**Theorem 3.** *In Policy T*


We refer to Appendix E for the details of Theorem 3. Results of Theorem 2,3 are similar. Overall, SW and GL are higher under the RS game under Policy RE and T. In all three policies, the subsidy rates are higher under the MS game, but it does not guarantee the higher SW and GL. Therefore, a shift of power from the manufacturer to the retailer in a CLSC always encourages in the perspective of achieving the sustainability goal. By combining concavity conditions, one can conclude that the profits of CLSC members and SW are concave in three subsidy policies under the MS and RS games if *M*<sup>1</sup> > 0, *N*<sup>1</sup> > 0 and Δ<sup>1</sup> > 0. To obtain an overview of increment or decrements in GL, used product collection rate, profits of CLSC members, and SW, we derive the optimal decision in the absence of a subsidy.

#### *3.4. Optimal Decisions in Absence of Subsidy*

By substituting *ρ j <sup>i</sup>* <sup>=</sup> 0, or *<sup>η</sup><sup>j</sup> <sup>i</sup>* <sup>=</sup> 0, or *<sup>μ</sup><sup>j</sup> <sup>i</sup>* = 0 in Equation (1),(2); (4),(5); (7),(8), respectively, one can obtain the profit functions of CLSC members in the absence of a subsidy. The corresponding optimal decisions under the MS and RS games are presented in Lemma 7,8, respectively.

\*\*Lemma 7.\*\* \*\*Optimal decision\*\* in the absence of a subsidy under the MS game is as follows:\*\*  $\pi\_{m}^{N} = \frac{aN\_{1}\lambda\_{2} + (YZ + 4c\_{m}\lambda\_{2})\kappa}{\Delta\_{4m}}$ ;  $p\_{m}^{N} = \frac{aN\_{2}\lambda\_{2} + (YZ + 2c\_{m}\lambda\_{2})\kappa}{\Delta\_{4m}}$ ;  $\theta\_{m}^{N} = \frac{(a - c\_{m})Z\kappa}{\Delta\_{4m}}$ ;  $\pi\_{m}^{N} = \frac{(a - c\_{m})Z\kappa}{\Delta\_{4m}}$ ;  $\pi\_{m}^{N} = \frac{(a - c\_{m})X\lambda\_{2}}{\Delta\_{4m}}$ ;  $\pi\_{m}^{N} = \frac{(a - c\_{m})X\lambda\_{2}}{\Delta\_{4m}}$ ;  $\pi\_{m}^{N} = N\_{4}\lambda\_{2} - \kappa Z^{2}$ .

\*\*Lemma 8.\*\* \*\*Optimal decision\*\* in the absence of a subsidy under the RS game is as follows:\*\*  $w\_{r}^{N} = \frac{(a + c\_{m})\mathcal{N}\_{1}\lambda\_{2} + (YZ + c\_{m}M\_{2})\mathbb{k}}{2\Delta\_{4r}}$ ;  $p\_{r}^{N} = \frac{2a\lambda\_{1}\lambda\_{2} + (aM\_{1} + YZ + 2c\_{m}\lambda\_{2})\mathbb{k}}{2\Delta\_{4r}}$ ;  $\theta\_{r}^{N} = \frac{(a - c\_{m})\mathcal{Z}\lambda\_{2}}{2\Delta\_{4r}}$ ;  $\theta\_{r}^{N} = \frac{(a - c\_{m})\mathcal{Z}\lambda\_{2}}{2\Delta\_{4r}}$ ;  $\Omega\_{r}^{N} = \frac{(a - c\_{m})\mathcal{Z}\lambda\_{2}}{2\Delta\_{4r}}$ ;  $Q\_{r}^{N} = \frac{(a - c\_{m})\mathcal{Z}\lambda\_{2}}{\Delta\_{4r}}$ ;  $\text{where } \Delta\_{4r} = N\_{2}\lambda\_{2} - \kappa Z^{2}$ .

Unlike optimal decisions in Policy C, the outcomes differ between the MS and RS game in the absence of a subsidy. In this regards, one can conclude that government can weaken the effect of power of CLSc members by implementing Policy C. We use results of above two lemmas as the benchmark to compare outcomes.

#### **4. Analysis and Discussions**

In the previous section, results were compared to highlight the behavior of optimal decisions between the two games. In the following subsections, we evaluate gains and losses in the perspective of consumer, CLSC members, and the government organization.
