} \right] \hfill \\ \, \quad \quad \; - \frac{1}{{2k_{2} }}\left[ {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right]^{2} \hfill \\ \quad \quad \; + \phi_{R} \left\{ \begin{gathered} \rho_{1} \left[ {\chi \left( {\alpha \left( {\frac{1}{{k_{1} }}\left( {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime}} } \right) - \xi_{1} G^{A} } \right) + \theta G^{A} } \right) + \left( {1 - \chi } \right)\left( {\beta \left( {\frac{1}{{k_{2} }}\left( {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right) - \xi_{2} G^{A} } \right) + \theta G^{A} } \right)} \right] \hfill \\ - \frac{1}{{2k_{1} }}\left[ {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime}} } \right]^{2} \hfill \\ \end{gathered} \right\} \hfill \\ \, \quad \quad \; + V_{R}^{A^{\prime} } \left[ {\frac{{\gamma_{1} }}{{k_{1} }}\left( {\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha + \gamma_{1} V_{M}^{A^{\prime}} } \right) + \frac{{\gamma_{2} }}{{k_{2} }}\left( {\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta + \gamma_{2} V_{R}^{A^{\prime} } } \right) - \delta G^{A} } \right] \hfill \\ \end{gathered} $$

(30)

According to the structure of (30), we assume the form of value functions of both manufacturer and retailer to be $V_{M}^{A} = h_{1} G^{A} + h_{2}$ and $V_{R}^{A} = m_{1} G^{A} + m_{2}$, where $h_{1} ,h_{2} ,m_{1} ,m_{2} > 0$ 为represent the undetermined coefficients. Substitute the value functions and the first order derivatives into (30), we can get the equation sets of the determined coefficients.

$$ \left\{ \begin{aligned} rh_{1} & = \rho _{1} \left[ {\chi \left( { - \alpha \xi _{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi _{2} + \theta } \right)} \right] + \phi _{M} \rho _{2} \left[ {\chi \left( { - \alpha \xi _{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi _{2} + \theta } \right)} \right] - h_{1} \delta \\ rh_{2} & = \rho _{1} \left[ {\chi \alpha \left( {\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} }}{{k_{2} }}} \right)} \right] - \frac{1}{{2k_{1} }}\left[ {\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} } \right]^{2} \\ & \quad + \phi _{M} \left\{ {\rho _{2} \left[ {\chi \alpha \left( {\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} }}{{k_{2} }}} \right)} \right] - \frac{1}{{2k_{2} }}\left[ {\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} } \right]^{2} } \right\} \\ & \quad + h_{1} \left[ {\frac{{\gamma _{1} }}{{k_{1} }}\left( {\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} } \right) + \frac{{\gamma _{2} }}{{k_{2} }}\left( {\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} } \right)} \right] \\ rm_{1} & = \rho _{2} \left[ {\chi \left( { - \alpha \xi _{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi _{2} + \theta } \right)} \right] + \phi _{R} \left\{ {\rho _{1} \left[ {\chi \left( { - \alpha \xi _{1} + \theta } \right) + \left( {1 - \chi } \right)\left( { - \beta \xi _{2} + \theta } \right)} \right]} \right\} - m_{1} \delta \\ rm_{2} & = \rho _{2} \left[ {\chi \alpha \left( {\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} }}{{k_{2} }}} \right)} \right] - \frac{1}{{2k_{2} }}\left[ {\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} } \right]^{2} \\ & \quad + \phi _{R} \left\{ {\rho _{1} \left[ {\chi \alpha \left( {\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} }}{{k_{1} }}} \right) + \left( {1 - \chi } \right)\beta \left( {\frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} }}{{k_{2} }}} \right)} \right] - \frac{1}{{2k_{1} }}\left[ {\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} } \right]^{2} } \right\} \\ & \quad + m_{1} \left[ {\frac{{\gamma _{1} }}{{k_{1} }}\left( {\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\chi \alpha + \gamma _{1} h_{1} } \right) + \frac{{\gamma _{2} }}{{k_{2} }}\left( {\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {1 - \chi } \right)\beta + \gamma _{2} m_{1} } \right)} \right] \\ \end{aligned} \right. $$

Solving the equation sets above we can get the explicit expression of the undetermined coefficients.

$$ \left\{ \begin{aligned} h_{1} & = \frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)}}{{r + \delta }}{\rm M} \\ h_{2} & = \left( {\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)^{2} \gamma _{1} ^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\gamma _{2} ^{2} \left( {\left( {2 - \phi _{M} \phi _{R} } \right)\rho _{1} + \phi _{M} \rho _{2} } \right)}}{{2r\left( {r + \delta } \right)^{2} k_{2} }}} \right){\rm M}^{2} \\ & \quad + \left( {\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)^{2} \gamma _{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\left( {2 - \phi _{M} \phi _{R} } \right)\rho _{1} + \phi _{M} \rho _{2} } \right)\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\gamma _{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right){\rm M} \\ & \quad {\text{ + }}\frac{{\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)^{2} \left( {\chi \alpha } \right)^{2} }}{{2rk_{1} }} + \frac{{\left( {\left( {2 - \phi _{M} \phi _{R} } \right)\rho _{1} + \phi _{M} \rho _{2} } \right)\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)\left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{2rk_{2} }} \\ m_{1} & = \frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)}}{{r + \delta }}{\rm M} \\ m_{2} & = \left( {\frac{{\left( {\phi _{R} \rho _{1} + \left( {2 - \phi _{M} \phi _{R} } \right)\rho _{2} } \right)\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\gamma _{1} ^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)^{2} \gamma _{2} ^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{2} }}} \right){\rm M}^{2} \\ & \quad + \left( {\frac{{\left( {\phi _{R} \rho _{1} + \left( {2 - \phi _{M} \phi _{R} } \right)\rho _{2} } \right)\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\gamma _{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)^{2} \gamma _{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right){\rm M} \\ & \quad + \frac{{\left( {\phi _{R} \rho _{1} + \left( {2 - \phi _{M} \phi _{R} } \right)\rho _{2} } \right)\left( {\rho _{1} + \phi _{M} \rho _{2} } \right)\left( {\chi \alpha } \right)^{2} }}{{2rk_{1} }} + \frac{{\left( {\rho _{2} + \phi _{R} \rho _{1} } \right)^{2} \left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{2rk_{2} }} \\ \end{aligned} \right. $$

(31)

Substituting (31) into (27) and (29) we can get the optimal strategies of supply chain members

$$ \begin{gathered} q^{A} = \frac{{\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\chi \alpha \left( {r + \delta } \right) + \left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\gamma_{1} {\rm M}}}{{\left( {r + \delta } \right)k_{1} }} \hfill \\ s^{A} = \frac{{\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\gamma_{2} {\rm M}}}{{\left( {r + \delta } \right)k_{2} }} \hfill \\ \end{gathered} $$

(32)

Substituting (32) into trajectory path of goodwill in objective functional we can get the differential equation of goodwill with regard to time $t$.

$$ \dot{G}^{A} (t) = \frac{{\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }} - \delta G^{A} (t), \, G^{A} (0) = G_{0} $$

(33)

Solving the differential equation above we can get the time trajectory of goodwill under pattern A. Substituting the goodwill and (31) into the value function we can get the manufacturer’s profit and retailer’s utility. According to (2) and (3), we can get the time trajectory of both reference quality and reference service. Proposition 3 is proved.

1.4 Proof of Proposition 4

The states under the three patterns are all globally evolutionary steady states. Let $G_{\infty }^{C}$, $G_{\infty }^{N}$, and $G_{\infty }^{A}$ stand for the steady state of goodwill under the three decision models. We have

$$ \begin{gathered} G_{\infty }^{C} = \frac{1}{\delta }\left( {\frac{{\left( {\rho_{1} + \rho_{2} } \right)\gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\rho_{1} + \rho_{2} } \right)\gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }}} \right) \hfill \\ G_{\infty }^{N} = \frac{1}{\delta }\left( {\frac{{\rho_{1} \gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\rho_{2} \gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }}} \right) \hfill \\ G_{\infty }^{A} = \frac{1}{\delta }\left( {\frac{{\left( {\rho_{1} + \phi_{M} \rho_{2} } \right)\gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\rho_{2} + \phi_{R} \rho_{1} } \right)\gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }}} \right) \hfill \\ \end{gathered} $$

Thus, we have

$$ \begin{gathered} G_{\infty }^{C} - G_{\infty }^{A} = \frac{1}{\delta }\left( {\frac{{\left( {1 - \phi_{M} } \right)\rho_{2} \gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {1 - \phi_{R} } \right)\rho_{1} \gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }}} \right) \ge 0 \hfill \\ G_{\infty }^{A} - G_{\infty }^{N} = \frac{1}{\delta }\left( {\frac{{\phi_{M} \rho_{2} \gamma_{1} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} + \frac{{\phi_{R} \rho_{1} \gamma_{2} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }}} \right) \ge 0 \hfill \\ \end{gathered} $$

1.5 Proof of Proposition 5

According to Propositions 1–3, we have

$$ q^{C} - q^{A} = \frac{{\left( {1 - \phi_{M} } \right)\rho_{2} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} \ge 0,\quad q^{A} - q^{N} = \frac{{\phi_{M} \rho_{2} \left( {\chi \alpha \left( {r + \delta } \right) + \gamma_{1} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{1} }} \ge 0, $$

$$ s^{C} - s^{A} = \frac{{\left( {1 - \phi_{R} } \right)\rho_{1} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }} \ge 0,\quad s^{A} - s^{N} = \frac{{\phi_{R} \rho_{1} \left( {\left( {1 - \chi } \right)\beta \left( {r + \delta } \right) + \gamma_{2} {\rm M}} \right)}}{{\left( {r + \delta } \right)k_{2} }} \ge 0. $$

1.6 Proof of Proposition 6

Under both centralized and decentralized scenarios, the objective of the supply chain members is to maximize their profits. The value functions represent the discounted value of profits. Under the altruistic and reciprocal patterns, the supply chain members aim to optimize their utility, as expressed by the value function. Their total profit can be derived by discounting the profit to present time, which is ${{\left( {\pi_{M}^{A} + \pi_{R}^{A} } \right)} \mathord{\left/ {\vphantom {{\left( {\pi_{M}^{A} + \pi_{R}^{A} } \right)} r}} \right. \kern-\nulldelimiterspace} r}$. Thus, we have

$$ \begin{gathered} V_{\infty }^{C} - \frac{{\pi_{M\infty }^{A} + \pi_{R\infty }^{A} }}{r} = \frac{{\left( {\rho_{1} + \rho_{2} } \right)M}}{r + \delta }\left( {G_{\infty }^{C} - G_{\infty }^{A} } \right) \hfill \\ + \left( {\frac{{\left( {\left( {1 - \phi_{M}^{2} } \right)\rho_{2}^{2} + 2\left( {1 - \phi_{M} } \right)\rho_{1} \rho_{2} } \right)\gamma_{1}^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\left( {\left( {1 - \phi_{R}^{2} } \right)\rho_{1}^{2} + 2\left( {1 - \phi_{R} } \right)\rho_{1} \rho_{2} } \right)\gamma_{2}^{2} }}{{2r\left( {r + \delta } \right)^{2} k_{2} }}} \right)M^{2} \hfill \\ { + }\left( {\frac{{\left( {\left( {1 - \phi_{M}^{2} } \right)\rho_{2}^{2} + 2\left( {1 - \phi_{M} } \right)\rho_{1} \rho_{2} } \right)\gamma_{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + \frac{{\left( {\left( {1 - \phi_{R}^{2} } \right)\rho_{1}^{2} + 2\left( {1 - \phi_{R} } \right)\rho_{1} \rho_{2} } \right)\gamma_{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right)M \hfill \\ + \frac{{\left( {\left( {1 - \phi_{M}^{2} } \right)\rho_{2}^{2} + 2\left( {1 - \phi_{M} } \right)\rho_{1} \rho_{2} } \right)\left( {\chi \alpha } \right)^{2} }}{{2rk_{1} }} + \frac{{\left( {\left( {1 - \phi_{R}^{2} } \right)\rho_{1}^{2} + 2\left( {1 - \phi_{R} } \right)\rho_{1} \rho_{2} } \right)\left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{2rk_{2} }} \ge 0 \hfill \\ \end{gathered} $$

$$ V_{\infty }^{C} = {{\left( {\pi_{M\infty }^{A} + \pi_{R\infty }^{A} } \right)} \mathord{\left/ {\vphantom {{\left( {\pi_{M\infty }^{A} + \pi_{R\infty }^{A} } \right)} r}} \right. \kern-\nulldelimiterspace} r}\quad {\text{only when}}\quad \phi_{M} = \phi_{R} = 0 $$

$$ \begin{gathered} \frac{{\pi_{M}^{A} + \pi_{R}^{A} }}{r} - \left( {V_{M}^{N} + V_{R}^{N} } \right) = \frac{{\left( {\rho_{1} + \rho_{2} } \right){\rm M}}}{r + \delta }\left( {G_{\infty }^{A} - G_{\infty }^{N} } \right) + \left( {\frac{{\phi_{M} \rho_{1} \rho_{2} \gamma_{1}^{2} }}{{r\left( {r + \delta } \right)^{2} k_{1} }} + \frac{{\phi_{R} \rho_{1} \rho_{2} \gamma_{2}^{2} }}{{r\left( {r + \delta } \right)^{2} k_{2} }}} \right){\rm M}^{2} \hfill \\ + \left( {\frac{{2\phi_{M} \rho_{1} \rho_{2} \gamma_{1} \chi \alpha }}{{r\left( {r + \delta } \right)k_{1} }} + + \frac{{2\phi_{R} \rho_{1} \rho_{2} \gamma_{2} \left( {1 - \chi } \right)\beta }}{{r\left( {r + \delta } \right)k_{2} }}} \right){\rm M} + \frac{{\phi_{M} \rho_{1} \rho_{2} \left( {\chi \alpha } \right)^{2} }}{{rk_{1} }} + \frac{{\phi_{R} \rho_{1} \rho_{2} \left( {\left( {1 - \chi } \right)\beta } \right)^{2} }}{{rk_{2} }} \ge 0 \hfill \\ \end{gathered} $$

$$ {{\left( {\pi_{M}^{A} + \pi_{R}^{A} } \right)} \mathord{\left/ {\vphantom {{\left( {\pi_{M}^{A} + \pi_{R}^{A} } \right)} r}} \right. \kern-\nulldelimiterspace} r} = V_{M}^{N} + V_{R}^{N} \quad {\text{only when}}\quad \phi_{M} = \phi_{R} = 0. $$

1.7 Proof of Proposition 7

Under the centralized and the decentralized patterns, the target of the supply chain members is to maximize their profits. Their value functions represent the discounted profit during the planning horizon. Under the altruistic and reciprocal patterns, the players aim to maximize their total utility during the planning horizon, which takes into consideration their own revenue and that of their partner, as represented by the value functions. When $\phi_{M} ,\phi_{R} \in (0,1]$, similar to the proof of Proposition 6, we have $V_{M}^{A} + V_{R}^{A} \ge V^{C} > V_{M}^{N} + V_{R}^{N}$ and $\phi_{M} = 1,\phi_{R} = 0$ or $\phi_{M} = 0,\phi_{R} = 1$ and $V_{M}^{A} + V_{R}^{A} = V^{C}$. If $\phi_{M} ,\phi_{R} = 0$, there is no altruistic and reciprocal behavior, and the performance of the supply chain under this pattern is identical to that of the decentralized pattern, which means that $V^{C} > V_{M}^{A} + V_{R}^{A} = V_{M}^{N} + V_{R}^{N}$.