*4.1. Consumer's Perspective*

The following theorem highlights consumers preference among three subsidy policies.

### **Theorem 4.** *Irrespective of game structures, the greening level and sales volume are higher in Policy C.*

We refer to Appendix F for the details of Theorem 4. Theorem 4 demonstrates that consumers always receive product at a higher GL in Policy C, where the government can penetrate consumers directly. Therefore, direct monetary gains stimulate consumers to buy more product. If sales volumes increase, the manufacturer can compensate investment cost, and GL is also consequently increased. Recall that GL, collection rate, and retail price are identical under both games in Policy C. Therefore, Policy C outperforms others in the perspective of consumer benefit and green product consumption. Graphical representation of GLs, sales volumes, effective prices consumer needs to pay, and ratios of GLs with effective retail price in six scenarios is presented in Figure 1a–d. Parameter values are used for numerical examples as follows: *a* = 500, *β* = 0.6, *cm* = 50(\$/unit), *cr* = 20(\$/unit), *α* = 10(\$/unit), *δ* = 0.7, *κ* = 1500, *λ*<sup>1</sup> = 0.3, and *λ*<sup>2</sup> = 1. Note that technical restrictions on parameters values are considered to ensure optimal conditions.

Figure 1a,b justify the statement of Theorem 4. The effective retail price is less in Policy C compared to others, and GL is always higher in the RS game in all three policies. If the consumer perceives the retail price in their mind, then Policy C outperforms others because the ratio of GLs with retail price is maximum under Policy C. By comparing the ratios, consumers can figure out how much they need to pay to procure the product. Figure 1c. demonstrates that the consumer needs to pay a lesser price under Policy C. One can observe that the GLs are lower in Policy RE compared to others, and the consumer needs to pay a higher price in Policy T.

**Figure 1.** Graphical representation of (**a**) greening levels, (**b**) sales volumes, (**c**) effective retail prices, and (**d**) ratio of greening levels with effective retail prices in Scenarios MC, RC, MRE, RRE, MT, and RT.

#### *4.2. Retailer and Manufacturer Perspectives*

The following theorem is proposed to highlight the pros and cons for three policies in the perspective of CLSC members.

**Theorem 5.** *Under both games, the collection rate of used product and profits for each member are always higher in Policy C.*

We refer to Appendix G for the details of Theorem 5. The outcomes of Theorem 5 are consistent with Theorem 4. GL and sales volume are both higher in Policy C; consequently, CLSC members receive a higher profit in a green-sensitive market. Flexibility of investment for the manufacturer in improving GL and used product return is increased with market demand. In such a scenario, the retailer can also get benefited. The results demonstrate that fact. Graphical representation of the profits for the retailer and manufacturer, collection rate, total investment for the manufacturer to produce product (MI) (*MI<sup>C</sup> <sup>i</sup>* = *<sup>λ</sup>*1*θ<sup>C</sup> <sup>i</sup> <sup>D</sup><sup>C</sup> <sup>i</sup>* + *<sup>λ</sup>*2*θ<sup>C</sup> i* 2 ; *MIRE <sup>i</sup>* = *<sup>λ</sup>*1*θ<sup>R</sup> <sup>i</sup> EDRE <sup>i</sup>* + *<sup>λ</sup>*2*θRE i* 2 ; and *MI<sup>T</sup> <sup>i</sup>* = *<sup>λ</sup>*1*θ<sup>T</sup> <sup>i</sup> <sup>D</sup><sup>T</sup> <sup>i</sup>* + (<sup>1</sup> − *<sup>μ</sup><sup>T</sup> <sup>i</sup>* )*λ*2*θ<sup>T</sup> i* 2 ), investment for the manufacturer in encouraging used product return (RI) (*RI<sup>C</sup> <sup>i</sup>* = *ατ<sup>C</sup> <sup>i</sup> <sup>D</sup><sup>C</sup> <sup>i</sup>* + *κτ<sup>C</sup> i* 2 ; *RIRE <sup>i</sup>* = *ατRE <sup>i</sup> <sup>D</sup>RE <sup>i</sup>* + (<sup>1</sup> − *<sup>η</sup>RE <sup>i</sup>* )*κτRE i* 2 ; and *RI<sup>T</sup> <sup>i</sup>* = *ατ<sup>T</sup> <sup>i</sup> <sup>D</sup><sup>T</sup> <sup>i</sup>* + *κτ<sup>T</sup> i* 2 ), ratios of relative change of GL with investment to produce products (Δ*θM<sup>j</sup> <sup>i</sup>* <sup>=</sup> *<sup>θ</sup> j i* <sup>−</sup>*θ<sup>N</sup> i MI<sup>j</sup> <sup>i</sup>* <sup>−</sup>*MI<sup>N</sup> i* ), and ratios of relative change of the collection rate with the investment effort to stimulate used product return (*τR<sup>j</sup> <sup>i</sup>* <sup>=</sup> *<sup>τ</sup><sup>j</sup> <sup>i</sup>* <sup>−</sup>*τ<sup>N</sup> i RIj <sup>i</sup>* <sup>−</sup>*RI<sup>N</sup> i* ) in six scenarios is presented in Figure 2a–g.

Figure 2a–g support the statement of Theorem 5. In Policy RE, the manufacturer receives a subsidy to encourage the product collection, but it does not yield a higher return compared to Policy C. Note that the used product collection rates are less in Policy T compared to other two. Figure 2e demonstrates that the investment effort to stimulate used product return for the manufacturer is less in Policy RE; however, due to the government support, collection rate improved. In Policy T, the manufacturer has more flexibility in the R&D investment until the manufacturer can invest more in

Policy C. However, Figure 2f exhibits some doubts about the efficiency of Policy C. In Policy C, the total amount of investment for the manufacturer to improve GL is maximum, but the ratio of relative change in GL improvement is less. Therefore, higher investment does not ensure higher GL, especially in Policy C. Figure 2g demonstrates a noteworthy outcome in the perspective of designing subsidy policy. It demonstrates that the power of CLSC members should be considered before implementation of the subsidy policy. Interestingly, the manufacturer reduces the investment effort considerably under the RS game, but a reverse trend is observed under the MS game. Overall, investment efficiency in producing greener product and used product return reduced in Policy C.

**Figure 2.** Graphical representation of (**a**) profit of manufacturer, (**b**) profit of retailer, (**c**) used product return, (**d**) manufacturer's R&D investment to produce product, (**e**) investment effort in encouraging used product collection, (**f**) ratios of relative change of greening levels with investment to produce greener product, and (**g**) ratios of relative change of return rate with investment effort to stimulate product return for the manufacturer in Scenarios MC, RC, MRE, RRE, MT, and RT.

#### *4.3. Government Perspective*

In this subsection, we compare SWs and the amount of government subsidy to explore consequence in the perspective of government organizations.

**Theorem 6.** *The social welfare and the amount of government subsidy is higher in Policy C in both the games.*

We refer Appendix H for the proof of Theorem 6. Theorem 6 demonstrates that the government expenditure and SW are always higher in Policy C. Therefore, the outcomes of Theorem 4–6 are very much alike. Higher subsidy cause higher profits, as well as GL, and SW consequently increased. Graphical representation of SW, the amount of government subsidy (GI) (*G I<sup>C</sup> <sup>i</sup>* = *<sup>ρ</sup><sup>C</sup> <sup>i</sup> <sup>D</sup><sup>C</sup> i* ; *G IRE <sup>i</sup>* = *<sup>η</sup>RE <sup>i</sup> κτRE i* 2 ; and *G I<sup>T</sup> <sup>i</sup>* = *<sup>μ</sup><sup>T</sup> <sup>i</sup> <sup>λ</sup>*2*θ<sup>T</sup> i* 2 ), and the ratios of relative change of GL with total amount of government subsidy (Δ*θG<sup>j</sup> <sup>i</sup>* <sup>=</sup> *<sup>θ</sup> j i* <sup>−</sup>*θ<sup>N</sup> i G I<sup>j</sup>* ) in six scenarios is depicted in Figure 3a–c.

*i*

**Figure 3.** Graphical representation of (**a**) social welfare, (**b**) total amount of government subsidy in each scenario, and (**c**) ratios of relative change of greening levels with total amount of government subsidy in Scenarios MC, RC, MRE, RRE, MT, and RT.

The above figures support the statement of Theorem 6. However, if we investigate at the macro level, then one cannot draw a straightforward conclusion in favor of Policy C. By correlating Figure 2e with Figure 3c, the ratio of investment efficiency reflects the different consequence. In the perspective of the manufacturer and government organization, Policy C may lead to an inadequate investment decision. However, one cannot ignore the influence of consumers until they enjoy the higher privilege in Policy C because government support directly passes to the consumer. The total amount of expenditure for the government is too high in Policy C, which does not yield a higher relative improvement in GL.

#### *4.4. When Manufacturer Produces Only DIGPs*

In this study, it is assumed that the manufacturer produces MDIGPs but does not recover the cost for used product. Therefore, we conduct numerical experiment where the manufacturer produces DIGPs(*λ*<sup>1</sup> = 0), which is more predominant in existing literature. The following figures represent the relative change of profits of the SC members (Δ*π<sup>j</sup> ki* <sup>=</sup> *<sup>π</sup><sup>j</sup> ki*|*λ*1=0−*π<sup>j</sup> ki πj ki* , *k* = *m*,*r*); collection rates (Δ*τ<sup>j</sup> <sup>i</sup>* <sup>=</sup> *<sup>τ</sup><sup>j</sup> <sup>i</sup>* <sup>|</sup>*λ*1=0−*τ<sup>j</sup> i τj i* ); GLs (Δ*θ j <sup>i</sup>* <sup>=</sup> *<sup>θ</sup> j i* |*λ*1=0−*<sup>θ</sup> j i θ j i* ); total amount of government subsidies (Δ*G I<sup>j</sup> <sup>i</sup>* <sup>=</sup> *G I<sup>j</sup> <sup>i</sup>* <sup>|</sup>*λ*1=0−*G I<sup>j</sup> i G I<sup>j</sup> i* ); and SWs (Δ*π<sup>j</sup> gi* <sup>=</sup> *<sup>π</sup><sup>j</sup> gi*|*λ*1=0−*π<sup>j</sup> gi πj gi* ).

It is expected that the CLSC members receive higher profits if unit production cost decreased. Figure 4a,b reflect that nature changes profits for CLSC members, SW, and GL, which also supports the expectation. CLSC members always receive a higher profit in Policy C, and GL is always maximum. However, the nature of used product collection and the amount of government subsidy changes significantly. Increment in the used the product collection rate and the amount of government subsidy are maximum in the RS game and in Policy T. Therefore, the government needs to examine the product type to frame an effective policy.

**Figure 4.** Graphical representation of (**a**) change in profits for the manufacturer, (**b**) change in profits for the retailer, (**c**) change in product collection rates, (**d**) change in greening levels (*λ*<sup>1</sup> = 0), (**e**) change in amount of government subsidies, and (**f**) change in social welfare in Scenarios MC, RC, MRE, RRE, MT, and RT (*λ*<sup>1</sup> = 0).

#### *4.5. Overall Implications*

The preceding discussion offers a rich amount of contextual detail based on the analytical and numerical evaluation. Subsidies make sense to encourage R&D activities in areas that would benefit society, stimulate greener product consumption with a society's environmental objectives, such as less contamination, cleaner air, etc., provide much-needed help to innovative startups, or support a manufacturer in surviving financial losses due to high R&D investment. However, there has been little discussion on comparative analysis among outcomes under the government SW optimization goal.

The present study discloses some eye opening issues. It has always been a topic of interest to consider which subsidy policy can lead to a pragmatic CLSC business model, or which is both environmentally and economically worthwhile for participating member and government organization. Based on the discussion, one can articulate that the optimal decision, preference, and implications of subsidy policies significantly change between MS and RS games. To maintain goodwill and dominate a green-sensitive market, it is always imperative for the retailer to sell greener product. However, the manufacturer receives a higher subsidy in the MS game; yet, GL and SW is higher in the RS game. Therefore, the power of a CLSC member adds a degree of conflict, and government organizations needs figure out the dynamics of power before implementing subsidy policies. Overall, Policy C under the RS game can drive toward encouraging outcomes in the perspective of consumers, retailer, and government organization.

It is commonly believed that a subsidy assists manufactures to produce greener products and trade them at low price to the consumers. To some extent, the results of the present study support the convention, but in the presence of government subsidy policy, CLSC members need to be prepared for sudden operational changes. In practice, government organizations sometimes commit to environmental policies for several years but afterward renege on their commitments. For example, the government of China recently recommended to withdraw a subsidy from the EV battery industry (https://chinaeconomicreview.com/subsidy-withdrawal-to-decimate-chinas-ev-industry/), and the government of India recently revised the amount of subsidy for the scheme "Faster Adoption and Manufacturing of Hybrid and Electric Vehicles" (https://energy.economictimes.indiatimes.com/ news/power/govt-withdraws-sops-to-conventional-battery-vehicles-under-fame/65990495). If the government suddenly revises a subsidy amount due to a sudden fall of market demand, the manufacturer needs to adjust its production rate and to be prepared for adjustment of the entire operations and marketing activities. For example, when the Indian government revised the scheme and reduced the direct subsidy to consumers, car manufacturers faced market fall. A similar situation also reported in the UK is that "Subsidy cuts blamed for fall in UK sales of electrified vehicles" (www.theguardian.com/business/2019/jul/04/subsidy-cuts-blamed-for-fall-in-uk-salesof-electrified-vehicles). Examples are similar with Policy C because, in the car industry, consumers directly receive a subsidy from government. Results indicate that Policy C clearly becomes a financial liability for government. Due to direct cash-transfer in Policy C, all the consumers enjoy a subsidy irrespective of income groups. Therefore, there is a possibility that government resources might become a drain, especially if high-income consumers take the subsidy. As observed earlier, GL and GIs are maximum in Policy C; therefore, this is where to find answer of the question how much additional amount needs to be paid for improving GL? Figure 5a,b, representing the ratio of GLs and GIs under Policy C, are drawn to obtain an overview in this direction.

**Figure 5.** (**a**) Ratio greening levels in different subsidy policies. (**b**) Ratios of amount of government subsidy in Policy C and RE. (**c**) Ratios of amount of government subsidy in Policy T and RE.

By observing the vertical axis of Figure 5a–c, one can recognize the additional financial burden associated with Policy C, especially under the MS game. It is also found that in Policy RE, the government needs to spend less and consequently also lessen GL. In practice, there is a possibility that the manufacturer can strategically reduce investment effort in presence of a subsidy. Our study also supports this fact because the manufacturer clearly reduce its investment effort, as depicted in Figure 2g. Figure 2g also demonstrates that if the intention of the government is to improve used product collection, then Policy T or RE under the MS game can drive to the desirable outcome. Finally, in Policy T, the government provides a subsidy and anticipates that the manufacturer can produce greener product, while consumers benefit from low prices. However, Policy T also becomes pricier in the perspective of consumers, but the improvement of GL is maximum under this policy.

#### **5. Conclusions**

The formation of a sustainable CLSC in the presence of a government subsidy is one of the key issues because it does not make sense to pollute the world for higher profits. One the other hand, it is infeasible in the perspective of a government organization to spend large amounts that fail to create value. Therefore, it is always challenging to design a subsidy policy that can lead to pragmatic outcomes. In literature, comparative studies on optimal outcomes in the presence of government SW optimization goal are scanty.

Motivated by emerging practice, we formulated eight CLSC models to compare outcomes of three subsidy policies. The central result emerging from the analysis reveals that in Policy C, CLSC members receive higher profits, SW of the government organization higher, and the consumers receive products at a higher GL. Characteristics of the optimal decision under the MS and RS games are not concurrent; GL and used product collection are always higher in the RS game, and the government subsidy rate is always higher in the MS game. Whatever the nature of game structures, the consumer always receives product at lower price in Policy C. However, Policy C still has shortcomings. GL does improve as the R&D investment or amount of the government subsidy increases, but the rate of change is lowest. It is found that Policy C can be a substantial financial burden without too much improvement in GL and used product collection. Because the amount of the subsidy is maximum in Policy C, our study contradicts conventional beliefs that a higher subsidy level always improves the performance of the CLSC members. The present study discloses that any straightforward conclusion on the optimal preferences in the perspective of CLSC members, consumers, and government organization is challenging to be made. If the government wants to improve green product consumption among its community, then the government can implement Policy C. However, expenditure as a subsidy will increase considerably. If the government aims to utilize a subsidy expenditure in an effective way, then Policy T can lead to a decent outcome. However, SW will be less and the consumer needs to pay more. In Policy RE, a strategic manufacturer can reduce the investment effort in the presence of a

subsidy. If the intention of government is to improve used product collection, then Policy T or RE under the MS game can drive to a desirable outcome. It is also observed that the manufacturer's decision to produce DIGPs or MDGIPs can also affect the outcomes of a subsidy policy. A retailer-dominated CLSC is always advantageous for the government; in that scenario, the government can reduce the amount of a subsidy, maximize SW, and the consumer receives greener products.

Therefore, this study can be extended in several directions. In practice, a retailer or third party is also involved in used product collection. Sometimes manufacturer and retailer can both be involved in collections. Therefore, one can examine optimal decisions in the different modes of collections i.e., manufacturer, retailer, third party, or their combined collection mode. We assume that the consumer cannot distinguish the difference between the new and re-manufactured products. However, consumers often value the re-manufactured product less than the new product [69]. Therefore, one can analyze the influence of subsidy policies where the CLSC members need to set different prices for new and re-manufactured product. We restricted our analysis under single period formulation; therefore, one can extend this analysis under two-period setting. Furthermore, it will be interesting to examine the behavior of a CLSC decision if the members agree to cooperate with each other through coordination contract mechanisms [70,71] under the influence of the government subsidy.

**Author Contributions:** S.S. and I.N. developed the concept. S.S. formulated the models. S.M. conducted all the numerical experiments. All the authors are equally contributed in manuscript writing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors deeply appreciate the valuable comments of three anonymous reviewers and the Associate Editor to improve this study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Optimal Decision in Scenario MC**

To obtain optimal response for the retailer in Equation (1), one needs to solve *<sup>d</sup>π<sup>C</sup> rm dp<sup>C</sup> m* = 0. On simplification, *p<sup>C</sup> m*(*w<sup>C</sup> <sup>m</sup>*, *θ<sup>C</sup> <sup>m</sup>*, *τ<sup>C</sup> <sup>m</sup>*, *ρ<sup>C</sup> <sup>m</sup>*) = *<sup>a</sup>*+*w<sup>C</sup> m*+*θ<sup>C</sup> mβ*+*ρ<sup>C</sup> m* <sup>2</sup> . Because *<sup>d</sup>*2*π<sup>C</sup> rm dp<sup>C</sup> m* <sup>2</sup> = −2 < 0, the profit function for the retailer is concave.

Therefore, substituting *p<sup>C</sup> <sup>m</sup>* in Equation (2), the profit function for the manufacturer is obtained as:

$$\pi\_{mn}^{\mathbb{C}}(w\_m^{\mathbb{C}}, \theta\_m^{\mathbb{C}}, \tau\_m^{\mathbb{C}}, \rho\_m^{\mathbb{C}}) = \frac{(w\_m^{\mathbb{C}} - \theta\_m^{\mathbb{C}}\lambda\_1 + X\tau\_m^{\mathbb{C}} - \mathfrak{c}\_m)(a - w\_m^{\mathbb{C}} + \beta\theta\_m^{\mathbb{C}} + \rho\_m^{\mathbb{C}}) - 2\kappa\tau\_m^{\mathbb{C}} - 2\lambda\_2\theta\_m^{\mathbb{C}}}{2}$$

To obtain optimal response for the manufacturer on wholesale price and investment efforts, we need to solve *<sup>d</sup>π<sup>C</sup> mm dw<sup>C</sup> m* <sup>=</sup> 0, *<sup>d</sup>π<sup>C</sup> mm dτ<sup>C</sup> m* <sup>=</sup> 0, and *<sup>d</sup>π<sup>C</sup> mm dθ<sup>C</sup> m* = 0, simultaneously. After simplification, the following response is obtained:

$$w\_{\mathfrak{m}}{}^{\mathbb{C}} = \frac{\kappa (a + c\_{\mathfrak{m}} + \rho\_{\mathfrak{m}}^{\mathbb{C}})(4\lambda\_2 - \beta(\beta - \lambda\_1)) + (\kappa(\beta^2 - \lambda\_1^2) - X^2\lambda\_2)(a + \rho\_{\mathfrak{m}}^{\mathbb{C}})}{\kappa(8\lambda\_2 - Z^2) - X^2\lambda\_2}$$

$$\tau\_{\mathfrak{m}}^{\mathbb{C}} = \frac{(a - c\_{\mathfrak{m}} + \rho\_{\mathfrak{m}}^{\mathbb{C}})X\lambda\_2}{\kappa(8\lambda\_2 - Z^2) - X^2\lambda\_2} \\ \text{and } \theta\_{\mathfrak{m}}^{\mathbb{C}} = \frac{\kappa Z(a - c\_{\mathfrak{m}} + \rho\_{\mathfrak{m}}^{\mathbb{C}})}{\kappa(8\lambda\_2 - Z^2) - X^2\lambda\_2}$$

Because, the manufacturer's profit function is a function of three variables, we compute the Hessian matrix(*H<sup>C</sup> <sup>m</sup>*) to verify concavity as follows:

*H<sup>C</sup> <sup>m</sup>* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *∂*2*π<sup>C</sup> mm ∂w<sup>C</sup> m* 2 *∂*2*π<sup>C</sup> mm ∂w<sup>C</sup> <sup>m</sup>∂τ<sup>C</sup> m ∂*2*π<sup>C</sup> mm ∂w<sup>C</sup> <sup>m</sup>∂θ<sup>C</sup> m ∂*2*π<sup>C</sup> mm ∂w<sup>C</sup> <sup>m</sup>∂τ<sup>C</sup> m ∂*2*π<sup>C</sup> mm ∂τ<sup>C</sup> m* 2 *∂*2*π<sup>C</sup> mm ∂τ<sup>C</sup> <sup>m</sup>∂θ<sup>C</sup> m ∂*2*π<sup>C</sup> mm ∂w<sup>C</sup> <sup>m</sup>∂θ<sup>C</sup> m ∂*2*π<sup>C</sup> mm ∂τ<sup>C</sup> <sup>m</sup>∂θ<sup>C</sup> m ∂*2*π<sup>C</sup> mm ∂θ<sup>C</sup> m* 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎣ <sup>−</sup><sup>1</sup> <sup>−</sup>*<sup>X</sup>* 2 *β*+*λ*<sup>1</sup> 2 −*X* <sup>2</sup> <sup>−</sup>2*<sup>κ</sup> <sup>β</sup><sup>X</sup>* <sup>2</sup> *<sup>β</sup>*+*λ*<sup>1</sup> 2 *βX* <sup>2</sup> −2*λ*<sup>2</sup> − *λ*1*β* ⎤ ⎥ ⎦

The values of first, second, and third principal minors are obtained as *H<sup>C</sup> <sup>m</sup>*<sup>1</sup> = −<sup>1</sup> < 0; *<sup>H</sup><sup>C</sup> <sup>m</sup>*<sup>2</sup> = <sup>8</sup>*κ*−*X*<sup>2</sup> <sup>4</sup> ; and *<sup>H</sup><sup>C</sup> <sup>m</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>(8*κ*−*X*2)*λ*2−*κ*(*β*−*λ*1)<sup>2</sup> <sup>2</sup> , respectively. Therefore, profit function for the manufacturer is concave if 8*<sup>κ</sup>* > *<sup>X</sup>*<sup>2</sup> and (8*<sup>κ</sup>* − *<sup>X</sup>*2)*λ*<sup>2</sup> − *<sup>κ</sup>*(*<sup>β</sup>* − *<sup>λ</sup>*1)<sup>2</sup> > 0.

Substituting optimal responses in Equation (3), the SW function for the government organization is obtained as *π<sup>C</sup> gm*(*ρ<sup>C</sup> <sup>m</sup>*) = *κλ*2(*a*−*cm*+*ρ<sup>C</sup> m*)((*a*−*cm*)((14*κ*−*X*2)*λ*2−*κZ*2)+*κZ*2*ρ<sup>C</sup> m*−(2*κ*−*X*2)*λ*2*ρ<sup>C</sup> m*) ((8*κ*−*X*2)*λ*2−*κ*(*β*−*λ*1)2)<sup>2</sup> . Therefore, one can obtain optimal subsidy rate by solving *<sup>d</sup>π<sup>C</sup> gr dρ<sup>C</sup> m* = 0. On Simplification, optimal subsidy rate is obtained as *ρC <sup>m</sup>* <sup>=</sup> <sup>6</sup>(*a*−*cm*)*κλ*<sup>2</sup> *<sup>κ</sup>*(2*λ*2−*Z*2)−*X*2*λ*<sup>2</sup> . Note that *<sup>d</sup>*2*π<sup>C</sup> rm dρ<sup>C</sup> m* <sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>2</sup>*κλ*2Δ<sup>1</sup> (*κ*(8*λ*2−(*β*−*λ*1)2)−*X*2*λ*2)<sup>2</sup> <sup>&</sup>lt; 0, i.e., *<sup>π</sup><sup>C</sup> gm* is concave with respect to subsidy rate if <sup>Δ</sup><sup>1</sup> = *<sup>κ</sup>*(2*λ*<sup>2</sup> − (*<sup>β</sup>* − *<sup>λ</sup>*1)2) − *<sup>X</sup>*2*λ*2. By using back substitution, we obtain optimal decision as presented in Lemma 1.

The following additional notations are used throughout the article for simplicity: *<sup>M</sup>*<sup>1</sup> = <sup>2</sup>*λ*<sup>2</sup> − *<sup>Z</sup>*2, *<sup>M</sup>*<sup>2</sup> = <sup>4</sup>*λ*<sup>2</sup> − *<sup>Z</sup>*2, *<sup>M</sup>*<sup>3</sup> = <sup>8</sup>*λ*<sup>2</sup> − *<sup>Z</sup>*2, *<sup>M</sup>*<sup>4</sup> = <sup>14</sup>*λ*<sup>2</sup> − *<sup>Z</sup>*2, *<sup>N</sup>*<sup>1</sup> = <sup>2</sup>*<sup>κ</sup>* − *<sup>X</sup>*2, *<sup>N</sup>*<sup>2</sup> = <sup>4</sup>*<sup>κ</sup>* − *<sup>X</sup>*2, *<sup>N</sup>*<sup>3</sup> = <sup>6</sup>*<sup>κ</sup>* − *<sup>X</sup>*2, *<sup>N</sup>*<sup>4</sup> = <sup>8</sup>*<sup>κ</sup>* − *<sup>X</sup>*2, *<sup>N</sup>*<sup>5</sup> = <sup>14</sup>*<sup>κ</sup>* − *<sup>X</sup>*2, *<sup>X</sup>* = *cm<sup>δ</sup>* − *cr* − *<sup>α</sup>*, *<sup>Y</sup>* = *<sup>a</sup>λ*<sup>1</sup> − *cmβ*, *<sup>Z</sup>* = *<sup>β</sup>* − *<sup>λ</sup>*1.

### **Appendix B. Optimal Decision in Scenario RC**

First, we substitute *m<sup>C</sup> <sup>r</sup>* = *p<sup>C</sup> <sup>r</sup>* − *<sup>w</sup><sup>C</sup> <sup>r</sup>* in Equation (1)–(3) to obtain optimal decision in the RS game. To obtain the manufacturer response first, one needs to solve *<sup>d</sup>π<sup>C</sup> mr dw<sup>C</sup> r* <sup>=</sup> 0, *<sup>d</sup>π<sup>C</sup> mr dτ<sup>C</sup> r* <sup>=</sup> 0, and *<sup>d</sup>π<sup>C</sup> mr dθ<sup>C</sup> r* = 0 simultaneously. Therefore, the manufacturer's response is obtained as follows:

$$w\_r^\mathbb{C} = \frac{(\kappa(2\lambda\_2 + Z\lambda\_1) - X^2\lambda\_2)(a - m\_r^\mathbb{C} + \rho\_r^\mathbb{C}) + c\_m\kappa(2\lambda\_2 - \beta(\beta - \lambda\_1))}{\kappa(4\lambda\_2 - Z^2) - X^2\lambda\_2}$$

$$\tau = \frac{(a - c\_m - m\_r^\mathbb{C} + \rho\_r^\mathbb{C})X\lambda\_2}{\kappa(4\lambda\_2 - Z^2) - X^2\lambda\_2} \text{and} \\ \theta = \frac{(a - c\_m - m\_r^\mathbb{C} + \rho\_r^\mathbb{C})\kappa Z}{\kappa(4\lambda\_2 - (\beta - \lambda\_1)^2) - X^2\lambda\_2}$$

Because, the profit function for the manufacturer is a function of three variables, therefore, we compute corresponding Hessian matrix(*H<sup>C</sup> <sup>r</sup>* ) for the manufacturer profit function is as follows:

$$H\_r^C = \begin{bmatrix} \frac{\partial^2 \pi\_{\rm MF}^L}{\partial w\_r^C} & \frac{\partial^2 \pi\_{\rm MF}^L}{\partial w\_r^C \partial w\_r^L} & \frac{\partial^2 \pi\_{\rm MF}^L}{\partial w\_r^C \partial \theta\_r^L} \\ \frac{\partial^2 \pi\_{\rm MF}^C}{\partial w\_r^C \partial w\_r^L} & \frac{\partial^2 \pi\_{\rm MF}^C}{\partial r\_r^C} & \frac{\partial^2 \pi\_{\rm MF}^C}{\partial r\_r^C \partial \theta\_r^L} \\ \frac{\partial^2 \pi\_{\rm MF}^C}{\partial w\_r^C \partial \theta\_r^L} & \frac{\partial^2 \pi\_{\rm MF}^C}{\partial r\_r^C \partial \theta\_r^L} & \frac{\partial^2 \pi\_{\rm MF}^C}{\partial \theta\_r^C} \end{bmatrix} = \begin{vmatrix} -2 & -X & \notin + \lambda\_1 & & & \\ -X & -2x & \notin X & & & \\ \notin + \lambda\_1 & \notin X & -2(\lambda\_2 + \mathcal{J}\lambda\_1) & & \\ \notin + \lambda\_1 & \notin X & -2(\lambda\_2 + \mathcal{J}\lambda\_1) & & \\ \end{vmatrix}$$

The principal minors of above Hessian matrix are *H<sup>C</sup> <sup>r</sup>*<sup>1</sup> = −<sup>2</sup> < 0; *<sup>H</sup><sup>T</sup> <sup>r</sup>*<sup>2</sup> = <sup>4</sup>*<sup>κ</sup>* − *<sup>X</sup>*<sup>2</sup> > 0; and *<sup>H</sup><sup>T</sup> <sup>r</sup>*<sup>3</sup> = −2(*κ*(4*λ*<sup>2</sup> + *<sup>Z</sup>*2) − *<sup>X</sup>*2*λ*2), respectively. Consequently, the profit function will be concave if 4*<sup>κ</sup>* > *<sup>X</sup>*<sup>2</sup> and *κ*(4*λ*<sup>2</sup> + *Z*2) > *X*2*λ*2.

Substituting optimal response for the manufacturer in Equation (1), profit function for the retailer is obtained as *π<sup>C</sup> rr*(*m<sup>C</sup> <sup>r</sup>* ) = <sup>2</sup>*m<sup>C</sup> <sup>r</sup> κλ*2(*a*−*cm*−*m<sup>C</sup> <sup>r</sup>* +*ρ<sup>C</sup> r* ) *<sup>κ</sup>*(4*λ*2−*Z*2)−*X*2*λ*<sup>2</sup> . Therefore, the optimal response for the retailer is obtained by solving *<sup>d</sup>π<sup>C</sup> rr dm<sup>C</sup> r* = 0. After simplification, *m<sup>C</sup> <sup>r</sup>* <sup>=</sup> *<sup>a</sup>*−*cm*+*ρ<sup>C</sup> r* <sup>2</sup> . The profit function of the retailer is also concave because *<sup>d</sup>*2*π<sup>C</sup> rr dm<sup>C</sup> r* <sup>2</sup> = <sup>−</sup>4*κλ*<sup>2</sup> *<sup>κ</sup>*(4*λ*2−*Z*2)−*X*2*λ*<sup>2</sup> . Substituting optimal responses in Equation (3), the simplified value of the SW in Scenario RC is obtained as follows:

$$\pi\_{\mathcal{S}^r}^{\mathbb{C}}(\rho\_r^{\mathbb{C}}) = \frac{\kappa \lambda\_2 (a - c\_m + \rho\_r^{\mathbb{C}}) ((a - c\_m)(14\kappa \lambda\_2 - 3\kappa Z^2 - 3X^2 \lambda\_2) + \kappa Z^2 \rho\_r^{\mathbb{C}} - (2\kappa - X^2)\lambda\_2 \rho\_r^{\mathbb{C}})}{4(\kappa (4\lambda\_2 - Z^2) - X^2 \lambda\_2)^2}.$$

Therefore, the optimal subsidy rate will be obtained by solving *<sup>d</sup>π<sup>C</sup> gr dρ<sup>C</sup> r* = 0. On simplification, we obtain the value of *ρ<sup>C</sup> <sup>r</sup>* as presented in Proposition (2). The SW under the RS game is concave because, *<sup>d</sup>*2*π<sup>C</sup> rm dρ<sup>C</sup> r* <sup>2</sup> <sup>=</sup> <sup>−</sup> *κλ*2Δ<sup>1</sup> <sup>2</sup>(*κ*(4*λ*2−(*β*−*λ*1)2)−*X*2*λ*2)<sup>2</sup> <sup>&</sup>lt; 0 where <sup>Δ</sup><sup>1</sup> <sup>=</sup> *<sup>κ</sup>*(2*λ*<sup>2</sup> <sup>−</sup> (*<sup>β</sup>* <sup>−</sup> *<sup>λ</sup>*1)2) <sup>−</sup> *<sup>X</sup>*2*λ*2. By using back substitution, we obtain optimal decision as presented in Lemma 2.

#### **Appendix C. Proof of Theorem 1**

The following inequalities ensure the proof of first part of Theorem 1:

*θC <sup>m</sup>* − *<sup>θ</sup><sup>C</sup> <sup>r</sup>* = 0; *τ<sup>C</sup> <sup>m</sup>* − *<sup>τ</sup><sup>C</sup> <sup>r</sup>* = 0; *π<sup>C</sup> gm* − *<sup>π</sup><sup>C</sup> gr* = 0; *ρ<sup>C</sup>* <sup>0</sup>*<sup>m</sup>* − *<sup>ρ</sup><sup>C</sup>* <sup>0</sup>*<sup>r</sup>* <sup>=</sup> (*a*−*cm*)(*κZ*2+*X*2*λ*2) <sup>Δ</sup><sup>1</sup> > 0;

Differentiating optimal decisions in Lemma 1,2, with respect to *λ*<sup>1</sup> and *λ*2, the following relations are obtained:

*dτ<sup>C</sup> m <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> *<sup>d</sup>τ<sup>C</sup> r <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> (*a*−*cm*)*XZκλ*<sup>2</sup> Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>τ<sup>C</sup> m <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> *<sup>d</sup>τ<sup>C</sup> r <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*XZ*2*<sup>κ</sup>* Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>θ<sup>C</sup> m <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> *<sup>d</sup>θ<sup>C</sup> r <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*κ*(2*Z*2*κ*+Δ1) Δ1 <sup>2</sup> < 0; *dθ<sup>C</sup> m <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> *<sup>d</sup>θ<sup>C</sup> r <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*N*1*Z<sup>κ</sup>* Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>π<sup>C</sup> gm <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> *<sup>d</sup>π<sup>C</sup> gr <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>2(*a*−*cm*)2*Zκ*2*λ*<sup>2</sup> Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>π<sup>C</sup> gm <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> *<sup>d</sup>π<sup>C</sup> gr <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)2*Z*2*κ*<sup>2</sup> Δ1 <sup>2</sup> < 0; *dρ<sup>C</sup> m <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>12(*a*−*cm*)2*Zκ*2*λ*<sup>2</sup> Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>ρ<sup>C</sup> m <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>6(*a*−*cm*)2*Z*2*κ*<sup>2</sup> Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>ρ<sup>C</sup> r <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>8(*a*−*cm*)2*Zκ*2*λ*<sup>2</sup> Δ1 <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>ρ<sup>C</sup> r <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>4(*a*−*cm*)2*Z*2*κ*<sup>2</sup> Δ1 <sup>2</sup> < 0. The above inequalities supports the claim in Theorem 1.

#### **Appendix D. Proof of Theorem 2**

$$\begin{aligned} &\text{The following inequalities ensure the proof of first part of Theorem 2:}\\ \theta\_r^R - \theta\_m^{RE} = \frac{(a - c\_m)(Z^2(X^4 + 16\lambda\_1\lambda\_1)\lambda\_2^2 + Z^4\kappa(Z^2\kappa - 2\lambda\_3\lambda\_2) + 2X^2\lambda\_2^2(5M\_1\kappa + 7\lambda\_3\lambda\_2))}{2\Delta\_{21}\Delta\_{21}} > 0;\\ \pi\_r^{RE} - \pi\_m^{RE} &= \frac{(a - c\_m)X\lambda\_2^2\left(244\lambda\_2\kappa\lambda\_2 + 4\lambda\_3\lambda\_2^2\kappa + 4\lambda\_4\lambda\_2^2\lambda\_2\right)}{4\Delta\_{21}\Delta\_{21}} > 0; \eta\_r^{RE} - \eta\_m^{RE} = \frac{2(6\lambda\_1 - X^2)\lambda\_2^2 - 4\lambda\_4\lambda\_2^2\lambda\_2 + \kappa Z^4}{2\kappa(5\lambda\_2 - Z^2)(44\lambda\_2 - Z^2)} > 0;\\ \pi\_{gr}^{RE} - \pi\_{gm}^{RE} &= \frac{(a - c\_m)^2\lambda\_2(2X^2(10\mathbf{2}\mathbf{r} - \mathbf{T}^2)\lambda\_2^3 + (9\mathbf{\alpha}^2 - 40\mathbf{\lambda}^2\mathbf{r} + \mathbf{X}^4)Z^2\lambda\_2^2 - 2(13\mathbf{\alpha} - \mathbf{X}^2)\mathbf{\alpha}Z^4\lambda\_2 + \kappa^2 Z^6)}{4\Delta\_{21}\Delta\_{21}} > 0. \end{aligned}$$

Differentiating optimal decisions in Lemma 3,4, with respect to *λ*<sup>1</sup> and *λ*2, the following relations are obtained:

*dτRE m <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>2(*a*−*cm*)(*M*4+6*λ*2)*M*3*XZκλ*<sup>2</sup> <sup>Δ</sup>2*m*<sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>τRE m <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*κXZ*2(48*λ*<sup>2</sup> <sup>2</sup>+28*M*2*λ*2+*Z*4) <sup>Δ</sup>2*m*<sup>2</sup> <sup>&</sup>lt; 0; *dθRE m <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*κ*(2(*M*4+6*λ*2)*Z*2*M*<sup>3</sup> <sup>2</sup>*κ*+(2(17*M*1+22*λ*2)*λ*2+*Z*4)Δ2*m*) *<sup>M</sup>*4Δ2*m*<sup>2</sup> <sup>&</sup>lt; 0; *dθRE m <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*κZ*(*M*4*X*2*Z*2+2*M*3(4*M*2*κ*+7*N*1*λ*2+2*κλ*2)) *<sup>M</sup>*4Δ2*m*<sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>πRE gm <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>2(*a*−*cm*)2*κ*2*Zλ*<sup>2</sup> <sup>Δ</sup>2*m*<sup>2</sup> <sup>&</sup>lt; 0; *dπRE gm <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)2*Z*2*κ*2(48*λ*<sup>2</sup> <sup>2</sup>+28*M*2*λ*2+*Z*4) <sup>Δ</sup>2*m*<sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>τRE r <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>2(*a*−*cm*)*XZλ*2((4*κ*+*X*2)*λ*<sup>2</sup> <sup>2</sup>+10*M*1*κλ*2+*κZ*4) Δ2*<sup>r</sup>* <sup>2</sup> < 0; *dτRE r <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*XZ*2((4*κ*+*X*2)*λ*<sup>2</sup> <sup>2</sup>+*M*1*κλ*2+*κZ*4) Δ2*<sup>r</sup>* <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>θRE r <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)*Z*(2*κ*Δ2*r*+*X*2(*M*2*Z*2*κ*+(3*N*1+2*κ*)*λ*<sup>2</sup> <sup>2</sup>)) 2Δ2*<sup>r</sup>* <sup>2</sup> < 0; *dθRE r <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)(*N*1*λ*<sup>2</sup> 2(16*κλ*2+*X*2(*Z*2+*λ*2))+4(*κ*2−*X*4)*λ*<sup>3</sup> 2+*N*3*X*2*λ*<sup>3</sup> 2+4*κ*2*λ*<sup>2</sup> <sup>2</sup>(7*λ*2−4*Z*2)+*Z*2*κ*(*Z*4*κ*−4*Z*2*κλ*2+4*X*2*λ*2(*M*2+*λ*2))) 2Δ2*<sup>r</sup>* <sup>2</sup> < 0; *dπRE gr <sup>d</sup>λ*<sup>1</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)2*Z*2(3*M*1*κ*+*N*2*λ*2)(*M*1*κ*+*X*2*λ*2) Δ2*<sup>r</sup>* <sup>2</sup> <sup>&</sup>lt; 0; *<sup>d</sup>πRE gr <sup>d</sup>λ*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*a*−*cm*)2*Zλ*2(3*M*1*κ*+*N*2*λ*2)(*M*1*κ*+*X*2*λ*2) Δ2*<sup>r</sup>* <sup>2</sup> < 0. The above inequalities supports the claim in Theorem 2.
