Channel structure and evolutionary stability analysis between traditional and green service supply chains

Abstract

This paper aims to explore the optimal pricing and green service decisions and discuss the evolutionary stability strategy (ESS) of vertical channel structure strategic interaction between the traditional and green service supply chains (TSSC and GSSC). Considering these two supply chains could choose between the centralized (C) and decentralized (D) channel structures, current research establishes four channel models, namely, Models DD, DC, CD and CC, wherein Model DD(CC) means that both supply chains adopt channel D(C) and Model DC(CD) refers to GSSC adopting channel D(C) while TSSC using channel C(D). Furthermore, an evolutionary game is developed to explore the ESSs of the dynamic competitive system. The research results show that the stronger the integration between upstream and downstream firms of GSSC is, the higher green service the supply chain would provide when TSSC adopts channel D. Besides, when the market competition is sufficiently low, only point (0,0) is the ESS; when it is moderate, there exist two ESSs, i.e., ESS (0,0) and ESS (1,1); when it is extremely high, only point (1,1) is the ESS. The numerical examples show that the green service level increases in market competition while some retailing and wholesale prices under specific models would not be affected by it or show an inverted U shape, and the initial states of two supply chains’ channel strategies significantly impact the system’s ESSs.

Appendix

Proof for Proposition 1 Comparing the optimal green service levels under different models yields the following results. ${S}_{1}^{CC*}-{S}_{1}^{CD*}=-\frac{a\beta }{4}<0$; ${S}_{1}^{DD*}-{S}_{1}^{DC*}=\frac{a\beta \left(2+\beta \right)\left(12+2\beta -8{\beta }^{2}+{\beta }^{4}\right)}{2\left(6-{\beta }^{2}\right)\left(3-{\beta }^{2}\right)\left(8-7{\beta }^{2}+{\beta }^{4}\right)}>0$; ${S}_{1}^{CC*}-{S}_{1}^{DD*}=\frac{a\left(2+\beta \right)\left(16-2\beta -21{\beta }^{2}+{\beta }^{3}+8{\beta }^{4}-{\beta }^{6}\right)}{2\left(3-{\beta }^{2}\right)\left(8-7{\beta }^{2}+{\beta }^{4}\right)}>0$; according to the above equations, we can derive Proposition 1.

Proof for Proposition 2 Comparing the optimal profits of GSSC under different models yields the following results. ${\Pi }_{SC1}^{DD*}-{\Pi }_{SC1}^{CD*}=\frac{-{a}^{2}}{(32{(3-{\beta }^{2})}^{2}{(8-7{\beta }^{2}+{\beta }^{4})}^{2})}\left\{(4096+6144\beta -12032{\beta }^{2}-21824{\beta }^{3}+8272{\beta }^{4}+25240{\beta }^{5}+705{\beta }^{6}-13488{\beta }^{7}-2688{\beta }^{8}+3648{\beta }^{9}+1054{\beta }^{10}-480{\beta }^{11}-164{\beta }^{12}+24{\beta }^{13}+9{\beta }^{14})\right\}$; ${\Pi }_{SC1}^{CC*}-{\Pi }_{SC1}^{DC*}=\frac{{a}^{2}(2-\beta ){(2+\beta )}^{3}(8-10{\beta }^{2}+{\beta }^{4})}{8{\left(6-{\beta }^{2}\right)}^{2}(2-{\beta }^{2})}$; based on these equations, we define ${f}_{1}\left(\beta \right)=4096+6144\beta -12032{\beta }^{2}-21824{\beta }^{3}+8272{\beta }^{4}+25240{\beta }^{5}+705{\beta }^{6}-13488{\beta }^{7}-2688{\beta }^{8}+3648{\beta }^{9}+1054{\beta }^{10}-480{\beta }^{11}-164{\beta }^{12}+24{\beta }^{13}+9{\beta }^{14}$ and ${f}_{2}\left(\beta \right)=8-10{\beta }^{2}+{\beta }^{4}$. Solving ${f}_{1}\left(\beta \right)=0$ and ${f}_{2}\left(\beta \right)=0$, we can obtain ${\overline{\beta }}_{1}$ and ${\overline{\beta }}_{2}$.

Proof for Proposition 3 comparing the optimal profits of TSSC under different models yields the following results. ${\Pi }_{SC2}^{DC*}-{\Pi }_{SC2}^{DD*}=\frac{{a}^{2}{\left(4-{\beta }^{2}\right)}^{2}{\left(2{\beta }^{2}+{\beta }^{3}-6-4\beta \right)}^{2}(6-14{\beta }^{2}+8{\beta }^{4}-{\beta }^{6})}{2(3-{\beta }^{2}){(12-8{\beta }^{2}+{\beta }^{4})}^{2}{(8-7{\beta }^{2}+{\beta }^{4})}^{2}}$; ${\Pi }_{SC2}^{CC*}-{\Pi }_{SC2}^{CD*}=\frac{{a}^{2}}{16}>0$; based on the above equations, we define ${f}_{3}\left(\beta \right)=6-14{\beta }^{2}+8{\beta }^{4}-{\beta }^{6}$. Solving ${f}_{3}\left(\beta \right)=0$, we can obtain ${\overline{\beta }}_{3}$.

Proof for Proposition 4 The conditions for different ESSs are presented in Tables

Table 4 Stability of points when $0<\beta \le {\overline{\beta }}_{1}$

4,

Table 5 Stability of points when ${\overline{\beta }}_{1}<\beta \le {\overline{\beta }}_{3}$

5 and

Table 6 Stability of points when ${\overline{\beta }}_{2}<\beta \le 1$

6.