<table><caption>**Table 3**
The optimal margins in both monopoly and oligopoly markets.. (Ω = 0.5, ρ<sub>G</sub> = 0.5)</caption><thead><tr><th rowspan="2">Policy</th><th colspan="4">Monopoly</th><th colspan="4">Oligopoly</th></tr><tr><th>Sce.</th><th>m<sub>M<sub>1</sub></sub></th><th>m<sub>M<sub>2</sub></sub></th><th>m<sub>R<sub>1</sub></sub></th><th>m<sub>R<sub>2</sub></sub></th><th>Sce.</th><th>m<sub>M<sub>1</sub></sub></th><th>m<sub>M<sub>2</sub></sub></th><th>m<sub>R<sub>1</sub></sub></th><th>m<sub>R<sub>2</sub></sub></th></tr></thead><tbody><tr><td>Drg.</td><td>1</td><td>19.010</td><td>18.081</td><td>7.225</td><td>6.684</td><td>2</td><td>12.860</td><td>11.525</td><td>6.162</td><td>5.522</td></tr><tr><td>DTM</td><td>3</td><td><strong>59.806</strong></td><td><strong>57.529</strong></td><td><strong>36.783</strong></td><td><strong>35.853</strong></td><td>4</td><td><strong>18.753</strong></td><td><strong>16.533</strong></td><td><strong>8.986</strong></td><td><strong>7.922</strong></td></tr><tr><td>SP&amp;SC</td><td>5</td><td>21.053</td><td>20.125</td><td>7.992</td><td>7.450</td><td>6</td><td>16.563</td><td>14.926</td><td>13.226</td><td>13.26</td></tr><tr><td>DLS</td><td>7</td><td>19.010</td><td>18.081</td><td>7.225</td><td>6.684</td><td>8</td><td>12.860</td><td>11.525</td><td>6.162</td><td>5.522</td></tr><tr><td>GSCE</td><td>9</td><td>27.116</td><td>26.188</td><td>10.265</td><td>9.724</td><td>10</td><td>15.117</td><td>13.509</td><td>7.243</td><td>6.473</td></tr></tbody></table>

<table><caption>**Table 4**
The optimal values in the monopoly market (Ω = 0.5, ρ<sub>G</sub> = 0.5)</caption><thead><tr><th>Sce.</th><th>p<sub>1</sub></th><th>p<sub>2</sub></th><th>s</th><th>w<sub>1</sub></th><th>w<sub>2</sub></th><th>q<sub>1</sub></th><th>q<sub>2</sub></th><th>u<sub>R</sub></th><th>u<sub>M</sub></th><th>RA's optimal decisions</th></tr></thead><tbody><tr><td>1</td><td>29.235</td><td>26.765</td><td>25.848</td><td>22.010</td><td>20.081</td><td>7.225</td><td>6.684</td><td>19.920</td><td>178.659</td><td>-</td></tr><tr><td>3</td><td>99.589</td><td>95.381</td><td><strong>76.001</strong></td><td>62.806</td><td>59.529</td><td><strong>36.783</strong></td><td><strong>35.853</strong></td><td><strong>1840.4</strong></td><td><strong>3456.81</strong></td><td>t<sub>1</sub> = 56.516, t<sub>2</sub>=53.818</td></tr><tr><td>5</td><td>32.045</td><td>29.575</td><td>34.023</td><td>24.053</td><td>22.125</td><td>7.992</td><td>7.450</td><td>0.000</td><td>195.411</td><td>x = 3.372, s<sub>T</sub> = 20.000</td></tr><tr><td>7</td><td>29.235</td><td>26.765</td><td>25.848</td><td>22.010</td><td>20.081</td><td>7.225</td><td>6.684</td><td>19.920</td><td>178.659</td><td>s<sub>L</sub> = 0.547</td></tr><tr><td>9</td><td>40.382</td><td>37.911</td><td>35.982</td><td>30.116</td><td>28.188</td><td>10.265</td><td>9.724</td><td>58.172</td><td>385.416</td><td>s<sub>G</sub> = 22.293</td></tr></tbody></table>

<table><caption>**Table 5**
The optimal values in the oligopoly market.. (Ω = 0.5, ρ<sub>G</sub> = 0.5)</caption><thead><tr><th>Sce.</th><th>p<sub>1</sub></th><th>p<sub>2</sub></th><th>s<sub>1</sub></th><th>s<sub>2</sub></th><th>w<sub>1</sub></th><th>w<sub>2</sub></th><th>q<sub>1</sub></th><th>q<sub>2</sub></th><th>u<sub>R<sub>1</sub></sub></th><th>u<sub>R<sub>2</sub></sub></th><th>u<sub>M<sub>1</sub></sub></th><th>u<sub>M<sub>2</sub></sub></th><th>RAs optimal decisions</th></tr></thead><tbody><tr><td>2</td><td>22.022</td><td>19.047</td><td>9.231</td><td>6.792</td><td>15.860</td><td>13.525</td><td>6.162</td><td>5.522</td><td>33.498</td><td>29.683</td><td>73.849</td><td>60.537</td><td>-</td></tr><tr><td>4</td><td>30.739</td><td>26.455</td><td>12.24</td><td>8.783</td><td>21.753</td><td>18.533</td><td>8.986</td><td>7.922</td><td>69.293</td><td>58.953</td><td><strong>155.84</strong></td><td><strong>123.59</strong></td><td>t<sub>1</sub> = 11.97, t<sub>2</sub>=10.60</td></tr><tr><td>6</td><td>32.789</td><td>30.192</td><td><strong>28.37</strong></td><td><strong>23.61</strong></td><td>19.563</td><td>16.926</td><td><strong>13.23</strong></td><td><strong>13.27</strong></td><td><strong>200.68</strong></td><td><strong>201.27</strong></td><td>0.000</td><td>0.000</td><td>x<sub>1</sub> = 10.35, x<sub>2</sub>=10.82,</td></tr><tr><td>8</td><td>22.022</td><td>19.047</td><td>9.231</td><td>6.792</td><td>15.860</td><td>13.525</td><td>6.162</td><td>5.522</td><td>33.498</td><td>29.683</td><td>73.849</td><td>60.537</td><td>s<sub>T<sub>1</sub></sub> = 5.0, s<sub>T<sub>2</sub></sub>=5.0</td></tr><tr><td>10</td><td>25.360</td><td>21.982</td><td>10.38</td><td>7.581</td><td>18.117</td><td>15.509</td><td>7.243</td><td>6.473</td><td>55.214</td><td>48.496</td><td>110.14</td><td>90.380</td><td>s<sub>L<sub>1</sub></sub> = 0.86, s<sub>L<sub>2</sub></sub>=1.03</td></tr><tr><td>10</td><td>25.360</td><td>21.982</td><td>10.38</td><td>7.581</td><td>18.117</td><td>15.509</td><td>7.243</td><td>6.473</td><td>55.214</td><td>48.496</td><td>110.14</td><td>90.380</td><td>s<sub>G<sub>1</sub></sub> = 10.63, s<sub>G<sub>2</sub></sub>=8.89</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** The MS of the sustainable products under various trade-off coefficients of GBU and CBU.</figcaption></figure>
<table><caption>**Table 3**
The optimal margins in both monopoly and oligopoly markets..(Ω = 0.5, ρ<sub>G</sub> = 0.5)</caption><thead><tr><th rowspan="2">Policy</th><th colspan="4">Monopoly</th><th colspan="4">Oligopoly</th></tr><tr><th>Sce.</th><th>m<sub>M<sub>1</sub></sub></th><th>m<sub>M<sub>2</sub></sub></th><th>m<sub>R<sub>1</sub></sub></th><th>m<sub>R<sub>2</sub></sub></th><th>Sce.</th><th>m<sub>M<sub>1</sub></sub></th><th>m<sub>M<sub>2</sub></sub></th><th>m<sub>R<sub>1</sub></sub></th><th>m<sub>R<sub>2</sub></sub></th></tr></thead><tbody><tr><td>Drg.</td><td>1</td><td>19.010</td><td>18.081</td><td>7.225</td><td>6.684</td><td>2</td><td>12.860</td><td>11.525</td><td>6.162</td><td>5.522</td></tr><tr><td>DTM</td><td>3</td><td><strong>59.806</strong></td><td><strong>57.529</strong></td><td><strong>36.783</strong></td><td><strong>35.853</strong></td><td>4</td><td><strong>18.753</strong></td><td><strong>16.533</strong></td><td><strong>8.986</strong></td><td><strong>7.922</strong></td></tr><tr><td>SP&amp;SC</td><td>5</td><td>21.053</td><td>20.125</td><td>7.992</td><td>7.450</td><td>6</td><td>16.563</td><td>14.926</td><td>13.226</td><td>13.26</td></tr><tr><td>DLS</td><td>7</td><td>19.010</td><td>18.081</td><td>7.225</td><td>6.684</td><td>8</td><td>12.860</td><td>11.525</td><td>6.162</td><td>5.522</td></tr><tr><td>GSCE</td><td>9</td><td>27.116</td><td>26.188</td><td>10.265</td><td>9.724</td><td>10</td><td>15.117</td><td>13.509</td><td>7.243</td><td>6.473</td></tr></tbody></table>

<table><caption>**Table 4**
The optimal values in the monopoly market (Ω = 0.5, ρ<sub>G</sub> = 0.5)</caption><thead><tr><th>Sce.</th><th>p<sub>1</sub></th><th>p<sub>2</sub></th><th>s</th><th>w<sub>1</sub></th><th>w<sub>2</sub></th><th>q<sub>1</sub></th><th>q<sub>2</sub></th><th>u<sub>R</sub></th><th>u<sub>M</sub></th><th>RA's optimal decisions</th></tr></thead><tbody><tr><td>1</td><td>29.235</td><td>26.765</td><td>25.848</td><td>22.010</td><td>20.081</td><td>7.225</td><td>6.684</td><td>19.920</td><td>178.659</td><td>-</td></tr><tr><td>3</td><td>99.589</td><td>95.381</td><td><strong>76.001</strong></td><td>62.806</td><td>59.529</td><td><strong>36.783</strong></td><td><strong>35.853</strong></td><td><strong>1840.4</strong></td><td><strong>3456.81</strong></td><td>t<sub>1</sub> = 56.516, t<sub>2</sub>=53.818</td></tr><tr><td>5</td><td>32.045</td><td>29.575</td><td>34.023</td><td>24.053</td><td>22.125</td><td>7.992</td><td>7.450</td><td>0.000</td><td>195.411</td><td>x = 3.372, s<sub>T</sub> = 20.000</td></tr><tr><td>7</td><td>29.235</td><td>26.765</td><td>25.848</td><td>22.010</td><td>20.081</td><td>7.225</td><td>6.684</td><td>19.920</td><td>178.659</td><td>s<sub>L</sub> = 0.547</td></tr><tr><td>9</td><td>40.382</td><td>37.911</td><td>35.982</td><td>30.116</td><td>28.188</td><td>10.265</td><td>9.724</td><td>58.172</td><td>385.416</td><td>s<sub>G</sub> = 22.293</td></tr></tbody></table>

<table><caption>**Table 5**
The optimal values in the oligopoly market..(Ω = 0.5, ρ<sub>G</sub> = 0.5)</caption><thead><tr><th>Sce.</th><th>p<sub>1</sub></th><th>p<sub>2</sub></th><th>s<sub>1</sub></th><th>s<sub>2</sub></th><th>w<sub>1</sub></th><th>w<sub>2</sub></th><th>q<sub>1</sub></th><th>q<sub>2</sub></th><th>u<sub>R<sub>1</sub></sub></th><th>u<sub>R<sub>2</sub></sub></th><th>u<sub>M<sub>1</sub></sub></th><th>u<sub>M<sub>2</sub></sub></th><th>RAs optimal decisions</th></tr></thead><tbody><tr><td>2</td><td>22.022</td><td>19.047</td><td>9.231</td><td>6.792</td><td>15.860</td><td>13.525</td><td>6.162</td><td>5.522</td><td>33.498</td><td>29.683</td><td>73.849</td><td>60.537</td><td>-</td></tr><tr><td>4</td><td>30.739</td><td>26.455</td><td>12.24</td><td>8.783</td><td>21.753</td><td>18.533</td><td>8.986</td><td>7.922</td><td>69.293</td><td>58.953</td><td><strong>155.84</strong></td><td><strong>123.59</strong></td><td>t<sub>1</sub> = 11.97, t<sub>2</sub>=10.60</td></tr><tr><td>6</td><td>32.789</td><td>30.192</td><td><strong>28.37</strong></td><td><strong>23.61</strong></td><td>19.563</td><td>16.926</td><td><strong>13.23</strong></td><td><strong>13.27</strong></td><td><strong>200.68</strong></td><td><strong>201.27</strong></td><td>0.000</td><td>0.000</td><td>x<sub>1</sub> = 10.35, x<sub>2</sub>=10.82,</td></tr><tr><td>8</td><td>22.022</td><td>19.047</td><td>9.231</td><td>6.792</td><td>15.860</td><td>13.525</td><td>6.162</td><td>5.522</td><td>33.498</td><td>29.683</td><td>73.849</td><td>60.537</td><td>s<sub>T<sub>1</sub></sub> = 5.0, s<sub>T<sub>2</sub></sub>=5.0</td></tr><tr><td>10</td><td>25.360</td><td>21.982</td><td>10.38</td><td>7.581</td><td>18.117</td><td>15.509</td><td>7.243</td><td>6.473</td><td>55.214</td><td>48.496</td><td>110.14</td><td>90.380</td><td>s<sub>L<sub>1</sub></sub> = 0.86, s<sub>L<sub>2</sub></sub>=1.03</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>s<sub>G<sub>1</sub></sub> = 10.63, s<sub>G<sub>2</sub></sub>=8.89</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** The MS of the sustainable products under various trade-off coefficients of GBU and CBU.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. The optimal CBU and TU under various trade-off coefficient of GBU and CBU.</figcaption></figure>

policy secures higher utility for the government. Therefore, firms in an oligopolistic market can maximize their utility by adopting these policies.

Fig. 8 presents a comparison of optimal TSE values under various regulatory policies of the RA for different degrees of substitutability in both monopoly and oligopoly markets. Results shown in Fig. 8.a, 8.c, and 8.d indicate that the degree of product substitutability significantly

affects the choice of regulatory policies and optimal TSE values. Specifically, in a monopoly market, when the degree of substitutability is low ($d \le 0.4$), the DTM, DLS, and GCSE policies lead to higher optimal TSE values. Among these policies, the DTM strategy is found to be the most effective. Therefore, monopolistic firms can benefit from adopting these regulatory policies, especially when the degree of substitutability among their products is low. In contrast, in an oligopoly market, the
<figure><img src="image_6.png" /><figcaption>Fig. 4. The optimal CBU and TU under various trade-off coefficient of GBU and CBU.</figcaption></figure>

policy secures higher utility for the government. Therefore, firms in an oligopolistic market can maximize their utility by adopting these policies.

Fig. 8 presents a comparison of optimal TSE values under various regulatory policies of the RA for different degrees of substitutability in both monopoly and oligopoly markets. Results shown in Fig. 8.a, 8.c, and 8.d indicate that the degree of product substitutability significantly

affects the choice of regulatory policies and optimal TSE values. Specifically, in a monopoly market, when the degree of substitutability is low ($d \le 0.4$), the DTM, DLS, and GCSE policies lead to higher optimal TSE values. Among these policies, the DTM strategy is found to be the most effective. Therefore, monopolistic firms can benefit from adopting these regulatory policies, especially when the degree of substitutability among their products is low. In contrast, in an oligopoly market, the
<figure><img src="image_7.png" /><figcaption>Fig. 5. The optimal TSE under various trade-off coefficient of GNI and TSE.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. The MS of the sustainable products under different substitutability degrees of products.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. The optimal TSE under various trade-off coefficient of GNI and TSE.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. The MS of the sustainable products under different substitutability degrees of products.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>(a) The optimal CBU and TU in Scenario 3</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) The optimal CBU and TU in Scenario 5</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(e) The optimal CBU and TU in Scenario 7</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(g) The optimal CBU and TU in Scenario 9</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b) The optimal CBU and TU in Scenario 4</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) The optimal CBU and TU in Scenario 6</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(f) The optimal CBU and TU in Scenario 8</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(h) The optimal CBU and TU in Scenario 10</figcaption></figure>

<caption>Fig. 7. The optimal CBU and TU under different substitutability degrees between products.</caption>

SP&SC policy is identified as the superior strategy for maximizing the optimal TSE value, regardless of the degree of substitutability among their products. Additionally, the figure suggests that higher degrees of substitutability among products lead to increased substitutability efforts in the market.

## 5. Conclusion

This study has provided a thorough examination of the interplay between CSR and SM within the context of SSCM. We have proposed five CSR regulatory policies and applied a Stackelberg game theoretical

framework to analyze their impact on the performance of SSCs that sell substitutable products under monopoly and oligopoly market structures. We highlighted the crucial role that governments play in selecting the appropriate policy as the basis for effective regulations. We introduced and evaluated the impacts of different CSR regulatory policies, namely Deregulation, Direct Tariff on Market, Sustainability Penalty & Sustainability Credits, Direct Limitation on Sustainability, and Government Cooperative Sustainability Efforts, on the performance of SSCs in both monopoly and oligopoly markets. We found that members of an SSC may not simply be profit-seeking, instead, they may increase their sustainability levels based on ethical and CSR strategy considerations.
<figure><img src="image_9.png" /><figcaption>(a) The optimal CBU and TU in Scenario 3</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) The optimal CBU and TU in Scenario 5</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(e) The optimal CBU and TU in Scenario 7</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(g) The optimal CBU and TU in Scenario 9</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b) The optimal CBU and TU in Scenario 4</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) The optimal CBU and TU in Scenario 6</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(f) The optimal CBU and TU in Scenario 8</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(h) The optimal CBU and TU in Scenario 10</figcaption></figure>

<caption>Fig. 7. The optimal CBU and TU under different substitutability degrees between products.</caption>

SP&SC policy is identified as the superior strategy for maximizing the optimal TSE value, regardless of the degree of substitutability among their products. Additionally, the figure suggests that higher degrees of substitutability among products lead to increased substitutability efforts in the market.

## 5. Conclusion

This study has provided a thorough examination of the interplay between CSR and SM within the context of SSCM. We have proposed five CSR regulatory policies and applied a Stackelberg game theoretical

framework to analyze their impact on the performance of SSCs that sell substitutable products under monopoly and oligopoly market structures. We highlighted the crucial role that governments play in selecting the appropriate policy as the basis for effective regulations. We introduced and evaluated the impacts of different CSR regulatory policies, namely Deregulation, Direct Tariff on Market, Sustainability Penalty & Sustainability Credits, Direct Limitation on Sustainability, and Government Cooperative Sustainability Efforts, on the performance of SSCs in both monopoly and oligopoly markets. We found that members of an SSC may not simply be profit-seeking, instead, they may increase their sustainability levels based on ethical and CSR strategy considerations.
<figure><img src="image_17.png" /><figcaption>Fig. 8. The optimal TSE under different substitutability degrees between products.</figcaption></figure>

Through a numerical example and a comprehensive sensitivity analysis, important managerial insights were provided to SSCs on how to effectively adopt CSR and SM principles to promote decarbonization and other environmental benefits. Numerical analysis suggested that the DTM policy of the RA in the monopoly market is the superior strategy from the perspectives of chain members' and government's total utilities and optimal sustainability efforts. We also find that the DTM policy in the oligopoly market and DLS policy in the monopoly market yield a higher market share of the sustainable product, which the DTM in the superior policy. From the viewpoints of the market demand and sustainability level in monopoly market the DTM policy is the best, wherein the SP&SC policy is superior in the oligopoly market. Our findings have uncovered the connection between government CSR regulations, market structure, and environmental sustainability, providing valuable insights for SSC managers. The successful implementation of CSR and SM approaches can lead to economic, social, and environmental well-being for all stakeholders involved. As such, it is of paramount importance for governments to implement appropriate policies and regulations to promote the principles of CSR and SM, particularly with regards to environmental sustainability.

There are several interesting and challenging future research directions that can be considered. For example, examining the impact of asymmetric information sharing in real-world scenarios, as many enterprises may be hesitant to share necessary information. Another suggestion is to investigate alternative governmental policies for CSR, such

as mandatory CSR information disclosure (see Liu et al., 2023). Additionally, incorporating green and smart manufacturing parameters into the model at the same time and examining the joint effects of governmental smart-green policies would provide valuable insights.

### CRediT authorship contribution statement

**Ashkan Hafezalkotob:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing, Project administration. **Sobhan Arisian:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing, Project administration. **Raziyeh Reza-Gharehbagh:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing. **Lia Nersesian:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

The data that has been used is confidential.

## Appendix: Mathematical proofs

### Proposition 1 proof

In a Stackelberg game model under Drg. policy in Scenario 1, to obtain the value of $p_i^{(1)}$, $\frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}}$ and $\frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}}$ must be obtained ($m_{R_i}$ represents the retailer's
<figure><img src="image_17.png" /><figcaption>Fig. 8. The optimal TSE under different substitutability degrees between products.</figcaption></figure>

Through a numerical example and a comprehensive sensitivity analysis, important managerial insights were provided to SSCs on how to effectively adopt CSR and SM principles to promote decarbonization and other environmental benefits. Numerical analysis suggested that the DTM policy of the RA in the monopoly market is the superior strategy from the perspectives of chain members' and government's total utilities and optimal sustainability efforts. We also find that the DTM policy in the oligopoly market and DLS policy in the monopoly market yield a higher market share of the sustainable product, which the DTM in the superior policy. From the viewpoints of the market demand and sustainability level in monopoly market the DTM policy is the best, wherein the SP&SC policy is superior in the oligopoly market. Our findings have uncovered the connection between government CSR regulations, market structure, and environmental sustainability, providing valuable insights for SSC managers. The successful implementation of CSR and SM approaches can lead to economic, social, and environmental well-being for all stakeholders involved. As such, it is of paramount importance for governments to implement appropriate policies and regulations to promote the principles of CSR and SM, particularly with regards to environmental sustainability.

There are several interesting and challenging future research directions that can be considered. For example, examining the impact of asymmetric information sharing in real-world scenarios, as many enterprises may be hesitant to share necessary information. Another suggestion is to investigate alternative governmental policies for CSR, such

as mandatory CSR information disclosure (see Liu et al., 2023). Additionally, incorporating green and smart manufacturing parameters into the model at the same time and examining the joint effects of governmental smart-green policies would provide valuable insights.

### CRediT authorship contribution statement

**Ashkan Hafezalkotob:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing, Project administration. **Sobhan Arisian:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing, Project administration.

**Raziyeh Reza-Gharehbagh:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing. **Lia Nersesian:** Investigation, Validation, Writing – original draft, Formal analysis, Writing – review & editing.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

The data that has been used is confidential.

## Appendix: Mathematical proofs

### Proposition 1 proof

In a Stackelberg game model under Drg. policy in Scenario 1, to obtain the value of $p_i^{(1)}$, $\frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}}$ and $\frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}}$ must be obtained ($m_{R_i}$ represents the retailer's
unit marginal profit). In addition, with considering $\frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}} = 0$ and $\frac{\partial U_{R_j}^{(1)}}{\partial m_{R_j}^{(1)}} = 0$ the optimal retailer's price will be calculated:

$$ \frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}} = \alpha_i - w_i^{(1)} + dw_j^{(1)} - 2m_{R_i}^{(1)} + dm_{R_j}^{(1)} + \beta s^{(1)} = 0; i = 1, 2; j = 3 - i, \quad (A.1) $$

$$ \frac{\partial U_{R_i}^{(1)}}{\partial m_{R_j}^{(1)}} = \alpha_j - w_j^{(1)} + dw_i^{(1)} - 2m_{R_j}^{(1)} + dm_{R_i}^{(1)} + \beta s^{(1)} = 0; j = 1, 2; i = 3 - j, \quad (A.2) $$

$$ m_{R_i}^{(1)} = \frac{[2(\alpha_i - w_i^{(1)} + dw_j^{(1)}) + d(\alpha_j - w_j^{(1)} + dw_i^{(1)}) + \beta(2 + d)s^{(1)}]}{V}; i = 1, 2; j = 3 - i. \quad (A.3) $$

By substituting $m_{R_i}^{(1)}$ into following equation the optimal retailer's price will be obtained:

$$ p_i^{(1)} = w_i^{(1)} + m_{R_i}^{(1)}; i = 1, 2; j = 3 - i. \quad (A.4) $$

To obtain optimal manufacturer's prices according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(1)}}{\partial w_i^{(1)}}=0$ and $\frac{\partial U_{M_j}^{(1)}}{\partial w_j^{(1)}}=0$ must be solved. With considering $w_i^{(1)} - c_i^{(1)} = m_{M_i}^{(1)}$ and $w_j^{(1)} - c_j^{(1)} = m_{M_j}^{(1)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(1)}}{\partial m_{M_i}^{(1)}} = 0$ and $\frac{\partial U_{M_j}^{(1)}}{\partial m_{M_j}^{(1)}} = 0$ are as follows:

$$ \frac{\partial U_{M_i}^{(1)}}{\partial m_{M_i}^{(1)}} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2+d)\rho_M}{2\eta\tau V} + [2(d^2 - 2) + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_i}^{(1)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_j}^{(1)} = 0, \quad (A.5) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ \frac{\partial U_{M_j}^{(1)}}{\partial m_{M_j}^{(1)}} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2+d)\rho_M}{2\eta\tau V} + [2(d^2 - 2) + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_j}^{(1)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_i}^{(1)} = 0, \quad (A.6) $$

where $j = 1, 2$ and $i = 3 - j$.

By solving the equations above the optimal $m_{M_i}^{(1)}$ is achieved:

$$ m_{M_i}^{(1)} = \frac{[E_i^{(1)}B^{(1)} - E_j^{(1)}A^{(1)}]}{[(A^{(1)})^2 - (B^{(1)})^2]; i = 1, 2; j = 3 - i} \quad (A.7) $$

To simplify the model, it is considered:

$$ A^{(1)} = 2d + \frac{\beta^2(2+d)^2}{2\eta\tau V}, \quad (A.8) $$

$$ B^{(1)} = -2(2-d^2) + \frac{\beta^2(2+d)^2}{2\eta\tau V}, \quad (A.9) $$

$$ E_i^{(1)} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2+d)\rho_M}{2\eta\tau V}; i = 1, 2; j = 3 - i, \quad (A.10) $$

$$ E_j^{(1)} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2+d)\rho_M}{2\eta\tau V}; j = 1, 2; i = 3 - j, \quad (A.11) $$

$$ V = 4 - d^2. \quad (A.12) $$

By substituting $m_{M_i}^{(1)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$ w_i^{(1)} = c_i + m_{M_i}^{(1)}; i = 1, 2; j = 3 - i. \quad (A.13) $$

$\frac{\partial U_{M_i}^{(1)}}{\partial s^{(1)}} = \sum_{i=1}^{2} m_{M_i}^{(1)} \frac{\beta(2+d)}{V} - 2\eta\tau s + \rho_M = 0$, By calculating $\frac{\partial U_{M_i}^{(1)}}{\partial s_i^{(1)}} = 0$, the optimal amount of $s_i^{(1)}$ is achieved:

$$ \frac{\partial U_{M_i}^{(1)}}{\partial s_i^{(1)}} = \sum_{i=1}^{2} m_{M_i}^{(1)} \frac{\beta(2+d)}{V} - 2\eta\tau s + \rho_M = 0 \quad (A.15) $$

$$ s^{(1)} = \frac{\sum_{i=1}^{2} m_{M_i}^{(1)} \beta(2+d)}{2\eta\tau V} + \frac{\rho_M}{2\eta\tau}, \quad (A.16) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ H(U_{M_i}^{1}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix} $$

Considering $H(U_{M_i}^{(1)}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,
unit marginal profit). In addition, with considering $\frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}} = 0$ and $\frac{\partial U_{R_j}^{(1)}}{\partial m_{R_j}^{(1)}} = 0$ the optimal retailer's price will be calculated:

$$ \frac{\partial U_{R_i}^{(1)}}{\partial m_{R_i}^{(1)}} = \alpha_i - w_i^{(1)} + dw_j^{(1)} - 2m_{R_i}^{(1)} + dm_{R_j}^{(1)} + \beta s^{(1)} = 0; i = 1, 2; j = 3 - i, \quad (A.1) $$

$$ \frac{\partial U_{R_i}^{(1)}}{\partial m_{R_j}^{(1)}} = \alpha_j - w_j^{(1)} + dw_i^{(1)} - 2m_{R_j}^{(1)} + dm_{R_i}^{(1)} + \beta s^{(1)} = 0; j = 1, 2; i = 3 - j, \quad (A.2) $$

$$ m_{R_i}^{(1)} = \frac{[2(\alpha_i - w_i^{(1)} + dw_j^{(1)}) + d(\alpha_j - w_j^{(1)} + dw_i^{(1)}) + \beta(2+d)s^{(1)}]}{V}; i = 1, 2; j = 3 - i. \quad (A.3) $$

By substituting $m_{R_i}^{(1)}$ into following equation the optimal retailer's price will be obtained:

$$ p_i^{(1)} = w_i^{(1)} + m_{R_i}^{(1)}; i = 1, 2; j = 3 - i. \quad (A.4) $$

To obtain optimal manufacturer's prices according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(1)}}{\partial w_i^{(1)}}=0$ and $\frac{\partial U_{M_j}^{(1)}}{\partial w_j^{(1)}}=0$ must be solved. With considering $w_i^{(1)} - c_i^{(1)} = m_{M_i}^{(1)}$ and $w_j^{(1)} - c_j^{(1)} = m_{M_j}^{(1)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(1)}}{\partial m_{M_i}^{(1)}} = 0$ and $\frac{\partial U_{M_j}^{(1)}}{\partial m_{M_j}^{(1)}} = 0$ are as follows:

$$ \frac{\partial U_{M_i}^{(1)}}{\partial m_{M_i}^{(1)}} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2+d)\rho_M}{2\eta\tau V} + [2(d^2 - 2) + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_i}^{(1)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_j}^{(1)} = 0, \quad (A.5) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ \frac{\partial U_{M_j}^{(1)}}{\partial m_{M_j}^{(1)}} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2+d)\rho_M}{2\eta\tau V} + [2(d^2 - 2) + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_j}^{(1)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V}]m_{M_i}^{(1)} = 0, \quad (A.6) $$

where $j = 1, 2$ and $i = 3 - j$.

By solving the equations above the optimal $m_{M_i}^{(1)}$ is achieved:

$$ m_{M_i}^{(1)} = \frac{[E_i^{(1)}B^{(1)} - E_j^{(1)}A^{(1)}]}{[(A^{(1)})^2 - (B^{(1)})^2]; i = 1, 2; j = 3 - i} \quad (A.7) $$

To simplify the model, it is considered:

$$ A^{(1)} = 2d + \frac{\beta^2(2+d)^2}{2\eta\tau V}, \quad (A.8) $$

$$ B^{(1)} = -2(2-d^2) + \frac{\beta^2(2+d)^2}{2\eta\tau V}, \quad (A.9) $$

$$ E_i^{(1)} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2+d)\rho_M}{2\eta\tau}; i = 1, 2; j = 3 - i, \quad (A.10) $$

$$ E_j^{(1)} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2+d)\rho_M}{2\eta\tau}; j = 1, 2; i = 3 - j, \quad (A.11) $$

$$ V = 4 - d^2. \quad (A.12) $$

By substituting $m_{M_i}^{(1)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$ w_i^{(1)} = c_i + m_{M_i}^{(1)}; i = 1, 2; j = 3 - i. \quad (A.13) $$

$\frac{\partial U_{M_i}^{(1)}}{\partial s_i^{(1)}} = \sum_{i=1}^{2} m_{M_i}^{(1)} \frac{\beta(2+d)}{V} - 2\eta\tau s + \rho_M = 0$, By calculating $\frac{\partial U_{M_i}^{(1)}}{\partial s_i^{(1)}} = 0$, the optimal amount of $s_i^{(1)}$ is achieved:

$$ \frac{\partial U_{M_i}^{(1)}}{\partial s_i^{(1)}} = \sum_{i=1}^{2} m_{M_i}^{(1)} \frac{\beta(2+d)}{V} - 2\eta\tau s + \rho_M = 0 \quad (A.15) $$

$$ s^{(1)} = \frac{\sum_{i=1}^{2} m_{M_i}^{(1)} \beta(2+d)}{2\eta\tau V} + \frac{\rho_M}{2\eta\tau}, \quad (A.16) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ H(U_{M_i}^{1}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix} $$

Considering $H(U_{M_i}^{(1)}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,
$H(U_{M_i}^{(1)})$ is negatively defined. It is necessary to mention that if $η ≥ β^2(2-d)/(2+d)[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $det(H(U_{M_i}^{(1)})) > 0$, the concavity of the profit function holds. □

**Proposition 2 proof**

To obtain the retailer price Under Drg. policy in Scenario 2 $\frac{\partial U_{R_i}^{(2)}}{\partial m_{R_i}^{(2)}}$ and $\frac{\partial U_{R_j}^{(2)}}{\partial m_{R_j}^{(2)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(2)}}{\partial m_{R_i}^{(2)}} = 0$ and $\frac{\partial U_{R_j}^{(2)}}{\partial m_{R_j}^{(2)}} = 0$ the optimal retailer's price will be calculated:

$$ \frac{\partial U_{R_i}^{(2)}}{\partial m_{R_i}^{(2)}} = \alpha_i - w_i^{(2)} + dw_j^{(2)} - 2m_{R_i}^{(2)} + dm_{R_j}^{(2)} + \beta(s_i^{(2)} - \theta s_j^{(2)}) = 0; i = 1, 2j = 3 - i, \quad (A.17) $$

$$ \frac{\partial U_{R_j}^{(2)}}{\partial m_{R_j}^{(2)}} = \alpha_j - w_j^{(2)} + dw_i^{(2)} - 2m_{R_j}^{(2)} + dm_{R_i}^{(2)} + \beta(s_j^{(2)} - \theta s_i^{(2)}) = 0; j = 1, 2i = 3 - j, \quad (A.18) $$

$$ m_{R_i}^{(2)} = \left\{ \begin{array}{l} 2(\alpha_i - w_i^{(2)} + dw_j^{(2)} + \beta(s_i^{(2)} - \theta s_j^{(2)})) + \\ d(\alpha_j - w_j^{(2)} + dw_i^{(2)} + \beta(s_j^{(2)} - \theta s_i^{(2)})) \end{array} \right\} / V; \quad i = 1, 2j = 3 - i. \quad (A.19) $$

By substituting $m_{R_i}^{(2)}$ into the following equation the optimal retailer's price will be obtained:

$$ p_i^{(2)} = w_i^{(2)} + m_{R_i}^{(2)}; \quad i = 1, 2. \quad (A.20) $$

To obtain optimal manufacturer's prices according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(2)}}{\partial w_i^{(2)}}=0$ and $\frac{\partial U_{M_j}^{(2)}}{\partial w_j^{(2)}} = 0$ must be solved. With considering $w_i^{(2)} - c_i^{(2)} = m_{M_i}$ and $w_j^{(2)} - c_j^{(2)} = m_{M_j}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(2)}}{\partial m_{M_i}} = 0$ and $\frac{\partial U_{M_j}^{(2)}}{\partial m_{M_j}}$ are as follows:

$$ \frac{\partial U_{M_i}^{(2)}}{\partial m_{M_i}} = [2(\alpha_i - c_i + dc_j + \frac{\rho_M \beta}{2\eta_i \tau_i} - \theta \frac{\rho_M \beta}{2\eta_j \tau_j}) + d(\alpha_j - c_j + dc_i + \frac{\rho_M \beta}{2\eta_j \tau_j} - \theta \frac{\rho_M \beta}{2\eta_i \tau_i})] + [\frac{2\beta^2(2-d\theta)}{2\eta_i \tau_i V} - \theta \frac{d\beta^2(2-d\theta)}{2\eta_i \tau_i V} - 2(2-d^2)]m_{M_i}^{(2)} + [\frac{d\beta^2(2-d\theta)}{2\eta_j \tau_j V} - \theta \frac{2\beta^2(2-d\theta)}{2\eta_j \tau_j V} + d]m_{M_j}^{(2)} = 0, \quad (A.21) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ \frac{\partial U_{M_i}^{(2)}}{\partial m_{M_i}} = [2(\alpha_j - c_j + dc_i + \frac{\rho_M \beta}{2\eta_j \tau_j} - \theta \frac{\rho_M \beta}{2\eta_i \tau_i}) + d(\alpha_i - c_i + dc_j + \frac{\rho_M \beta}{2\eta_i \tau_i} - \theta \frac{\rho_M \beta}{2\eta_j \tau_j})] + [\frac{2\beta^2(2-d\theta)}{2\eta_j \tau_j V} - \theta \frac{d\beta^2(2-d\theta)}{2\eta_j \tau_j V} - 2(2-d^2)]m_{M_i}^{(2)} + [\frac{d\beta^2(2-d\theta)}{2\eta_i \tau_i V} - \theta \frac{2\beta^2(2-d\theta)}{2\eta_i \tau_i V} + d]m_{M_j}^{(2)} = 0, \quad (A.22) $$

where $j = 1, 2$ and $i = 3 - j$.

By solving the equations above the optimal $m_{M_i}^{(2)}$ is achieved:

$$ m_{M_i}^{(2)} = \frac{[E_i^{(2)} B_j^{(2)} - E_j^{(2)} A_j^{(2)}]}{[A_i^{(2)} A_j^{(2)} - B_i^{(2)} B_j^{(2)}]}; \quad i = 1, 2j = 3 - i. \quad (A.23) $$

To simplify the model, it is considered:

$$ E_i^{(2)} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2 - d\theta)\rho_M}{2\eta_i \tau_i} + \frac{\beta(d - 2\theta)\rho_{M_i}}{2\eta_j \tau_j}, \quad (A.24) $$

$$ A_i^{(2)} = d + \beta^2(2 - d\theta)(d - 2\theta)/2\eta_i \tau_i V, \quad (A.25) $$

$$ B_i^{(2)} = \beta^2(2 - d\theta)^2/2\eta_i \tau_i V - 2(2 - d^2), \quad (A.26) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ E_j^{(2)} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2 - d\theta)\rho_M}{2\eta_j \tau_j} + \frac{\beta(d - 2\theta)\rho_{M_i}}{2\eta_i \tau_i}, \quad (A.27) $$

$$ A_j^{(2)} = d + \beta^2(2 - d\theta)(d - 2\theta)/2\eta_j \tau_j V, \quad (A.28) $$

$$ B_j^{(2)} = \beta^2(2 - d\theta)^2/2\eta_j \tau_j V - 2(2 - d^2), \quad (A.29) $$

where $j = 1, 2$ and $i = 3 - j$. By substituting $m_{M_i}^{(2)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$H(U_{M_i}^{(1)})$ is negatively defined. It is necessary to mention that if $η≥β^2(2-d)/(2+d)[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $det(H(U_{M_i}^{(1)})) > 0$, the concavity of the profit function holds. □

**Proposition 2 proof**

To obtain the retailer price Under Drg. policy in Scenario 2 $\frac{\partial U_{R_i}^{(2)}}{\partial m_{R_i}^{(2)}}$ and $\frac{\partial U_{R_j}^{(2)}}{\partial m_{R_j}^{(2)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(2)}}{\partial m_{R_i}^{(2)}} = 0$ and $\frac{\partial U_{R_j}^{(2)}}{\partial m_{R_j}^{(2)}} = 0$ the optimal retailer's price will be calculated:

$$ \frac{\partial U_{R_i}^{(2)}}{\partial m_{R_i}^{(2)}} = \alpha_i - w_i^{(2)} + dw_j^{(2)} - 2m_{R_i}^{(2)} + dm_{R_j}^{(2)} + \beta(s_i^{(2)} - \theta s_j^{(2)}) = 0; i = 1, 2j = 3 - i, \quad (A.17) $$

$$ \frac{\partial U_{R_j}^{(2)}}{\partial m_{R_j}^{(2)}} = \alpha_j - w_j^{(2)} + dw_i^{(2)} - 2m_{R_j}^{(2)} + dm_{R_i}^{(2)} + \beta(s_j^{(2)} - \theta s_i^{(2)}) = 0; j = 1, 2i = 3 - j, \quad (A.18) $$

$$ m_{R_i}^{(2)} = \left\{ \begin{array}{l} 2(\alpha_i - w_i^{(2)} + dw_j^{(2)} + \beta(s_i^{(2)} - \theta s_j^{(2)})) + \\ d(\alpha_j - w_j^{(2)} + dw_i^{(2)} + \beta(s_j^{(2)} - \theta s_i^{(2)})) \end{array} \right\} / V; \quad i = 1, 2j = 3 - i. \quad (A.19) $$

By substituting $m_{R_i}^{(2)}$ into the following equation the optimal retailer's price will be obtained:

$$ p_i^{(2)} = w_i^{(2)} + m_{R_i}^{(2)}; \quad i = 1, 2. \quad (A.20) $$

To obtain optimal manufacturer's prices according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(2)}}{\partial w_i^{(2)}}=0$ and $\frac{\partial U_{M_j}^{(2)}}{\partial w_j^{(2)}} = 0$ must be solved. With considering $w_i^{(2)} - c_i^{(2)} = m_{M_i}$ and $w_j^{(2)} - c_j^{(2)} = m_{M_j}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(2)}}{\partial m_{M_i}} = 0$ and $\frac{\partial U_{M_j}^{(2)}}{\partial m_{M_j}}$ are as follows:

$$ \frac{\partial U_{M_i}^{(2)}}{\partial m_{M_i}} = [2(\alpha_i - c_i + dc_j + \frac{\rho_M \beta}{2\eta_i \tau_i} - \theta \frac{\rho_M \beta}{2\eta_j \tau_j}) + d(\alpha_j - c_j + dc_i + \frac{\rho_M \beta}{2\eta_j \tau_j} - \theta \frac{\rho_M \beta}{2\eta_i \tau_i})] + [\frac{2\beta^2(2-d\theta)}{2\eta_i \tau_i V} - \theta \frac{d\beta^2(2-d\theta)}{2\eta_i \tau_i V} - 2(2-d^2)]m_{M_i}^{(2)} + [\frac{d\beta^2(2-d\theta)}{2\eta_j \tau_j V} - \theta \frac{2\beta^2(2-d\theta)}{2\eta_j \tau_j V} + d]m_{M_j}^{(2)} = 0, \quad (A.21) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ \frac{\partial U_{M_i}^{(2)}}{\partial m_{M_i}} = [2(\alpha_j - c_j + dc_i + \frac{\rho_M \beta}{2\eta_j \tau_j} - \theta \frac{\rho_M \beta}{2\eta_i \tau_i}) + d(\alpha_i - c_i + dc_j + \frac{\rho_M \beta}{2\eta_i \tau_i} - \theta \frac{\rho_M \beta}{2\eta_j \tau_j})] + [\frac{2\beta^2(2-d\theta)}{2\eta_j \tau_j V} - \theta \frac{d\beta^2(2-d\theta)}{2\eta_j \tau_j V} - 2(2-d^2)]m_{M_i}^{(2)} + [\frac{d\beta^2(2-d\theta)}{2\eta_i \tau_i V} - \theta \frac{2\beta^2(2-d\theta)}{2\eta_i \tau_i V} + d]m_{M_j}^{(2)} = 0, \quad (A.22) $$

where $j = 1, 2$ and $i = 3 - j$.

By solving the equations above the optimal $m_{M_i}^{(2)}$ is achieved:

$$ m_{M_i}^{(2)} = \frac{[E_i^{(2)}B_j^{(2)} - E_j^{(2)}A_j^{(2)}]}{[A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}]}; \quad i = 1, 2j = 3 - i. \quad (A.23) $$

To simplify the model, it is considered:

$$ E_i^{(2)} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2 - d\theta)\rho_M}{2\eta_i\tau_i} + \frac{\beta(d - 2\theta)\rho_{M_i}}{2\eta_j\tau_j}, \quad (A.24) $$

$$ A_i^{(2)} = d + \beta^2(2 - d\theta)(d - 2\theta)/2\eta_i\tau_i V, \quad (A.25) $$

$$ B_i^{(2)} = \beta^2(2 - d\theta)^2/2\eta_i\tau_i V - 2(2 - d^2), \quad (A.26) $$

where $i = 1, 2$ and $j = 3 - i$.

$$ E_j^{(2)} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2 - d\theta)\rho_M}{2\eta_j\tau_j} + \frac{\beta(d - 2\theta)\rho_{M_i}}{2\eta_i\tau_i}, \quad (A.27) $$

$$ A_j^{(2)} = d + \beta^2(2 - d\theta)(d - 2\theta)/2\eta_j\tau_j V, \quad (A.28) $$

$$ B_j^{(2)} = \beta^2(2 - d\theta)^2/2\eta_j\tau_j V - 2(2 - d^2), \quad (A.29) $$

where $j = 1, 2$ and $i = 3 - j$. By substituting $m_{M_i}^{(2)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(2)} = c_i + m_{M_i}^{(2)}; i = 1, 2; j = 3 - i. \qquad (A.30)$$

By calculating $\frac{\partial U_{M_i}^{(2)}}{\partial s_i^{(2)}} = 0$, the optimal amount of $s_i^{(2)}$ is achieved:

$$\frac{\partial U_{M_i}^{(2)}}{\partial s_i^{(2)}} = (w_i^{(2)} - c_i^{(2)}) \frac{2\beta - d\beta\theta}{V} - 2\eta_i \tau_i s_i^{(2)} + \rho_{M_i} = 0, \qquad (A.31)$$

$$s_i^{(2)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(2)} + \rho_{M_i}V]}{2\eta_i\tau_i V}, \qquad (A.32)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(2)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta - d\beta\theta}{4-d^2} \\ \frac{2\beta - d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(2)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(2)})) > 0$, the concavity of the profit function holds. $\square$

**Proposition 3 proof**

To obtain the retailer's price Under DTM policy $t = (t_1, t_2)$ in Scenario 3 $\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_i}^{(3)}}$ and $\frac{\partial U_{R_j}^{(3)}}{\partial m_{R_j}^{(3)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_i}^{(3)}} = 0$ and $\frac{\partial U_{R_j}^{(3)}}{\partial m_{R_j}^{(3)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_i}^{(3)}} = \alpha_i - w_i^{(3)} + dw_j^{(3)} - 2m_{R_i}^{(3)} + dm_{R_i}^{(3)} + \beta s_i^{(3)} - t_i + dt_j = 0; i = 1, 2; j = 3 - i, \qquad (A.33)$$

$$\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_j}^{(3)}} = \alpha_j - w_j^{(3)} + dw_i^{(3)} - 2m_{R_j}^{(3)} + dm_{R_j}^{(3)} + \beta s_j^{(3)} - t_j + dt_i = 0; j = 1, 2; i = 3 - j, \qquad (A.34)$$

$$m_{R_i}^{(3)} = \frac{2(\alpha_i - w_i^{(3)} + dw_j^{(3)}) + d(\alpha_j - w_j^{(3)} + dw_i^{(3)}) + \beta(2+d)s_i^{(3)} - (2-d^2)t_i + t_j}{V}; i = 1, 2; j = 3 - i, \qquad (A.35)$$

By substituting $m_{R_i}^{(3)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(3)} = w_i^{(3)} + m_{R_i}^{(3)}; i = 1, 2; j = 3 - i. \qquad (A.36)$$

To obtain optimal manufacturer's prices according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(3)}}{\partial w_i^{(3)}} = 0$ and $\frac{\partial U_{M_i}^{(3)}}{\partial w_j^{(3)}} = 0$ must be solved. With considering $w_i^{(3)} - c_i^{(3)} = m_{M_i}^{(3)}$ and $w_j^{(3)} - c_j^{(3)} = m_{M_j}^{(3)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}^{(3)}} = 0$ and $\frac{\partial U_{M_j}^{(3)}}{\partial m_{M_j}^{(3)}}$ are as follows:

$$\frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}} = 2(\alpha_j - c_j + dc_i - m_{M_i} + dm_{M_i} - t_i + dt_j) + d(\alpha_j - c_j + dc_j - m_{M_i} + dm_{M_i} - t_j + dt_i) + \frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}} = 2(\alpha_i - c_i + dc_j - m_{M_i} + dm_{M_i} - t_i + dt_j) \\ + \frac{\beta^2(2+d)^2(m_{M_i}^{(3)} + m_{M_j}^{(3)})}{2\eta\tau V} + \frac{\beta(2+d)\rho_{M_i}}{2\eta\tau} + (d-2)m_{M_i}^{(3)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V} + d]m_{M_i}^{(3)} = 0, \\ +d(\alpha_j - c_j + dc_i - m_{M_i} + dm_{M_i} - t_j + dt_i) + \frac{\beta^2(2+d)^2(m_{M_i}^{(3)} + m_{M_j}^{(3)})}{2\eta\tau V} + \frac{\beta(2+d)\rho_{M_j}}{2\eta\tau} + (d-2)m_{M_j}^{(3)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V} + d]m_{M_j}^{(3)} = 0, \\ \text{where } i = 1, 2 \text{ and } j = 3 - i.$$

$$\frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}} = 2(\alpha_j - c_j + dc_i - m_{M_i} + dm_{M_i} - t_i + dt_j) + d(\alpha_i - c_i + dc_j - m_{M_i} + dm_{M_i} - t_i + dt_j) \\ + \frac{\beta^2(2+d)^2(m_{M_i}^{(3)} + m_{M_j}^{(3)})}{2\eta\tau V} + \frac{\beta(2+d)\rho_{M_i}}{2\eta\tau} + (d-2)m_{M_i}^{(3)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V} + d]m_{M_i}^{(3)} = 0, \qquad (A.38)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(3)}$ is achieved:

$$m_{M_i}^{(3)} = m_{M_i}^{(1)} - \frac{(2 - d^2)B^{(1)} + dA^{(1)}}{(A^{(1)})^2 - (B^{(1)})^2}t_i + \frac{dB^{(1)} + (2 - d^2)A^{(1)}}{(A^{(1)})^2 - (B^{(1)})^2}t_j; i = 1, 2; j = 3 - i. \qquad (A.39)$$

By substituting $m_{M_i}^{(3)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(2)} = c_i + m_{M_i}^{(2)}; i = 1, 2; j = 3 - i. \qquad (A.30)$$

By calculating $\frac{\partial U_{M_i}^{(2)}}{\partial s_i^{(2)}} = 0$, the optimal amount of $s_i^{(2)}$ is achieved:

$$\frac{\partial U_{M_i}^{(2)}}{\partial s_i^{(2)}} = (w_i^{(2)} - c_i^{(2)}) \frac{2\beta - d\beta\theta}{V} - 2\eta_i \tau_i s_i^{(2)} + \rho_{M_i} = 0, \qquad (A.31)$$

$$s_i^{(2)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(2)} + \rho_{M_i}V]}{2\eta_i\tau_i V}, \qquad (A.32)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(2)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(2)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(2)})) > 0$, the concavity of the profit function holds. $\square$

**Proposition 3 proof**

To obtain the retailer's price Under DTM policy $t = (t_1, t_2)$ in Scenario 3 $\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_i}^{(3)}}$ and $\frac{\partial U_{R_j}^{(3)}}{\partial m_{R_j}^{(3)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_i}^{(3)}} = 0$ and $\frac{\partial U_{R_j}^{(3)}}{\partial m_{R_j}^{(3)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_i}^{(3)}} = \alpha_i - w_i^{(3)} + dw_j^{(3)} - 2m_{R_i}^{(3)} + dm_{R_i}^{(3)} + \beta s_i^{(3)} - t_i + dt_j = 0; i = 1, 2; j = 3 - i, \qquad (A.33)$$

$$\frac{\partial U_{R_i}^{(3)}}{\partial m_{R_j}^{(3)}} = \alpha_j - w_j^{(3)} + dw_i^{(3)} - 2m_{R_j}^{(3)} + dm_{R_j}^{(3)} + \beta s_j^{(3)} - t_j + dt_i = 0; j = 1, 2; i = 3 - j, \qquad (A.34)$$

$$m_{R_i}^{(3)} = \frac{2(\alpha_i - w_i^{(3)} + dw_j^{(3)}) + d(\alpha_j - w_j^{(3)} + dw_i^{(3)}) + \beta(2+d)s_i^{(3)} - (2-d^2)t_i + t_j}{V}; i = 1, 2; j = 3 - i, \qquad (A.35)$$

By substituting $m_{R_i}^{(3)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(3)} = w_i^{(3)} + m_{R_i}^{(3)}; i = 1, 2; j = 3 - i. \qquad (A.36)$$

To obtain optimal manufacturer's prices according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(3)}}{\partial w_i^{(3)}} = 0$ and $\frac{\partial U_{M_j}^{(3)}}{\partial w_j^{(3)}} = 0$ must be solved. With considering $w_i^{(3)} - c_i^{(3)} = m_{M_i}^{(3)}$ and $w_j^{(3)} - c_j^{(3)} = m_{M_j}^{(3)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}^{(3)}} = 0$ and $\frac{\partial U_{M_j}^{(3)}}{\partial m_{M_j}^{(3)}}$ are as follows:

$$\frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}^{(3)}} = 2(\alpha_j - c_j + dc_i - m_{M_i} + dm_{M_i} - t_i + dt_j) + d(\alpha_j - c_j + dc_j - m_{M_i} + dm_{M_i} - t_j + dt_i) + \frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}^{(3)}} = 2(\alpha_i - c_i + dc_j - m_{M_i} + dm_{M_i} - t_i + dt_j) \\ + \left[ \frac{\beta^2(2+d)^2(m_{M_i}^{(3)} + m_{M_i}^{(3)})}{2\eta\tau V} + \frac{\beta(2+d)\rho_M}{2\eta\tau} + (d-2)m_{M_i}^{(3)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V} + d]m_{M_i}^{(3)} = 0, \\ +d(\alpha_j - c_j + dc_i - m_{M_i} + dm_{M_i} - t_j + dt_i) + \left[ \frac{\beta^2(2+d)^2(m_{M_i}^{(3)} + m_{M_i}^{(3)})}{2\eta\tau V} + \frac{\beta(2+d)\rho_M}{2\eta\tau} + (d-2)m_{M_i}^{(3)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V} + d]m_{M_i}^{(3)} = 0, \\ \text{where } i = 1, 2 \text{ and } j = 3 - i. \end{aligned}$$

$$\frac{\partial U_{M_i}^{(3)}}{\partial m_{M_i}^{(3)}} = 2(\alpha_j - c_j + dc_i - m_{M_i} + dm_{M_i} - t_i + dt_j) + d(\alpha_i - c_i + dc_j - m_{M_i} + dm_{M_i} - t_i + dt_j) \\ + \left[ \frac{\beta^2(2+d)^2(m_{M_i}^{(3)} + m_{M_i}^{(3)})}{2\eta\tau V} + \frac{\beta(2+d)\rho_M}{2\eta\tau} + (d-2)m_{M_i}^{(3)} + [2d + \frac{\beta^2(2+d)^2}{2\eta\tau V} + d]m_{M_i}^{(3)} = 0, \right] \qquad (A.38)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(3)}$ is achieved:

$$m_{M_i}^{(3)} = m_{M_i}^{(1)} - \frac{(2 - d^2)B^{(1)} + dA^{(1)}}{(A^{(1)})^2 - (B^{(1)})^2} t_i + \frac{dB^{(1)} + (2 - d^2)A^{(1)}}{(A^{(1)})^2 - (B^{(1)})^2} t_j; i = 1, 2; j = 3 - i. \qquad (A.39)$$

By substituting $m_{M_i}^{(3)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(3)} = c_i + m_{M_i}^{(3)}; i = 1, 2, j = 3 - i. \qquad (A.40)$$

By calculating $\frac{\partial U_{M_i}^{(3)}}{\partial s_i^{(3)}} = 0$, the optimal amount of $s_i^{(3)}$ is achieved:

$$\frac{\partial U_{M}^{(3)}}{\partial s^{(3)}} = \sum_{i=1}^{2} m_{M_i} \frac{\beta(2+d)}{V} - 2\eta \tau s^{(3)} + \rho_M = 0, \qquad (A.41)$$

$$s^{(3)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(3)} + \rho_M V \right] / 2\eta\tau V, \qquad (A.42)$$

where $i = 1, 2, j = 3 - i$.

Considering $H(U_{M_i}^{(3)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(3)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)$ $[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(3)})) > 0$, the concavity of the profit function holds. ☐

**Proposition 4 proof**

To obtain the retailer price under Drg. policy $t = (t_1, t_2)$ in Scenario 4 $\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}}$ and $\frac{\partial U_{R_j}^{(4)}}{\partial m_{R_j}^{(4)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}} = 0$ and $\frac{\partial U_{R_j}^{(4)}}{\partial m_{R_j}^{(4)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}} = \alpha_i - w_i^{(4)} + dw_j^{(4)} - 2m_{R_i}^{(4)} + dm_{R_i}^{(4)} + \beta(s_i^{(4)} - \theta s_j^{(4)}) - t_i + dt_j = 0; i = 1, 2j = 3 - i, \qquad (A.43)$$

$$\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}} = \alpha_j - w_j^{(4)} + dw_i^{(4)} - 2m_{R_j}^{(4)} + dm_{R_j}^{(4)} + \beta(s_j^{(4)} - \theta s_i^{(4)}) - t_j + dt_i = 0; j = 1, 2i = 3 - j, \qquad (A.44)$$

$$m_{R_i}^{(4)} = \frac{\left\{ 2(\alpha_i - w_i^{(4)} + dw_j^{(4)} + \beta(s_i^{(4)} - \theta s_j^{(4)})) + d(\alpha_j - w_j^{(4)} + dw_i^{(4)} + \beta(s_j^{(4)} - \theta s_i^{(4)})) - (2-d^2)t_i + dt_j \right\}}{V}; i = 1, 2j = 3 - i. \qquad (A.45)$$

By substituting $m_{R_i}^{(4)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(4)} = w_i^{(4)} + m_{R_i}^{(4)}; i = 1, 2j = 3 - i.$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(4)}}{\partial w_i^{(4)}} = 0$ and $\frac{\partial U_{M_j}^{(4)}}{\partial w_j^{(4)}} = 0$ must be solved. With considering $w_i^{(4)} - c_i^{(4)} = m_{M_i}^{(4)}$ and $w_j^{(4)} - c_j^{(4)} = m_{M_j}^{(4)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(4)}}{\partial m_{M_i}^{(4)}} = 0$ and $\frac{\partial U_{M_j}^{(4)}}{\partial m_{M_j}^{(4)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(4)}}{\partial m_{M_i}} = 2(a_i - c_i + dc_j - t_i + dt_j) + d(a_j - c_j + dc_i - t_j + dt_i) + (2\beta - d\beta\theta) \frac{\rho_{M_i}}{2\eta_i\tau_i} + (-2\beta\theta + d\beta) \frac{\rho_{M_j}}{2\eta_j\tau_j} + (-2 + d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_i\tau_i V} + d^2 - \frac{d\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V})m_{M_i}^{(4)} + (2d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} - d + \frac{d\beta^2(2-d\theta)}{2\eta_j\tau_j V})m_{M_j}^{(4)} = 0, \qquad (A.46)$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_j}^{(4)}}{\partial m_{M_j}} = 2(a_j - c_j + dc_i - t_i + dt_j) + d(a_i - c_i + dc_j - t_i + dt_j) + (2\beta - d\beta\theta) \frac{\rho_{M_j}}{2\eta_j\tau_j} + (-2\beta\theta + d\beta) \frac{\rho_{M_i}}{2\eta_i\tau_i} + (-2 + d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_j\tau_j V} + d^2 - \frac{d\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V})m_{M_j}^{(4)} + (2d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} - d + \frac{d\beta^2(2-d\theta)}{2\eta_i\tau_i V})m_{M_i}^{(4)} = 0, \qquad (A.47)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(4)}$ is achieved:

$$m_{M_i}^{(4)} = m_{M_i}^{(2)} - \frac{(2 - d^2)B_j^{(2)} + dA_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_j^{(2)}B_j^{(2)}}t_i + \frac{dB_j^{(2)} + (2 - d^2)A_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_j^{(2)}B_j^{(2)}}t_j; i = 1, 2j = 3 - i. \qquad (A.48)$$

By substituting $m_{M_i}^{(4)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(3)} = c_i + m_{M_i}^{(3)}; i = 1, 2, j = 3 - i. \qquad (A.40)$$

By calculating $\frac{\partial U_{M_i}^{(3)}}{\partial s_i^{(3)}} = 0$, the optimal amount of $s_i^{(3)}$ is achieved:

$$\frac{\partial U_{M}^{(3)}}{\partial s^{(3)}} = \sum_{i=1}^{2} m_{M_i} \frac{\beta(2+d)}{V} - 2\eta \tau s^{(3)} + \rho_M = 0, \qquad (A.41)$$

$$s^{(3)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(3)} + \rho_M V \right] / 2\eta\tau V, \qquad (A.42)$$

where $i = 1, 2, j = 3 - i$.

Considering $H(U_{M_i}^{(3)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(3)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(3)})) > 0$, the concavity of the profit function holds. □

**Proposition 4 proof**

To obtain the retailer price under Drg. policy $t = (t_1, t_2)$ in Scenario 4 $\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}}$ and $\frac{\partial U_{R_j}^{(4)}}{\partial m_{R_j}^{(4)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}} = 0$ and $\frac{\partial U_{R_j}^{(4)}}{\partial m_{R_j}^{(4)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}} = \alpha_i - w_i^{(4)} + dw_j^{(4)} - 2m_{R_i}^{(4)} + dm_{R_i}^{(4)} + \beta(s_i^{(4)} - \theta s_j^{(4)}) - t_i + dt_j = 0; i = 1, 2j = 3 - i, \qquad (A.43)$$

$$\frac{\partial U_{R_i}^{(4)}}{\partial m_{R_i}^{(4)}} = \alpha_j - w_j^{(4)} + dw_i^{(4)} - 2m_{R_j}^{(4)} + dm_{R_j}^{(4)} + \beta(s_j^{(4)} - \theta s_i^{(4)}) - t_j + dt_i = 0; j = 1, 2i = 3 - j, \qquad (A.44)$$

$$m_{R_i}^{(4)} = \frac{\left\{ 2(\alpha_i - w_i^{(4)} + dw_j^{(4)} + \beta(s_i^{(4)} - \theta s_j^{(4)})) + d(\alpha_j - w_j^{(4)} + dw_i^{(4)} + \beta(s_j^{(4)} - \theta s_i^{(4)})) - (2-d^2)t_i + dt_j \right\}}{V}; i = 1, 2j = 3 - i. \qquad (A.45)$$

By substituting $m_{R_i}^{(4)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(4)} = w_i^{(4)} + m_{R_i}^{(4)}; i = 1, 2j = 3 - i.$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(4)}}{\partial m_{M_i}^{(4)}} = 0$ and $\frac{\partial U_{M_j}^{(4)}}{\partial m_{M_j}^{(4)}} = 0$ must be solved. With considering $w_i^{(4)} - c_i^{(4)} = m_{M_i}^{(4)}$ and $w_j^{(4)} - c_j^{(4)} = m_{M_j}^{(4)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(4)}}{\partial m_{M_i}^{(4)}} = 0$ and $\frac{\partial U_{M_j}^{(4)}}{\partial m_{M_j}^{(4)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(4)}}{\partial m_{M_i}^{(4)}} = 2(a_i - c_i + dc_j - t_i + dt_j) + d(a_j - c_j + dc_i - t_j + dt_i) + (2\beta - d\beta\theta) \frac{\rho_{M_i}}{2\eta_i\tau_i} + (-2\beta\theta + d\beta) \frac{\rho_{M_j}}{2\eta_j\tau_j} + (-2 + d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_i\tau_i V} + d^2 - \frac{d\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V})m_{M_i}^{(4)} + (2d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} - d + \frac{d\beta^2(2-d\theta)}{2\eta_j\tau_j V})m_{M_j}^{(4)} = 0, \qquad (A.46)$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_j}^{(4)}}{\partial m_{M_j}^{(4)}} = 2(a_j - c_j + dc_i - t_j + dt_i) + d(a_i - c_i + dc_j - t_i + dt_j) + (2\beta - d\beta\theta) \frac{\rho_{M_j}}{2\eta_j\tau_j} + (-2\beta\theta + d\beta) \frac{\rho_{M_i}}{2\eta_i\tau_i} + (-2 + d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_j\tau_j V} + d^2 - \frac{d\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V})m_{M_j}^{(4)} + (2d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} - d + \frac{d\beta^2(2-d\theta)}{2\eta_i\tau_i V})m_{M_i}^{(4)} = 0, \qquad (A.47)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(4)}$ is achieved:

$$m_{M_i}^{(4)} = m_{M_i}^{(2)} - \frac{(2 - d^2)B_j^{(2)} + dA_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_j^{(2)}B_j^{(2)}}t_i + \frac{dB_j^{(2)} + (2 - d^2)A_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_j^{(2)}B_j^{(2)}}t_j; i = 1, 2j = 3 - i. \qquad (A.48)$$

By substituting $m_{M_i}^{(4)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(4)} = c_i + m_{M_i}^{(4)}; i = 1, 2j = 3 - i. \qquad (A.49)$$

By calculating $\frac{\partial U_{M_i}^{(4)}}{\partial s_i^{(4)}} = 0$, the optimal amount of $s_i^{(4)}$ is achieved:

$$\frac{\partial U_{M_i}^{(4)}}{\partial s_i^{(4)}} = m_{M_i}^{(4)} \frac{2\beta - d\beta\theta}{V} - 2\eta_i \tau_i s_i^{(4)} + \rho_{M_i} = 0, \qquad (A.50)$$

$$s_i^{(4)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(4)} + \rho_{M_i}V]}{2\eta_i\tau_i V}, \qquad (A.51)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(4)}) = \begin{bmatrix} -2 + d^2 + (d^2 - 2) & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(4)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(4)})) > 0$, the concavity of the profit function holds. ☐.

**Proposition 5 proof**

To obtain the retailer price Under SP&SC policy $(x, s_T)$ in Scenario 5 $\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}}$ and $\frac{\partial U_{R_i}^{(5)}}{\partial w_i^{(5)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}} = 0$ and $\frac{\partial U_{R_i}^{(5)}}{\partial w_i^{(5)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}} = \alpha_i - w_i^{(5)} + dw_j^{(5)} - 2m_{R_i}^{(5)} + dm_{R_i}^{(5)} + \beta s = 0; i = 1, 2j = 3 - i, \qquad (A.52)$$

$$\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}} = \alpha_j - w_j^{(5)} + dw_i^{(5)} - 2m_{R_i}^{(5)} + dm_{R_i}^{(5)} + \beta s = 0; j = 1, 2i = 3 - j, \qquad (A.53)$$

$$m_{R_i}^{(5)} = \left\{ \frac{2(\alpha_i - w_i^{(5)} + dw_j^{(5)}) + d(\alpha_j - w_j^{(5)} + dw_i^{(5)}) + \beta(2 + d)s^{(5)}}{V}; i = 1, 2j = 3 - i. \right. \qquad (A.54)$$

By substituting $m_{R_i}^{(5)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(5)} = w_i^{(5)} + m_{R_i}^{(5)}; i = 1, 2j = 3 - i. \qquad (A.55)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(5)}}{\partial w_i^{(5)}} = 0$ and $\frac{\partial U_{M_i}^{(5)}}{\partial w_j^{(5)}} = 0$ must be solved. With considering $w_i^{(5)} - c_i^{(5)} = m_{M_i}^{(5)}$ and $w_j^{(5)} - c_j^{(5)} = m_{M_j}^{(5)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(5)}}{\partial m_{M_i}^{(5)}} = 0$ and $\frac{\partial U_{M_j}^{(5)}}{\partial m_{M_j}^{(5)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(5)}}{\partial m_{M_i}^{(5)}} = 2(\alpha_i - c_i + dc_j - m_{M_i}^{(5)} + dm_{M_j}^{(5)}) + d(\alpha_j - c_j + dc_i - m_{M_j}^{(5)} + dm_{M_i}^{(5)}) + \frac{\beta(2+d)}{2\eta\tau} \left[ \frac{\beta(2+d)\sum_{i=1}^{2}(m_{M_i}^{(5)} + T_i)}{2\eta\tau V} + \rho_M \right] + (m_{M_i}^{(5)} + T_i)(d^2 - 2) + d(m_{M_j}^{(5)} + T_j) = 0, \qquad (A.56)$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(5)}}{\partial m_{M_i}^{(5)}} = 2(\alpha_j - c_j + dc_i - m_{M_j}^{(5)} + dm_{M_i}^{(5)}) + d(\alpha_i - c_i + dc_j - m_{M_j}^{(5)} + dm_{M_i}^{(5)}) + \frac{\beta(2+d)}{2\eta\tau} \left[ \frac{\beta(2+d)\sum_{j=1}^{2}(m_{M_j}^{(5)} + T_j)}{2\eta\tau V} + \rho_M \right] + (m_{M_j}^{(5)} + T_j)(d^2 - 2) + d(m_{M_i}^{(5)} + T_i) = 0, \qquad (A.57)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(5)}$ is achieved:

$$m_{M_i}^{(5)} = m_{M_i}^{(1)} - \frac{\beta(2+d)}{2\eta(A^{(1)} + B^{(1)})} x_i = 1, 2j = 3 - i \qquad (A.58)$$

By substituting $m_{M_i}^{(5)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(4)} = c_i + m_{M_i}^{(4)}; i = 1, 2j = 3 - i. \qquad (A.49)$$

By calculating $\frac{\partial U_{M_i}^{(4)}}{\partial s_i^{(4)}} = 0$, the optimal amount of $s_i^{(4)}$ is achieved:

$$\frac{\partial U_{M_i}^{(4)}}{\partial s_i^{(4)}} = m_{M_i}^{(4)} \frac{2\beta - d\beta\theta}{V} - 2\eta_i \tau_i s_i^{(4)} + \rho_{M_i} = 0, \qquad (A.50)$$

$$s_i^{(4)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(4)} + \rho_{M_i}V]}{2\eta_i\tau_i V}, \qquad (A.51)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(4)}) = \begin{bmatrix} -2 + d^2 + (d^2 - 2) & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(4)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(4)})) > 0$, the concavity of the profit function holds. ☐.

**Proposition 5 proof**

To obtain the retailer price Under SP&SC policy $(x, s_T)$ in Scenario 5 $\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}}$ and $\frac{\partial U_{R_i}^{(5)}}{\partial w_i^{(5)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}} = 0$ and $\frac{\partial U_{R_i}^{(5)}}{\partial w_i^{(5)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}} = \alpha_i - w_i^{(5)} + dw_j^{(5)} - 2m_{R_i}^{(5)} + dm_{R_i}^{(5)} + \beta s = 0; i = 1, 2j = 3 - i, \qquad (A.52)$$

$$\frac{\partial U_{R_i}^{(5)}}{\partial m_{R_i}^{(5)}} = \alpha_j - w_j^{(5)} + dw_i^{(5)} - 2m_{R_i}^{(5)} + dm_{R_i}^{(5)} + \beta s = 0; j = 1, 2i = 3 - j, \qquad (A.53)$$

$$m_{R_i}^{(5)} = \left\{ \frac{2(\alpha_i - w_i^{(5)} + dw_j^{(5)}) + d(\alpha_j - w_j^{(5)} + dw_i^{(5)}) + \beta(2 + d)s^{(5)}}{V}; i = 1, 2j = 3 - i. \right. \qquad (A.54)$$

By substituting $m_{R_i}^{(5)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(5)} = w_i^{(5)} + m_{R_i}^{(5)}; i = 1, 2j = 3 - i. \qquad (A.55)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(5)}}{\partial w_i^{(5)}} = 0$ and $\frac{\partial U_{M_i}^{(5)}}{\partial w_j^{(5)}} = 0$ must be solved. With considering $w_i^{(5)} - c_i^{(5)} = m_{M_i}^{(5)}$ and $w_j^{(5)} - c_j^{(5)} = m_{M_j}^{(5)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(5)}}{\partial m_{M_i}^{(5)}} = 0$ and $\frac{\partial U_{M_j}^{(5)}}{\partial m_{M_j}^{(5)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(5)}}{\partial m_{M_i}^{(5)}} = 2(\alpha_i - c_i + dc_j - m_{M_i}^{(5)} + dm_{M_i}^{(5)}) + d(\alpha_j - c_j + dc_i - m_{M_j}^{(5)} + dm_{M_j}^{(5)}) + \frac{\beta(2+d)}{2\eta\tau} \left[ \frac{\beta(2+d)\sum_{i=1}^{2}(m_{M_i}^{(5)} + T_i)}{2\eta\tau V} + \rho_M \right] + (m_{M_i}^{(5)} + T_i)(d^2 - 2) + d(m_{M_j}^{(5)} + T_j) = 0, \qquad (A.56)$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(5)}}{\partial m_{M_i}^{(5)}} = 2(\alpha_j - c_j + dc_i - m_{M_j}^{(5)} + dm_{M_j}^{(5)}) + d(\alpha_i - c_i + dc_j - m_{M_i}^{(5)} + dm_{M_i}^{(5)}) + \frac{\beta(2+d)}{2\eta\tau} \left[ \frac{\beta(2+d)\sum_{j=1}^{2}(m_{M_j}^{(5)} + T_j)}{2\eta\tau V} + \rho_M \right] + (m_{M_j}^{(5)} + T_j)(d^2 - 2) + d(m_{M_i}^{(5)} + T_i) = 0, \qquad (A.57)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(5)}$ is achieved:

$$m_{M_i}^{(5)} = m_{M_i}^{(1)} - \frac{\beta(2+d)}{2\eta(A^{(1)} + B^{(1)})} x_i = 1, 2j = 3 - i \qquad (A.58)$$

By substituting $m_{M_i}^{(5)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(5)} = c_i + m_{M_i}^{(5)}, i = 1, 2j = 3 - i$$

By calculating $\frac{\partial U_{M_i}^{(5)}}{\partial s_i^{(5)}} = 0$, the optimal amount of $s_i^{(5)}$ is achieved:

$$\frac{\partial U_{M_i}^{(5)}}{\partial s_i^{(5)}} = \sum_{i=1}^{2} (m_{M_i}^{(5)} + T_i) \frac{\beta(2+d)}{2\eta V} - 2\eta \tau s_i^{(5)} + \rho_M = 0, \qquad (A.59)$$

$$s_i^{(5)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(5)} + (\rho_M + \tau x)V \right] / 2\eta\tau V, \qquad (A.60)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(5)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(5)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)$ $[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(5)})) > 0$, the concavity of the profit function holds. □.

**Proposition 6 proof**

To obtain the retailer price Under SP&SC ($x, s_T$) in Scenario 6 $\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}}$ and $\frac{\partial U_{R_j}^{(6)}}{\partial m_{R_j}^{(6)}}$ must be obtained. In addition with considering $\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}} = 0$ and $\frac{\partial U_{R_j}^{(6)}}{\partial m_{R_j}^{(6)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}} = \alpha_i - w_i^{(6)} + dw_j^{(6)} - 2m_{R_i}^{(6)} - dm_{R_i}^{(6)} + \beta(s_i^{(6)} - \theta s_j^{(6)}) = 0; i = 1, 2j = 3 - i, \qquad (A.61)$$

$$\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}} = \alpha_j - w_j^{(6)} + dw_i^{(6)} - 2m_{R_j}^{(6)} - dm_{R_j}^{(6)} + \beta(s_j^{(6)} - \theta s_i^{(6)}) = 0; j = 1, 2i = 3 - j, \qquad (A.62)$$

$$m_{R_i}^{(6)} = \left\{ \begin{array}{l} 2(\alpha_i - w_i^{(6)} + dw_j^{(6)} + \beta(s_i^{(6)} - \theta s_j^{(6)})) \\ + dw_i^{(6)} + \beta(s_j^{(6)} - \theta s_i^{(6)}) \end{array} \right\} / V; i = 1, 2j = 3 - i. \qquad (A.63)$$

By substituting $m_{R_i}^{(6)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(6)} = w_i^{(6)} + m_{R_i}^{(6)}; i = 1, 2j = 3 - i. \qquad (A.64)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(6)}}{\partial w_i^{(6)}} = 0$ and $\frac{\partial U_{M_i}^{(6)}}{\partial w_j^{(6)}} = 0$ must be solved. With considering $w_i^{(6)} - c_i^{(6)} = m_{M_i}^{(6)}$ and $w_j^{(6)} - c_j^{(6)} = m_{M_j}^{(6)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(6)}}{\partial m_{M_i}^{(6)}} = 0$ and $\frac{\partial U_{M_j}^{(6)}}{\partial m_{M_j}^{(6)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(6)}}{\partial m_{M_i}^{(6)}} = 2(\alpha_i - c_i + dc_i + \frac{\beta\rho_{M_i}}{2\eta_i\tau_i} - \frac{\theta\beta\rho_{M_i}}{2\eta_i\tau_j}) + d(\alpha_j - c_j + dc_j + \frac{\beta\rho_{M_j}}{2\eta_j\tau_j} - \frac{\theta\beta\rho_{M_j}}{2\eta_j\tau_i}) + T_i[\frac{2\beta^2(2-d\theta)}{2\eta_i\tau_i V} - \frac{\beta^2 d\theta(2-d\theta)}{2\eta_i\tau_i V} + (d^2-2)] - T_j[\frac{2\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} - \frac{\beta^2 d(2-d\theta)}{2\eta_j\tau_i V}] + \qquad (A.65)$$

$$m_{M_i}^{(6)}[d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_i\tau_i V} - \frac{d\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} + (d^2-2)] + m_{M_j}^{(6)}[d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} + \frac{\beta^2\theta(2-d\theta)}{2\eta_j\tau_i V}] = 0,$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(6)}}{\partial m_{M_i}^{(6)}} = 2(\alpha_j - c_j + dc_j + \frac{\beta\rho_{M_j}}{2\eta_j\tau_j} - \frac{\theta\beta\rho_{M_j}}{2\eta_j\tau_i}) + d(\alpha_i - c_i + dc_i + \frac{\beta\rho_{M_i}}{2\eta_i\tau_i} - \frac{\theta\beta\rho_{M_i}}{2\eta_i\tau_j}) + T_j[\frac{2\beta^2(2-d\theta)}{2\eta_j\tau_j V} - \frac{\beta^2 d\theta(2-d\theta)}{2\eta_j\tau_i V} + (d^2-2)] - T_i[\frac{2\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} - \frac{\beta^2 d(2-d\theta)}{2\eta_j\tau_j V}] + m_{M_i}^{(6)}[d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_j\tau_j V} - \frac{d\beta^2\theta(2-d\theta)}{2\eta_j\tau_i V} + (d^2-2)] + m_{M_j}^{(6)}[d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} + \frac{\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V}] = 0, \qquad (A.66)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(6)}$ is achieved:
$$w_i^{(5)} = c_i + m_{M_i}^{(5)}; i = 1, 2j = 3 - i$$

By calculating $\frac{\partial U_{M_i}^{(5)}}{\partial s_i^{(5)}} = 0$, the optimal amount of $s_i^{(5)}$ is achieved:

$$\frac{\partial U_{M_i}^{(5)}}{\partial s_i^{(5)}} = \sum_{i=1}^{2} (m_{M_i}^{(5)} + T_i) \frac{\beta(2+d)}{2\eta V} - 2\eta \tau s_i^{(5)} + \rho_M = 0, \quad (A.59)$$

$$s^{(5)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(5)} + (\rho_M + \tau x)V \right] / 2\eta\tau V, \quad (A.60)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(5)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(5)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)$ $[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(5)})) > 0$, the concavity of the profit function holds. □

**Proposition 6 proof**

To obtain the retailer price Under SP&SC ($x, s_T$) in Scenario 6 $\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}}$ and $\frac{\partial U_{R_j}^{(6)}}{\partial m_{R_j}^{(6)}}$ must be obtained. In addition with considering $\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}} = 0$ and $\frac{\partial U_{R_j}^{(6)}}{\partial m_{R_j}^{(6)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}} = \alpha_i - w_i^{(6)} + dw_j^{(6)} - 2m_{R_i}^{(6)} - dm_{R_i}^{(6)} + \beta(s_i^{(6)} - \theta s_j^{(6)}) = 0; i = 1, 2j = 3 - i, \quad (A.61)$$

$$\frac{\partial U_{R_i}^{(6)}}{\partial m_{R_i}^{(6)}} = \alpha_j - w_j^{(6)} + dw_i^{(6)} - 2m_{R_j}^{(6)} - dm_{R_j}^{(6)} + \beta(s_j^{(6)} - \theta s_i^{(6)}) = 0; j = 1, 2i = 3 - j, \quad (A.62)$$

$$m_{R_i}^{(6)} = \left\{ \begin{array}{l} 2(\alpha_i - w_i^{(6)} + dw_j^{(6)} + \beta(s_i^{(6)} - \theta s_j^{(6)})) \\ + dw_i^{(6)} + \beta(s_j^{(6)} - \theta s_i^{(6)}) \end{array} \right\} / V; \quad i = 1, 2j = 3 - i. \quad (A.63)$$

By substituting $m_{R_i}^{(6)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(6)} = w_i^{(6)} + m_{R_i}^{(6)}; i = 1, 2j = 3 - i. \quad (A.64)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(6)}}{\partial w_i^{(6)}} = 0$ and $\frac{\partial U_{M_j}^{(6)}}{\partial w_j^{(6)}} = 0$ must be solved. With considering $w_i^{(6)} - c_i^{(6)} = m_{M_i}^{(6)}$ and $w_j^{(6)} - c_j^{(6)} = m_{M_j}^{(6)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(6)}}{\partial m_{M_i}^{(6)}} = 0$ and $\frac{\partial U_{M_j}^{(6)}}{\partial m_{M_j}^{(6)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(6)}}{\partial m_{M_i}^{(6)}} = 2(\alpha_i - c_i + dc_i + \frac{\beta\rho_{M_i}}{2\eta_i\tau_i} - \frac{\theta\beta\rho_{M_i}}{2\eta_i\tau_j}) + d(\alpha_j - c_j + dc_j + \frac{\beta\rho_{M_j}}{2\eta_j\tau_j} - \frac{\theta\beta\rho_{M_j}}{2\eta_j\tau_i}) + T_i[\frac{2\beta^2(2-d\theta)}{2\eta_i\tau_i V} - \frac{\beta^2 d\theta(2-d\theta)}{2\eta_i\tau_i V} + (d^2-2)] - T_j[\frac{2\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} - \frac{\beta^2 d(2-d\theta)}{2\eta_j\tau_i V}] + \quad (A.65)$$

$$m_{M_i}^{(6)}[d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_i\tau_i V} - \frac{d\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} + (d^2-2)] + m_{M_j}^{(6)}[d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} + \frac{\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V}] = 0,$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(6)}}{\partial m_{M_i}^{(6)}} = 2(\alpha_j - c_j + dc_j + \frac{\beta\rho_{M_j}}{2\eta_j\tau_j} - \frac{\theta\beta\rho_{M_j}}{2\eta_j\tau_i}) + d(\alpha_i - c_i + dc_i + \frac{\beta\rho_{M_i}}{2\eta_i\tau_i} - \frac{\theta\beta\rho_{M_i}}{2\eta_i\tau_j}) + T_j[\frac{2\beta^2(2-d\theta)}{2\eta_j\tau_j V} - \frac{\beta^2 d\theta(2-d\theta)}{2\eta_j\tau_j V} + (d^2-2)] - T_i[\frac{2\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} - \frac{\beta^2 d(2-d\theta)}{2\eta_i\tau_j V}] + m_{M_i}^{(6)}[d^2 - 2 + \frac{2\beta^2(2-d\theta)}{2\eta_j\tau_j V} - \frac{d\beta^2\theta(2-d\theta)}{2\eta_j\tau_j V} + (d^2-2)] + m_{M_j}^{(6)}[d - \frac{2\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V} + \frac{\beta^2\theta(2-d\theta)}{2\eta_i\tau_i V}] = 0, \quad (A.66)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above the optimal $m_{M_i}^{(6)}$ is achieved:
$$m_{M_i}^{(6)} = m_{M_i}^{(2)} + \frac{\beta}{2\eta_i} \frac{(2-d\theta)B_j^{(2)} - (d-2\theta)A_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} - x_i + \frac{\beta}{2\eta_j} \frac{(d-2\theta)B_j^{(2)} - (d-d\theta)A_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} - x_j; i = 1, 2, j = 3 - i. \quad (A.67)$$

By substituting $m_{M_i}^{(6)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$w_i^{(6)} = c_i + m_{M_i}^{(6)}; \quad i = 1, 2, j = 3 - i. \quad (A.68)$$

By calculating $\frac{\partial U_{M_i}^{(6)}}{\partial s_i^{(6)}} = 0$, the optimal amount of $s_i^{(6)}$ is achieved:

$$\frac{\partial U_{M_i}^{(6)}}{\partial s_i^{(6)}} = (m_{M_i}^{(6)} + T_i) \frac{2\beta - d\beta\theta}{V} - 2\eta_i\tau_i s_i^{(6)} + \rho_{M_i} = 0, \quad (A.69)$$

$$s_i^{(6)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(6)} + (\rho_{M_i} + \tau_i x_i)V]}{2\eta_i\tau_i V}, \quad (A.70)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(6)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta - d\beta\theta}{4-d^2} \\ \frac{2\beta - d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le \eta \le 1$,

$H(U_{M_i}^{(6)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(6)})) > 0$, the concavity of the profit function holds. $\square$.

**Proposition 7 proof**

To obtain the retailer price Under DLS policy $s_L$ in Scenario 7 $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_i}^{(7)}}$ and $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_j}^{(7)}}$ must be obtained. In addition with considering $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_i}^{(7)}} = 0$ and $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_j}^{(7)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_i}^{(7)}} = \alpha_i - w_i^{(7)} + dw_j^{(7)} - 2m_{R_i}^{(7)} + dm_{R_j}^{(7)} + \beta s_i^{(7)} = 0; \quad i = 1, 2, j = 3 - i, \quad (A.71)$$

$$\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_j}^{(7)}} = \alpha_j - w_j^{(7)} + dw_i^{(7)} - 2m_{R_j}^{(7)} + dm_{R_i}^{(7)} + \beta s_i^{(7)} = 0; \quad j = 1, 2, i = 3 - j, \quad (A.72)$$

$$m_{R_i}^{(7)} = \frac{[2(\alpha_i - w_i^{(7)} + dw_j^{(7)}) + d(\alpha_j - w_j^{(7)} + dw_i^{(7)}) + \beta(2+d)s_i^{(7)}]}{V}; \quad i = 1, 2, j = 3 - i. \quad (A.73)$$

By substituting $m_{R_i}^{(7)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(7)} = w_i^{(7)} + m_{R_i}^{(7)}; \quad i = 1, 2, j = 3 - i. \quad (A.74)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 0$ and $\frac{\partial U_{M_i}^{(7)}}{\partial w_j^{(7)}} = 0$ must be solved. With considering $w_i^{(7)} - c_i^{(7)} = m_{M_i}^{(7)}$ and $w_j^{(7)} - c_j^{(7)} = m_{M_j}^{(7)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 0$ and $\frac{\partial U_{M_j}^{(7)}}{\partial m_{M_j}^{(7)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 2(\alpha_i - c_i + dc_j - m_{M_i}^{(7)} + dm_{M_i}^{(7)}) + d(\alpha_j - c_j + dc_i - m_{M_j}^{(7)} + dm_{M_j}^{(7)}) + \beta(2+d) \quad (A.75)$$

$$\frac{\beta(2+d)\sum_{i=1}^{2} m_{M_i}^{(7)} + \rho_M + \tau x}{2\eta \tau V} + m_M(d^2 - 2) + dm_{M_i}^{(7)} = 0,$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 2(\alpha_j - c_j + dc_i - m_{M_i}^{(7)} + dm_{M_i}^{(7)}) + d(\alpha_i - c_i + dc_j - m_{M_j}^{(7)} + dm_{M_j}^{(7)}) + \beta(2+d) \quad (A.76)$$

$$\frac{\beta(2+d)\sum_{j=1}^{2} m_{M_j}^{(7)} + \rho_M + \tau x}{2\eta \tau V} + m_M(d^2 - 2) + dm_{M_j}^{(7)} = 0,$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above $m_{M_i}^{(7)}$ is achieved:

$$m_{M_i}^{(7)} = m_{M_i}^{(1)} - \lambda\beta(2+d)/2\eta\tau(A^{(1)} + B^{(1)}); \quad i = 1, 2, j = 3 - i. \quad (A.77)$$
$$m_{M_i}^{(6)} = m_{M_i}^{(2)} + \frac{\beta}{2\eta_i} \frac{(2-d\theta)B_j^{(2)} - (d-2\theta)A_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} - x_i + \frac{\beta}{2\eta_j} \frac{(d-2\theta)B_j^{(2)} - (d-d\theta)A_j^{(2)}}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} - x_j; i = 1, 2, j = 3 - i. \quad (A.67)$$

By substituting $m_{M_i}^{(6)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$w_i^{(6)} = c_i + m_{M_i}^{(6)}; \quad i = 1, 2, j = 3 - i. \quad (A.68)$$

By calculating $\frac{\partial U_{M_i}^{(6)}}{\partial s_i^{(6)}} = 0$, the optimal amount of $s_i^{(6)}$ is achieved:

$$\frac{\partial U_{M_i}^{(6)}}{\partial s_i^{(6)}} = (m_{M_i}^{(6)} + T_i) \frac{2\beta - d\beta\theta}{V} - 2\eta_i\tau_i s_i^{(6)} + \rho_{M_i} = 0, \quad (A.69)$$

$$s_i^{(6)} = \left[ \beta(2 - d\theta)m_{M_i}^{(6)} + (\rho_{M_i} + \tau_i x_i)V \right] / 2\eta_i\tau_i V, \quad (A.70)$$

where $i=1,2$ and $j=3-i$.

Considering $H(U_{M_i}^{(6)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le \eta \le 1$,

$H(U_{M_i}^{(6)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2-d\theta)^2/4(2-d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(6)})) > 0$, the concavity of the profit function holds. ☐

**Proposition 7 proof**

To obtain the retailer price Under DLS policy $s_L$ in Scenario 7 $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_i}^{(7)}}$ and $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_j}^{(7)}}$ must be obtained. In addition with considering $\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_i}^{(7)}} = 0$ and $\frac{\partial U_{R_j}^{(7)}}{\partial m_{R_j}^{(7)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_i}^{(7)}} = \alpha_i - w_i^{(7)} + dw_j^{(7)} - 2m_{R_i}^{(7)} + dm_{R_j}^{(7)} + \beta s_i^{(7)} = 0; \quad i = 1, 2, j = 3 - i, \quad (A.71)$$

$$\frac{\partial U_{R_i}^{(7)}}{\partial m_{R_j}^{(7)}} = \alpha_j - w_j^{(7)} + dw_i^{(7)} - 2m_{R_j}^{(7)} + dm_{R_i}^{(7)} + \beta s_j^{(7)} = 0; \quad j = 1, 2, i = 3 - j, \quad (A.72)$$

$$m_{R_i}^{(7)} = \left[ 2(\alpha_i - w_i^{(7)} + dw_j^{(7)}) + d(\alpha_j - w_j^{(7)} + dw_i^{(7)}) + \beta(2+d)s_i^{(7)} \right] / V; \quad i = 1, 2, j = 3 - i. \quad (A.73)$$

By substituting $m_{R_i}^{(7)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(7)} = w_i^{(7)} + m_{R_i}^{(7)}; \quad i = 1, 2, j = 3 - i. \quad (A.74)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 0$ and $\frac{\partial U_{M_i}^{(7)}}{\partial w_j^{(7)}} = 0$ must be solved. With considering $w_i^{(7)} - c_i^{(7)} = m_{M_i}^{(7)}$ and $w_j^{(7)} - c_j^{(7)} = m_{M_j}^{(7)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 0$ and $\frac{\partial U_{M_j}^{(7)}}{\partial m_{M_j}^{(7)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 2(\alpha_i - c_i + dc_j - m_{M_i}^{(7)} + dm_{M_i}^{(7)}) + d(\alpha_j - c_j + dc_i - m_{M_j}^{(7)} + dm_{M_j}^{(7)}) + \beta(2+d) \quad (A.75)$$

$$\left[ \frac{\beta(2+d)\sum_{i=1}^{2} m_{M_i}^{(7)}}{2\eta \tau V} + \frac{\rho_M + \tau x_i}{2\eta \tau} \right] + m_{M_i}(d^2 - 2) + dm_{M_i}^{(7)} = 0,$$

where $i=1,2$ and $j=3-i$.

$$\frac{\partial U_{M_i}^{(7)}}{\partial m_{M_i}^{(7)}} = 2(\alpha_j - c_j + dc_i - m_{M_i}^{(7)} + dm_{M_i}^{(7)}) + d(\alpha_i - c_i + dc_j - m_{M_j}^{(7)} + dm_{M_j}^{(7)}) + \beta(2+d) \quad (A.76)$$

$$\left[ \frac{\beta(2+d)\sum_{j=1}^{2} m_{M_j}^{(7)}}{2\eta \tau V} + \frac{\rho_M + \tau x_i}{2\eta \tau} \right] + m_{M_j}(d^2 - 2) + dm_{M_j}^{(7)} = 0,$$

where $j=1,2$ and $i=3-j$. By solving the equations above $m_{M_i}^{(7)}$ is achieved:

$$m_{M_i}^{(7)} = m_{M_i}^{(1)} - \lambda\beta(2+d)/2\eta\tau(A^{(1)} + B^{(1)}); \quad i = 1, 2, j = 3 - i. \quad (A.77)$$
By substituting $m_{M_i}^{(7)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$w_i^{(7)} = c_i + m_{M_i}^{(7)}; i = 1, 2, j = 3 - i. \qquad (A.78)$$

By calculating $\frac{\partial U_{M_i}^{(7)}}{\partial s_i^{(7)}} = 0$, the optimal amount of $s_i^{(7)}$ is achieved:

$$\frac{\partial U_M^{(7)}}{\partial s^{(7)}} = \frac{\beta(2+d)}{V} \sum_{i=1}^{2} m_{M_i}^{(7)} + \tau x - 2\eta \tau s^{(7)} + \rho_M = 0, \qquad (A.79)$$

$$s^{(7)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(7)} + (\rho_M + \lambda)V \right] / 2\eta\tau V, \qquad (A.80)$$

where $i=1,2$ and $j=3-i$.

Considering $H(U_{M_i}^{(7)}) = \begin{bmatrix} -2+d^2+(d^2-2) & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(7)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)$ $[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(7)})) > 0$, the concavity of the profit function holds. □.

### Proposition 8 proof

To obtain the retailer price Under DLS $s_L = (s_{L_1}, s_{L_2})$ in Scenario 8 $\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}}$ and $\frac{\partial U_{R_j}^{(8)}}{\partial m_{R_j}^{(8)}}$ must be obtained. In addition with considering $\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}} = 0$ and $\frac{\partial U_{R_j}^{(8)}}{\partial m_{R_j}^{(8)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}} = \alpha_i - w_i^{(8)} + dw_j^{(8)} - 2m_{R_i}^{(8)} + dm_{R_j}^{(8)} + \beta(s_i^{(8)} - \theta s_j^{(8)}) = 0; i = 1, 2j = 3 - i, \qquad (A.81)$$

$$\frac{\partial U_{R_j}^{(8)}}{\partial m_{R_j}^{(8)}} = \alpha_j - w_j^{(8)} + dw_i^{(8)} - 2m_{R_j}^{(8)} + dm_{R_i}^{(8)} + \beta(s_i^{(8)} - \theta s_j^{(8)})_i = 0; i = 1, 2j = 3 - i, \qquad (A.82)$$

$$m_{R_i}^{(8)} = \left[ 2(\alpha_i - w_i^{(8)} + dw_j^{(8)} + \beta(s_i^{(8)} - \theta s_j^{(8)})) + d(\alpha_j - w_j^{(8)} + dw_i^{(8)} + \beta(s_i^{(8)} - \theta s_i^{(8)})) \right] / V; i = 1, 2j = 3 - i. \qquad (A.83)$$

By substituting $m_{R_i}^{(8)}$ into following equation the optimal retailer's price will be obtained:

$$p_i^{(8)} = w_i^{(8)} + m_{R_i}^{(8)}; i = 1, 2j = 3 - i. \qquad (A.84)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(8)}}{\partial w_i^{(8)}}=0$ and $\frac{\partial U_{M_j}^{(8)}}{\partial w_j^{(8)}} = 0$ must be solved. With considering $w_i^{(8)} - c_i^{(8)} = m_{M_i}^{(8)}$ and $w_j^{(8)} - c_j^{(8)} = m_{M_j}^{(8)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(8)}}{\partial m_{M_i}^{(8)}} = 0$ and $\frac{\partial U_{M_j}^{(8)}}{\partial m_{M_j}^{(8)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(8)}}{\partial m_{M_i}^{(8)}} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + (2\beta - d\beta\theta)\frac{\rho_{M_i} + \tau_i x_i}{2\eta_i \tau_i} + (-2\beta\theta + d\beta)\frac{\rho_{M_j} + \tau_j x_j}{2\eta_j \tau_j} + \\ [-2 + d^2 - 2 + \frac{\beta^2(2-d\theta)^2}{2\eta_i \tau_i V} + (d^2 - 2)]m_{M_i}^{(8)} + (d + \beta^2(d - 2\theta)\frac{(2-d\theta)}{2\eta_j \tau_j V})m_{M_j}^{(8)} = 0, \qquad (A.85)$$

where $i=1,2$ and $j=3-i$.

$$\frac{\partial U_{M_i}^{(8)}}{\partial m_{M_i}^{(8)}} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + (2\beta - d\beta\theta)\frac{\rho_{M_i} + \tau_i x_i}{2\eta_j \tau_j} + (-2\beta\theta + d\beta)\frac{\rho_{M_j} + \tau_j x_j}{2\eta_i \tau_i} + \\ [-2 + d^2 - 2 + \frac{\beta^2(2-d\theta)^2}{2\eta_j \tau_j V} + (d^2 - 2)]m_{M_i}^{(8)} + (d + \beta^2(d - 2\theta)\frac{(2-d\theta)}{2\eta_i \tau_i V})m_{M_j}^{(8)} = 0, \qquad (A.86)$$

where $j=1,2$ and $i=3-j$. By solving the equations above $m_{M_i}^{(8)}$ is achieved:

$$m_{M_i}^{(8)} = m_{M_i}^{(2)} + \frac{\beta}{2\eta_i \tau_i} \frac{(2 - d\theta)B_j^{(2)} - (d - 2\theta)A_j^{(2)}}{A_i^{(2)} A_j^{(2)} - B_i^{(2)} B_j^{(2)}} \lambda_i + \frac{\beta}{2\eta_j \tau_j} \frac{(d - 2\theta)B_j^{(2)} - (2 - d\theta)A_j^{(2)}}{A_i^{(2)} A_j^{(2)} - B_i^{(2)} B_j^{(2)}} \lambda_j; i = 1, 2j = 3 - i. \qquad (A.87)$$

By substituting $m_{M_i}^{(8)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
By substituting $m_{M_i}^{(7)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$w_i^{(7)} = c_i + m_{M_i}^{(7)}; i = 1, 2, j = 3 - i. \qquad (A.78)$$

By calculating $\frac{\partial U_{M_i}^{(7)}}{\partial s_i^{(7)}} = 0$, the optimal amount of $s_i^{(7)}$ is achieved:

$$\frac{\partial U_M^{(7)}}{\partial s^{(7)}} = \frac{\beta(2+d)}{V} \sum_{i=1}^{2} m_{M_i}^{(7)} + \tau x - 2\eta \tau s^{(7)} + \rho_M = 0, \qquad (A.79)$$

$$s^{(7)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(7)} + (\rho_M + \lambda)V \right] / 2\eta\tau V, \qquad (A.80)$$

where $i=1,2$ and $j=3-i$.

Considering $H(U_{M_i}^{(7)}) = \begin{bmatrix} -2+d^2+(d^2-2) & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(7)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)$ $[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(7)})) > 0$, the concavity of the profit function holds. □.

### Proposition 8 proof

To obtain the retailer price Under DLS $s_L = (s_{L_1}, s_{L_2})$ in Scenario 8 $\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}}$ and $\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}}$ must be obtained. In addition with considering $\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}} = 0$ and $\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(8)}}{\partial m_{R_i}^{(8)}} = \alpha_i - w_i^{(8)} + dw_j^{(8)} - 2m_{R_i}^{(8)} + dm_{R_j}^{(8)} + \beta(s_i^{(8)} - \theta s_j^{(8)}) = 0; i = 1, 2j = 3 - i, \qquad (A.81)$$

$$\frac{\partial U_{R_j}^{(8)}}{\partial m_{R_j}^{(8)}} = \alpha_j - w_j^{(8)} + dw_i^{(8)} - 2m_{R_j}^{(8)} + dm_{R_i}^{(8)} + \beta(s_i^{(8)} - \theta s_j^{(8)})_i = 0; i = 1, 2j = 3 - i, \qquad (A.82)$$

$$m_{R_i}^{(8)} = \left[ 2(\alpha_i - w_i^{(8)} + dw_j^{(8)} + \beta(s_i^{(8)} - \theta s_j^{(8)})) + d(\alpha_j - w_j^{(8)} + dw_i^{(8)} + \beta(s_j^{(8)} - \theta s_i^{(8)})) \right] / V; i = 1, 2j = 3 - i. \qquad (A.83)$$

By substituting $m_{R_i}^{(8)}$ into following equation the optimal retailer's price will be obtained:

$$p_i^{(8)} = w_i^{(8)} + m_{R_i}^{(8)}; i = 1, 2j = 3 - i. \qquad (A.84)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(8)}}{\partial w_i^{(8)}}=0$ and $\frac{\partial U_{M_j}^{(8)}}{\partial w_j^{(8)}} = 0$ must be solved. With considering $w_i^{(8)} - c_i^{(8)} = m_{M_i}^{(8)}$ and $w_j^{(8)} - c_j^{(8)} = m_{M_j}^{(8)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(8)}}{\partial m_{M_i}^{(8)}} = 0$ and $\frac{\partial U_{M_j}^{(8)}}{\partial m_{M_j}^{(8)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(8)}}{\partial m_{M_i}^{(8)}} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + (2\beta - d\beta\theta)\frac{\rho_{M_i} + \tau_i x_i}{2\eta_i \tau_i} + (-2\beta\theta + d\beta)\frac{\rho_{M_j} + \tau_j x_j}{2\eta_j \tau_j} + \\ [-2 + d^2 - 2 + \frac{\beta^2(2 - d\theta)^2}{2\eta_i \tau_i V} + (d^2 - 2)]m_{M_i}^{(8)} + (d + \beta^2(d - 2\theta)\frac{(2 - d\theta)}{2\eta_j \tau_j V})m_{M_j}^{(8)} = 0, \qquad (A.85)$$

where $i=1,2$ and $j=3-i$.

$$\frac{\partial U_{M_i}^{(8)}}{\partial m_{M_i}^{(8)}} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + (2\beta - d\beta\theta)\frac{\rho_{M_i} + \tau_i x_i}{2\eta_j \tau_j} + (-2\beta\theta + d\beta)\frac{\rho_{M_j} + \tau_j x_j}{2\eta_i \tau_i} + \\ [-2 + d^2 - 2 + \frac{\beta^2(2 - d\theta)^2}{2\eta_j \tau_j V} + (d^2 - 2)]m_{M_i}^{(8)} + (d + \beta^2(d - 2\theta)\frac{(2 - d\theta)}{2\eta_i \tau_i V})m_{M_j}^{(8)} = 0, \qquad (A.86)$$

where $j=1,2$ and $i=3-j$. By solving the equations above $m_{M_i}^{(8)}$ is achieved:

$$m_{M_i}^{(8)} = m_{M_i}^{(2)} + \frac{\beta}{2\eta_i \tau_i} \frac{(2 - d\theta)B_j^{(2)} - (d - 2\theta)A_j^{(2)}}{A_i^{(2)} A_j^{(2)} - B_i^{(2)} B_j^{(2)}} \lambda_i + \frac{\beta}{2\eta_j \tau_j} \frac{(d - 2\theta)B_j^{(2)} - (2 - d\theta)A_j^{(2)}}{A_i^{(2)} A_j^{(2)} - B_i^{(2)} B_j^{(2)}} \lambda_j; i = 1, 2j = 3 - i. \qquad (A.87)$$

By substituting $m_{M_i}^{(8)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(8)} = c_i + m_{M_i}^{(8)}; i = 1, 2j = 3 - i. \tag{A.88}$$

By calculating $\frac{\partial U_{M_i}^{(8)}}{\partial s_i^{(8)}} = 0$, the optimal amount of $s_i^{(8)}$ is achieved:

$$\frac{\partial U_{M_i}^{(8)}}{\partial s_i^{(8)}} = m_{M_i}^{(8)} \frac{2\beta - d\beta\theta}{V} - 2\eta_i \tau_i s_i^{(8)} + \rho_{M_i} + \tau_i x_i = 0, \tag{A.89}$$

$$s_i^{(8)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(8)} + (\rho_{M_i} + \lambda_i)V]}{2\eta_i \tau_i V}, \tag{A.90}$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(8)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(8)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(8)})) > 0$, the concavity of the profit function holds. ☐

### Proposition 9 proof

To obtain the retailer price Under GCSE policy in scenario 9 $\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}}$ and $\frac{\partial U_{R_j}^{(9)}}{\partial m_{R_j}^{(9)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}} = 0$ and $\frac{\partial U_{R_j}^{(9)}}{\partial m_{R_j}^{(9)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}} = \alpha_i - w_i^{(8)} + dw_j^{(8)} - 2m_{R_i}^{(8)} + dm_{R_i}^{(8)} + \beta s_i^{(8)} = 0; i = 1, 2j = 3 - i, \tag{A.91}$$

$$\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}} = \alpha_j - w_j^{(9)} + dw_i^{(9)} - 2m_{R_i}^{(9)} + dm_{R_i}^{(9)} + \beta s_i^{(9)} = 0; j = 1, 2i = 3 - j, \tag{A.92}$$

$$m_{R_i}^{(9)} = \left\{ \begin{array}{l} 2(\alpha_i - w_i^{(9)} + dw_j^{(9)}) + \\ d(\alpha_j - w_j^{(9)} + dw_i^{(9)}) + \beta(2+d)(s_i^{(9)} + s_G) \end{array} \right\} / V; \quad i = 1, 2, j = 3 - i, \tag{A.93}$$

By substituting $m_{R_i}^{(9)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(9)} = w_i^{(9)} + m_{R_i}^{(9)}; i = 1, 2j = 3 - i. \tag{A.94}$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(9)}}{\partial w_i^{(9)}} = 0$ and $\frac{\partial U_{M_j}^{(9)}}{\partial w_j^{(9)}} = 0$ must be solved. With considering $w_i^{(9)} - c_i^{(9)} = m_{M_i}^{(9)}$ and $w_j^{(9)} - c_j^{(9)} = m_{M_j}^{(9)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(9)}}{\partial m_{M_i}^{(9)}} = 0$ and $\frac{\partial U_{M_j}^{(9)}}{\partial m_{M_j}^{(9)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(9)}}{\partial m_{M_i}^{(9)}} = 2(\alpha_i - c_i + dc_j - m_{M_i}^{(9)} + dm_{M_i}^{(9)}) + d(\alpha_j - c_j + dc_i - m_{M_j}^{(9)} + dm_{M_j}^{(9)}) + \beta(2+d) \tag{A.95}$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(9)}}{\partial m_{M_i}^{(9)}} = 2(\alpha_j - c_j + dc_i - m_{M_j}^{(9)} + dm_{M_j}^{(9)}) + d(\alpha_i - c_i + dc_j - m_{M_i}^{(9)} + dm_{M_i}^{(9)}) + \beta(2+d) \tag{A.96}$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above $m_{M_i}^{(9)}$ is achieved:

$$m_{M_i}^{(9)} = m_{M_i}^{(1)} - \frac{\beta(2+d)}{A^{(1)} + B^{(1)}s_G}, \quad i = 1, 2. \tag{A.97}$$

By substituting $m_{M_i}^{(9)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(8)} = c_i + m_{M_i}^{(8)}; i = 1, 2j = 3 - i. \qquad (A.88)$$

By calculating $\frac{\partial U_{M_i}^{(8)}}{\partial s_i^{(8)}} = 0$, the optimal amount of $s_i^{(8)}$ is achieved:

$$\frac{\partial U_{M_i}^{(8)}}{\partial s_i^{(8)}} = m_{M_i}^{(8)} \frac{2\beta - d\beta\theta}{V} - 2\eta_i \tau_i s_i^{(8)} + \rho_{M_i} + \tau_i x_i = 0, \qquad (A.89)$$

$$s_i^{(8)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(8)} + (\rho_{M_i} + \lambda_i)V]}{2\eta_i\tau_i V}, \qquad (A.90)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(8)}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(8)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(8)})) > 0$, the concavity of the profit function holds. ☐

### Proposition 9 proof

To obtain the retailer price Under GCSE policy in scenario 9 $\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}}$ and $\frac{\partial U_{R_j}^{(9)}}{\partial m_{R_j}^{(9)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}} = 0$ and $\frac{\partial U_{R_j}^{(9)}}{\partial m_{R_j}^{(9)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}} = \alpha_i - w_i^{(8)} + dw_j^{(8)} - 2m_{R_i}^{(8)} + dm_{R_i}^{(8)} + \beta s_i^{(8)} = 0; i = 1, 2j = 3 - i, \qquad (A.91)$$

$$\frac{\partial U_{R_i}^{(9)}}{\partial m_{R_i}^{(9)}} = \alpha_j - w_j^{(9)} + dw_i^{(9)} - 2m_{R_i}^{(9)} + dm_{R_i}^{(9)} + \beta s_i^{(9)} = 0; j = 1, 2i = 3 - j, \qquad (A.92)$$

$$m_{R_i}^{(9)} = \left\{ \begin{array}{l} 2(\alpha_i - w_i^{(9)} + dw_j^{(9)}) + \\ d(\alpha_j - w_j^{(9)} + dw_i^{(9)}) + \beta(2+d)(s_i^{(9)} + s_G) \end{array} \right\} / V; \quad i = 1, 2, j = 3 - i, \qquad (A.93)$$

By substituting $m_{R_i}^{(9)}$ into the following equation the optimal retailer's price will be obtained:

$$p_i^{(9)} = w_i^{(9)} + m_{R_i}^{(9)}; i = 1, 2j = 3 - i. \qquad (A.94)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(9)}}{\partial w_i^{(9)}} = 0$ and $\frac{\partial U_{M_j}^{(9)}}{\partial w_j^{(9)}} = 0$ must be solved. With considering $w_i^{(9)} - c_i^{(9)} = m_{M_i}^{(9)}$ and $w_j^{(9)} - c_j^{(9)} = m_{M_j}^{(9)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(9)}}{\partial m_{M_i}^{(9)}} = 0$ and $\frac{\partial U_{M_j}^{(9)}}{\partial m_{M_j}^{(9)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(9)}}{\partial m_{M_i}^{(9)}} = 2(\alpha_i - c_i + dc_j - m_{M_i}^{(9)} + dm_{M_i}^{(9)}) + d(\alpha_j - c_j + dc_i - m_{M_j}^{(9)} + dm_{M_j}^{(9)}) + \beta(2+d) \qquad (A.95)$$

where $i = 1, 2$ and $j = 3 - i$.

$$\frac{\partial U_{M_i}^{(9)}}{\partial m_{M_i}^{(9)}} = 2(\alpha_j - c_j + dc_i - m_{M_j}^{(9)} + dm_{M_j}^{(9)}) + d(\alpha_i - c_i + dc_j - m_{M_i}^{(9)} + dm_{M_i}^{(9)}) + \beta(2+d) \qquad (A.96)$$

where $j = 1, 2$ and $i = 3 - j$. By solving the equations above $m_{M_i}^{(9)}$ is achieved:

$$m_{M_i}^{(9)} = m_{M_i}^{(1)} - \frac{\beta(2+d)}{A^{(1)} + B^{(1)}s_G}, \quad i = 1, 2. \qquad (A.97)$$

By substituting $m_{M_i}^{(9)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:
$$w_i^{(9)} = c_i + m_{M_i}^{(9)}, i = 1, 2, j = 3 - i \qquad (A.98)$$

By calculating $\frac{\partial U_{M_i}^{(9)}}{\partial s_i^{(9)}} = 0$, the optimal amount of $s_i^{(9)}$ is achieved:

$$\frac{\partial U_M^{(9)}}{\partial s^{(9)}} = \frac{\beta(2+d)}{V} \sum_{i=1}^{2} m_{M_i}^{(9)} + \lambda - 2\eta \tau s^{(9)} + \rho_M = 0, \qquad (A.99)$$

$$s^{(9)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(9)} + \rho_M V \right] / 2\eta\tau V, \qquad (A.100)$$

where $i=1, 2$ and $j=3-i$.

Considering $H(U_{M_i}^{(9)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le \lambda \le 1$,

$H(U_{M_i}^{(9)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)$ $[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(9)})) > 0$, the concavity of the profit function holds. □

**Proposition 10 proof**

To obtain the retailer price When the RA establishes GCSE according to Scenario 10 $\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}}$ and $\frac{\partial U_{R_j}^{(10)}}{\partial m_{R_j}^{(10)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}} = 0$ and $\frac{\partial U_{R_j}^{(10)}}{\partial m_{R_j}^{(10)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}} = \alpha_i - w_i^{(10)} + dw_j^{(10)} - 2m_{R_i}^{(10)} + dm_{R_i}^{(10)} + \beta(s_i^{(10)} - \theta s_j^{(10)}) = 0; i = 1, 2, j = 3 - i, \qquad (A.101)$$

$$\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}} = \alpha_j - w_j^{(10)} + dw_i^{(10)} - 2m_{R_i}^{(10)} + dm_{R_i}^{(10)} + \beta(s_j^{(10)} - \theta s_i^{(10)}) = 0; j = 1, 2, i = 3 - j, \qquad (A.102)$$

$$m_{R_i}^{(10)} = \left\{ \begin{array}{ll} 2(\alpha_i - w_i^{(10)} + dw_j^{(10)} + \beta(s_i^{(10)} - \theta s_j^{(10)})) & \quad V; i = 1, 2, j = 3 - i. \\ +d(\alpha_j - w_j^{(10)} + dw_i^{(10)} + \beta(s_j^{(10)} - \theta s_i^{(10)})) + \beta(2 - d\theta)s_{G_i} + \beta(d - 2\theta)s_{G_j} & \quad \end{array} \right\} \qquad (A.103)$$

By substituting $m_{R_i}^{(10)}$ into following equation the optimal retailer's price will be obtained:

$$p_i^{(10)} = w_i^{(10)} + m_{R_i}^{(10)}; i = 1, 2, j = 3 - i. \qquad (A.104)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(10)}}{\partial w_i^{(10)}}=0$ and $\frac{\partial U_{M_j}^{(10)}}{\partial w_j^{(10)}} = 0$ must be solved. With considering $w_i^{(10)} - c_i^{(10)} = m_{M_i}^{(10)}$ and $w_j^{(10)} - c_j^{(10)} = m_{M_j}^{(10)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(10)}}{\partial m_{M_i}^{(10)}} = 0$ and $\frac{\partial U_{M_j}^{(10)}}{\partial m_{M_j}^{(10)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(10)}}{\partial m_{M_i}^{(10)}} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2-d\theta)\rho_{M_i}}{2\eta_i\tau_i V} + \frac{\beta(d-2\theta)\rho_{M_j}}{2\eta_j\tau_j V} + \frac{\beta(2-d\theta)\lambda_i}{2\eta_i\tau_i V} + \frac{\beta(d-2\theta)\lambda_j}{2\eta_j\tau_j V} + \frac{\beta^2(2-d\theta)^2}{2\eta_i\tau_i V - 2(2-d^2)}m_{M_i}^{(10)} + [d + \frac{\beta^2(2-d\theta)(d-2\theta)}{2\eta_j\tau_j V}]m_{M_j}^{(10)} = 0, \qquad (A.105)$$

where $i=1, 2$ and $j=3-i$.

$$\frac{\partial U_{M_j}^{(10)}}{\partial m_{M_j}^{(10)}} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2-d\theta)\rho_{M_j}}{2\eta_j\tau_j V} + \frac{\beta(d-2\theta)\rho_{M_i}}{2\eta_i\tau_i V} + \frac{\beta(2-d\theta)\lambda_j}{2\eta_j\tau_j V} + \frac{\beta(d-2\theta)\lambda_i}{2\eta_i\tau_i V} + \frac{\beta^2(2-d\theta)^2}{2\eta_j\tau_j V - 2(2-d^2)}m_{M_j}^{(10)} + [d + \frac{\beta^2(2-d\theta)(d-2\theta)}{2\eta_i\tau_i V}]m_{M_i}^{(10)} = 0, \qquad (A.106)$$

where $j=1, 2$ and $i=3-j$. By solving the equations above $m_{M_i}^{(10)}$ is achieved:

$$m_{M_i}^{(10)} = m_{M_i}^{(2)} + \frac{\beta[(2-d\theta)B_j^{(2)} - (d-2\theta)A_j^{(2)}]}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} s_{G_i} + \frac{\beta[(d-2\theta)B_j^{(2)} - (d-2\theta)A_j^{(2)}]}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} s_{G_j}; i = 1, 2, j = 3 - i. \qquad (A.107)$$
$$w_i^{(9)} = c_i + m_{M_i}^{(9)}, i = 1, 2, j = 3 - i \qquad (A.98)$$

By calculating $\frac{\partial U_{M_i}^{(9)}}{\partial s_i^{(9)}} = 0$, the optimal amount of $s_i^{(9)}$ is achieved:

$$\frac{\partial U_M^{(9)}}{\partial s^{(9)}} = \frac{\beta(2+d)}{V} \sum_{i=1}^{2} m_{M_i}^{(9)} + \lambda - 2\eta \tau s^{(9)} + \rho_M = 0, \qquad (A.99)$$

$$s^{(9)} = \left[ \beta(2+d) \sum_{i=1}^{2} m_{M_i}^{(9)} + \rho_M V \right] / 2\eta\tau V, \qquad (A.100)$$

where $i=1, 2$ and $j=3-i$.

Considering $H(U_{M_i}^{(9)}) = \begin{bmatrix} \frac{-2+d^2+(d^2-2)}{4-d^2} & \frac{2\beta-d\beta\theta}{4-d^2} \\ \frac{2\beta-d\beta\theta}{4-d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le \lambda \le 1$,

$H(U_{M_i}^{(9)})$ is negatively defined. It is necessary to mention that if $\eta \ge \beta^2(2-d)/(2+d)[2(2-d^2)-d]$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(9)})) > 0$, the concavity of the profit function holds. □

**Proposition 10 proof**

To obtain the retailer price When the RA establishes GCSE according to Scenario 10 $\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}}$ and $\frac{\partial U_{R_j}^{(10)}}{\partial m_{R_j}^{(10)}}$ must be obtained. In addition, with considering $\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}} = 0$ and $\frac{\partial U_{R_j}^{(10)}}{\partial m_{R_j}^{(10)}} = 0$ the optimal retailer's price will be calculated:

$$\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}} = \alpha_i - w_i^{(10)} + dw_j^{(10)} - 2m_{R_i}^{(10)} + dm_{R_i}^{(10)} + \beta(s_i^{(10)} - \theta s_j^{(10)}) = 0; i = 1, 2, j = 3 - i, \qquad (A.101)$$

$$\frac{\partial U_{R_i}^{(10)}}{\partial m_{R_i}^{(10)}} = \alpha_j - w_j^{(10)} + dw_i^{(10)} - 2m_{R_i}^{(10)} + dm_{R_i}^{(10)} + \beta(s_j^{(10)} - \theta s_i^{(10)}) = 0; j = 1, 2, i = 3 - j, \qquad (A.102)$$

$$m_{R_i}^{(10)} = \left\{ \begin{array}{ll} 2(\alpha_i - w_i^{(10)} + dw_j^{(10)} + \beta(s_i^{(10)} - \theta s_j^{(10)})) & \quad V; i = 1, 2, j = 3 - i. \\ +d(\alpha_j - w_j^{(10)} + dw_i^{(10)} + \beta(s_j^{(10)} - \theta s_i^{(10)})) + \beta(2 - d\theta)s_{G_i} + \beta(d - 2\theta)s_{G_j} & \quad \end{array} \right. \qquad (A.103)$$

By substituting $m_{R_i}^{(10)}$ into following equation the optimal retailer's price will be obtained:

$$p_i^{(10)} = w_i^{(10)} + m_{R_i}^{(10)}; i = 1, 2, j = 3 - i. \qquad (A.104)$$

To obtain optimal manufacturer's price according to the Stackelberg game model, $\frac{\partial U_{M_i}^{(10)}}{\partial w_i^{(10)}}=0$ and $\frac{\partial U_{M_j}^{(10)}}{\partial w_j^{(10)}} = 0$ must be solved. With considering $w_i^{(10)} - c_i^{(10)} = m_{M_i}^{(10)}$ and $w_j^{(10)} - c_j^{(10)} = m_{M_j}^{(10)}$ as the manufacturer's unit marginal profit $\frac{\partial U_{M_i}^{(10)}}{\partial m_{M_i}^{(10)}} = 0$ and $\frac{\partial U_{M_j}^{(10)}}{\partial m_{M_j}^{(10)}} = 0$ are as follows:

$$\frac{\partial U_{M_i}^{(10)}}{\partial m_{M_i}^{(10)}} = 2(\alpha_i - c_i + dc_j) + d(\alpha_j - c_j + dc_i) + \frac{\beta(2-d\theta)\rho_{M_i}}{2\eta_i\tau_i V} + \frac{\beta(d-2\theta)\rho_{M_j}}{2\eta_j\tau_j V} + \frac{\beta(2-d\theta)\lambda_i}{2\eta_i\tau_i V} + \frac{\beta(d-2\theta)\lambda_j}{2\eta_j\tau_j V} + \left[ \frac{\beta^2(2-d\theta)^2}{2\eta_i\tau_i V} - 2(2-d^2) \right] m_{M_i}^{(10)} + \left[ d + \frac{\beta^2(2-d\theta)(d-2\theta)}{2\eta_j\tau_j V} \right] m_{M_j}^{(10)} = 0, \qquad (A.105)$$

where $i=1, 2$ and $j=3-i$.

$$\frac{\partial U_{M_j}^{(10)}}{\partial m_{M_j}^{(10)}} = 2(\alpha_j - c_j + dc_i) + d(\alpha_i - c_i + dc_j) + \frac{\beta(2-d\theta)\rho_{M_j}}{2\eta_j\tau_j V} + \frac{\beta(d-2\theta)\rho_{M_j}}{2\eta_j\tau_j V} + \frac{\beta(2-d\theta)\lambda_j}{2\eta_j\tau_j V} + \frac{\beta(d-2\theta)\lambda_i}{2\eta_i\tau_i V} + \left[ \frac{\beta^2(2-d\theta)^2}{2\eta_j\tau_j V} - 2(2-d^2) \right] m_{M_j}^{(10)} + \left[ d + \frac{\beta^2(2-d\theta)(d-2\theta)}{2\eta_i\tau_i V} \right] m_{M_i}^{(10)} = 0, \qquad (A.106)$$

where $j=1, 2$ and $i=3-j$. By solving the equations above $m_{M_i}^{(10)}$ is achieved:

$$m_{M_i}^{(10)} = m_{M_i}^{(2)} + \frac{\beta[(2-d\theta)B_j^{(2)} - (d-2\theta)A_j^{(2)}]}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} s_{G_i} + \frac{\beta[(d-2\theta)B_j^{(2)} - (d-2\theta)A_j^{(2)}]}{A_i^{(2)}A_j^{(2)} - B_i^{(2)}B_j^{(2)}} s_{G_j}; i = 1, 2, j = 3 - i. \qquad (A.107)$$
By substituting $m_{M_i}^{(10)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$w_i^{(10)} = c_i + m_{M_i}^{(10)}; i = 1, 2; j = 3 - i. \qquad (A.108)$$

By calculating $\frac{\partial U_{M_i}^{(10)}}{\partial s_i^{(2)}} = 0$, the optimal amount of $s_i^{(2)}$ is achieved:

$$\frac{\partial U_{M_i}^{(10)}}{\partial s_i^{(10)}} = m_{M_i}^{(10)} \frac{2\beta - d\beta\theta}{V} - 2\eta_i\tau_i s_i^{(10)} + \rho_{M_i} = 0, \qquad (A.109)$$

$$s_i^{(10)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(10)} + \rho_{M_i}V]}{2\eta_i\tau_i V}, \qquad (A.110)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(10)}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(10)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $\det(H(U_{M_i}^{(10)})) > 0$, the concavity of the profit function holds. $\square$.

## References

Arena, M., Azzone, G., & Mapelli, F. (2018). What drives the evolution of Corporate Social Responsibility strategies? An institutional logics perspective. *J. Clean. Prod.*, **171**, 345–355. https://doi.org/10.1016/j.jclepro.2017.09.245

Arunmozhi, M., Venkatesh, V. G., Arisian, S., Shi, Y., & Sreedharan, V. R. (2022). Application of blockchain and smart contracts in autonomous vehicle supply chains: An experimental design. *Transp. Res. E: Logist. Transp. Rev.*, **165**, Article 102864.

Asian, S., Hafezalkotob, A., & John, J. J. (2019). Sharing economy in organic food supply chains: A pathway to sustainable development. *Int. J. Prod. Econ.*, **218**, 322–338.

Asian, S., Wang, J., & Dickson, G. (2020). Trade disruptions, behavioral biases, and social influences: Can luxury sporting goods supply chains be immunized? *Transp. Res. Part E: Logist. Transp. Rev.*, **143**, Article 102064.

Aoun, A., Ilinca, A., Ghandour, M., & Ibrahim, H. (2021). A review of Industry 4.0 characteristics and challenges, with potential improvements using blockchain technology. *Comput. Ind. Eng.*, **162**, Article 107746.

Basiri, Z., & Heydari, J. (2017). A mathematical model for green supply chain coordination with substitutable products. *J. Clean. Prod.*, **145**, 232–249. https://doi.org/10.1016/j.jclepro.2017.01.060

Bazan, E., Jaber, M. Y., & El Saadany, A. M. A. (2015a). Carbon Emissions and Energy Effects on Manufacturing-Remanufacturing Inventory Models. *Comput. Ind. Eng.*, **88**, 307–316. https://doi.org/10.1016/j.cie.2015.07.002

Bazan, E., Jaber, M. Y., & Zanoni, S. (2015b). Supply chain models with greenhouse gases emissions, energy usage and different coordination decisions. *App. Math. Model.*, **39**, 5131–5151. https://doi.org/10.1016/j.apm.2015.03.044

Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). *Nonlinear Programming: Theory and Algorithms*. Wiley.

Bhardwaj, B. R. (2016). Role of green policy on sustainable supply chain management. *Benchmarking Int. J.*, **23**, 456–468. https://doi.org/10.1108/BIJ-08-2013-0077

Brandenburg, M., Govindan, K., Sarkis, J., & Seuring, S. (2014). Quantitative models for sustainable supply chain management: Developments and directions. *Eur. J. Oper. Res.*, **233**, 299–312. https://doi.org/10.1016/j.ejor.2013.09.032

Bueno, A., Godinho Filho, M., & Frank, A. G. (2020). Smart production planning and control in the Industry 4.0 context: A systematic literature review. *Comput. Ind. Eng.*, **149**, Article 106774.

Cheng, H., & Ding, H. (2021). Dynamic game of corporate social responsibility in a supply chain with competition. *J. Clean. Prod.*, **317**, Article 128398.

Chen, X., Wang, C., & Li, S. (2022). The impact of supply chain finance on corporate social responsibility and creating shared value: A case from the emerging economy. *Supply Chain Management: An International Journal*.

Chen, P. K., & Ye, Y. (2022). Influence of creating an oligopoly through government intervention to improve partner collaboration intentions in the context of green supply chains. *Environ. Sci. Pollut. Res.*, **29(5)**, 6433–6448.

Dai, D., Wu, X., Si, F., Feng, Z., & Chu, W. (2023). The impact of tariff policies on vaccine supply chains: Short-term and evolutionary game behaviors based on uncertain utility. *App. Math. Model.*, **115**, 754–777.

Dong, C., Shen, B., Chow, P.-S., Yang, L., & Ng, C. T. (2016). Sustainability investment under cap-and-trade regulation. *Ann. Oper. Res.*, **240**, 509–531. https://doi.org/10.1007/s10479-013-1514-1

European Commission, 2011. Communication from the Commission to the European Parliament, the Council, the European Economic and Social Committee and the Committee of the Regions: A Renewed EU Strategy 2011-14 for Corporate Social Responsibility. COM(2011) 681 Final 1–15.

Feng, Y., Zhu, Q., & Lai, K.-H. (2017). Corporate social responsibility for supply chain management: A literature review and bibliometric analysis. *J. Clean. Prod.*, **158**, 296–307. https://doi.org/10.1016/j.jclepro.2017.05.018

Formentini, M., & Taticchi, P. (2016). Corporate sustainability approaches and governance mechanisms in sustainable supply chain management. *J. Clean. Prod.*, **112**, 1920–1933. https://doi.org/10.1016/j.jclepro.2014.12.072

Guo, D., He, Y., Wu, Y., & Xu, Q. (2016). Analysis of Supply Chain under Different Subsidy Policies of the Government. *Sustainability*, **8**, 1290. https://doi.org/10.3390/su8121290

Hafezalkotob, A. (2018). Direct and indirect intervention schemas of government in the competition between green and non-green supply chains. *J. Clean. Prod.*, **170**, 753–772. https://doi.org/10.1016/j.jclepro.2017.09.124

Hafezalkotob, A. (2017a). Competition, cooperation, and coopetition of green supply chains under regulations on energy saving levels. *Transp. Res. Part E Logist. Transp. Rev.*, **97**, 228–250. https://doi.org/10.1016/j.tre.2016.11.004

Hafezalkotob, A. (2017b). Competition of domestic manufacturer and foreign supplier under sustainable development objectives of government. *Appl. Math Comput.*, **292**, 294–308. https://doi.org/10.1016/j.amc.2016.07.007

Hafezalkotob, A. (2015). Competition of two green and regular supply chains under environmental protection and revenue seeking policies of government. *Comput. Ind. Eng.*, **82**, 103–114. https://doi.org/10.1016/j.cie.2015.01.016

Hafezalkotob, A., Nersesian, L., & Fardi, K. (2023). A policy-making model for evolutionary SME Behavior during a pandemic recession supported on Game Theory Approach. *Comput. Ind. Eng.*, **108975**.

He, X., Prasad, A., Sethi, S. P., & Gutierrez, G. J. (2007). A survey of Stackelberg differential game models in supply and marketing channels. *J. Syst. Sci. Syst. Eng.*, **16** (4), 385–413.

Hepburn, C. (2006). Regulation by Prices, Quantities, or Both: A Review of Instrument Choice. *Oxford Rev. Econ. Policy*, **22**, 226–247. https://doi.org/10.1093/oxrep/grj014

Heydari, J., Govindan, K., & Jafari, A. (2017). Reverse and closed loop supply chain coordination by considering government role. *Transp. Res. Part D: Transp. Environ.*, **52**, 379–398. https://doi.org/10.1016/j.trd.2017.03.008

Hsueh, C.-F. (2015). A bilevel programming model for corporate social responsibility collaboration in sustainable supply chain management. *Transp. Res. Part E Logist. Transp. Rev.*, **73**, 84–95. https://doi.org/10.1016/j.tre.2014.11.006

Hsueh, C. F. (2014). Improving corporate social responsibility in a supply chain through a new revenue sharing contract. *Int. J. Prod. Econ.*, **151**, 214–222. https://doi.org/10.1016/j.ije.2013.10.017

Javadi, T., Alizadeh-Basban, N., Asian, S., & Hafezalkotob, A. (2019). Pricing policies in a dual-channel supply chain considering flexible return and energy-saving regulations. *Comput. Ind. Eng.*, **135**, 655–674.

Kogg, B., & Mont, O. (2012). Environmental and social responsibility in supply chains: The practise of choice and inter-organisational management. *Ecol. Econ.*, **83**, 154–163. https://doi.org/10.1016/j.ecolecon.2011.08.023

Leo, Z., Anderson, T. D., & Cruz, J. M. (2012). Consumer environmental awareness and competition in two-stage supply chains. *Eur. J. Oper. Res.*, **218**, 602–613. https://doi.org/10.1016/j.ejor.2011.11.027

Li, G., Wu, H., Sethi, S. P., & Zhang, X. (2021a). Contracting green product supply chains considering marketing efforts in the circular economy era. *Int. J. Prod. Econ.*, **234**, Article 108041.

Li, G., Li, L., Choi, T. M., & Sethi, S. P. (2020a). Green supply chain management in Chinese firms: Innovative measures and the moderating role of quick response technology. *J. Oper. Manag.*, **66(7–8)**, 958–988.

Li, X., Li, Y., 2014. Chain-to-chain competition on product sustainability 1–8.

Li, G., Li, X., & Liu, M. (2023). Inducing supplier backup via manufacturer information sharing under supply disruption risk. *Comput. Ind. Eng.*, **176**, Article 108914.

Liu, B., Li, T., & Tsai, S. (2017). Low Carbon Strategy Analysis of Competing Supply Chains with Different Power Structures. *Sustainability*, **9**, 835. https://doi.org/10.3390/su9050835
By substituting $m_{M_i}^{(10)}$ in the following equation the optimal manufacturer's wholesale price will be obtained:

$$w_i^{(10)} = c_i + m_{M_i}^{(10)}; i = 1, 2; j = 3 - i. \qquad (A.108)$$

By calculating $\frac{\partial U_{M_i}^{(10)}}{\partial s_i^{(2)}} = 0$, the optimal amount of $s_i^{(2)}$ is achieved:

$$\frac{\partial U_{M_i}^{(10)}}{\partial s_i^{(10)}} = m_{M_i}^{(10)} \frac{2\beta - d\beta\theta}{V} - 2\eta_i\tau_i s_i^{(10)} + \rho_{M_i} = 0, \qquad (A.109)$$

$$s_i^{(10)} = \frac{[\beta(2 - d\theta)m_{M_i}^{(10)} + \rho_{M_i}V]}{2\eta_i\tau_i V}, \qquad (A.110)$$

where $i = 1, 2$ and $j = 3 - i$.

Considering $H(U_{M_i}^{(10)}) = \begin{bmatrix} \frac{-2 + d^2 + (d^2 - 2)}{4 - d^2} & \frac{2\beta - d\beta\theta}{4 - d^2} \\ \frac{2\beta - d\beta\theta}{4 - d^2} & -2\eta \end{bmatrix}$ as the Hessian matrix and $0 \le d \le 1$,

$H(U_{M_i}^{(10)})$ is negatively defined. It is necessary to mention that if $\eta_i \ge \beta^2(2 - d\theta)^2/4(2 - d^2)$ the determinant of Hessian matrix takes a positive amount $det(H(U_{M_i}^{(10)})) > 0$, the concavity of the profit function holds. $\square$.

## References

Arena, M., Azzone, G., & Mapelli, F. (2018). What drives the evolution of Corporate Social Responsibility strategies? An institutional logics perspective. *J. Clean. Prod.*, **171**, 345–355. https://doi.org/10.1016/j.jclepro.2017.09.245

Arunmozhi, M., Venkatesh, V. G., Arisian, S., Shi, Y., & Sreedharan, V. R. (2022). Application of blockchain and smart contracts in autonomous vehicle supply chains: An experimental design. *Transp. Res. E: Logist. Transp. Rev.*, **165**, Article 102864.

Asian, S., Hafezalkotob, A., & John, J. J. (2019). Sharing economy in organic food supply chains: A pathway to sustainable development. *Int. J. Prod. Econ.*, **218**, 322–338.

Asian, S., Wang, J., & Dickson, G. (2020). Trade disruptions, behavioral biases, and social influences: Can luxury sporting goods supply chains be immunized? *Transp. Res. Part E: Logist. Transp. Rev.*, **143**, Article 102064.

Aoun, A., Ilinca, A., Ghandour, M., & Ibrahim, H. (2021). A review of Industry 4.0 characteristics and challenges, with potential improvements using blockchain technology. *Comput. Ind. Eng.*, **162**, Article 107746.

Basiri, Z., & Heydari, J. (2017). A mathematical model for green supply chain coordination with substitutable products. *J. Clean. Prod.*, **145**, 232–249. https://doi.org/10.1016/j.jclepro.2017.01.060

Bazan, E., Jaber, M. Y., & El Saadany, A. M. A. (2015a). Carbon Emissions and Energy Effects on Manufacturing-Remanufacturing Inventory Models. *Comput. Ind. Eng.*, **88**, 307–316. https://doi.org/10.1016/j.cie.2015.07.002

Bazan, E., Jaber, M. Y., & Zanoni, S. (2015b). Supply chain models with greenhouse gases emissions, energy usage and different coordination decisions. *App. Math. Model.*, **39**, 5131–5151. https://doi.org/10.1016/j.apm.2015.03.044

Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). *Nonlinear Programming: Theory and Algorithms*. Wiley.

Bhardwaj, B. R. (2016). Role of green policy on sustainable supply chain management. *Benchmarking Int. J.*, **23**, 456–468. https://doi.org/10.1108/BIJ-08-2013-0077

Brandenburg, M., Govindan, K., Sarkis, J., & Seuring, S. (2014). Quantitative models for sustainable supply chain management: Developments and directions. *Eur. J. Oper. Res.*, **233**, 299–312. https://doi.org/10.1016/j.ejor.2013.09.032

Bueno, A., Godinho Filho, M., & Frank, A. G. (2020). Smart production planning and control in the Industry 4.0 context: A systematic literature review. *Comput. Ind. Eng.*, **149**, Article 106774.

Cheng, H., & Ding, H. (2021). Dynamic game of corporate social responsibility in a supply chain with competition. *J. Clean. Prod.*, **317**, Article 128398.

Chen, X., Wang, C., & Li, S. (2022). The impact of supply chain finance on corporate social responsibility and creating shared value: A case from the emerging economy. *Supply Chain Management: An International Journal*.

Chen, P. K., & Ye, Y. (2022). Influence of creating an oligopoly through government intervention to improve partner collaboration intentions in the context of green supply chains. *Environ. Sci. Pollut. Res.*, **29(5)**, 6433–6448.

Dai, D., Wu, X., Si, F., Feng, Z., & Chu, W. (2023). The impact of tariff policies on vaccine supply chains: Short-term and evolutionary game behaviors based on uncertain utility. *App. Math. Model.*, **115**, 754–777.

Dong, C., Shen, B., Chow, P.-S., Yang, L., & Ng, C. T. (2016). Sustainability investment under cap-and-trade regulation. *Ann. Oper. Res.*, **240**, 509–531. https://doi.org/10.1007/s10479-013-1514-1

European Commission, 2011. Communication from the Commission to the European Parliament, the Council, the European Economic and Social Committee and the Committee of the Regions: A Renewed EU Strategy 2011-14 for Corporate Social Responsibility. COM(2011) 681 Final 1–15.

Feng, Y., Zhu, Q., & Lai, K.-H. (2017). Corporate social responsibility for supply chain management: A literature review and bibliometric analysis. *J. Clean. Prod.*, **158**, 296–307. https://doi.org/10.1016/j.jclepro.2017.05.018

Formentini, M., & Taticchi, P. (2016). Corporate sustainability approaches and governance mechanisms in sustainable supply chain management. *J. Clean. Prod.*, **112**, 1920–1933. https://doi.org/10.1016/j.jclepro.2014.12.072

Guo, D., He, Y., Wu, Y., & Xu, Q. (2016). Analysis of Supply Chain under Different Subsidy Policies of the Government. *Sustainability*, **8**, 1290. https://doi.org/10.3390/su8121290

Hafezalkotob, A. (2018). Direct and indirect intervention schemas of government in the competition between green and non-green supply chains. *J. Clean. Prod.*, **170**, 753–772. https://doi.org/10.1016/j.jclepro.2017.09.124

Hafezalkotob, A. (2017a). Competition, cooperation, and coopetition of green supply chains under regulations on energy saving levels. *Transp. Res. Part E Logist. Transp. Rev.*, **97**, 228–250. https://doi.org/10.1016/j.tre.2016.11.004

Hafezalkotob, A. (2017b). Competition of domestic manufacturer and foreign supplier under sustainable development objectives of government. *Appl. Math Comput.*, **292**, 294–308. https://doi.org/10.1016/j.amc.2016.07.007

Hafezalkotob, A. (2015). Competition of two green and regular supply chains under environmental protection and revenue seeking policies of government. *Comput. Ind. Eng.*, **82**, 103–114. https://doi.org/10.1016/j.cie.2015.01.016

Hafezalkotob, A., Nersesian, L., & Fardi, K. (2023). A policy-making model for evolutionary SME Behavior during a pandemic recession supported on Game Theory Approach. *Comput. Ind. Eng.*, **108975**.

He, X., Prasad, A., Sethi, S. P., & Gutierrez, G. J. (2007). A survey of Stackelberg differential game models in supply and marketing channels. *J. Syst. Sci. Syst. Eng.*, **16** (4), 385–413.

Hepburn, C. (2006). Regulation by Prices, Quantities, or Both: A Review of Instrument Choice. *Oxford Rev. Econ. Policy*, **22**, 226–247. https://doi.org/10.1093/oxrep/grj014

Heydari, J., Govindan, K., & Jafari, A. (2017). Reverse and closed loop supply chain coordination by considering government role. *Transp. Res. Part D: Transp. Environ.*, **52**, 379–398. https://doi.org/10.1016/j.trd.2017.03.008

Hsueh, C.-F. (2015). A bilevel programming model for corporate social responsibility collaboration in sustainable supply chain management. *Transp. Res. Part E Logist. Transp. Rev.*, **73**, 84–95. https://doi.org/10.1016/j.tre.2014.11.006

Hsueh, C. F. (2014). Improving corporate social responsibility in a supply chain through a new revenue sharing contract. *Int. J. Prod. Econ.*, **151**, 214–222. https://doi.org/10.1016/j.ijepr.2013.10.017

Javadi, T., Alizadeh-Basban, N., Asian, S., & Hafezalkotob, A. (2019). Pricing policies in a dual-channel supply chain considering flexible return and energy-saving regulations. *Comput. Ind. Eng.*, **135**, 655–674.

Kogg, B., & Mont, O. (2012). Environmental and social responsibility in supply chains: The practise of choice and inter-organisational management. *Ecol. Econ.*, **83**, 154–163. https://doi.org/10.1016/j.ecolecon.2011.08.023

Leo, Z., Anderson, T. D., & Cruz, J. M. (2012). Consumer environmental awareness and competition in two-stage supply chains. *Eur. J. Oper. Res.*, **218**, 602–613. https://doi.org/10.1016/j.ejor.2011.11.027

Li, G., Wu, H., Sethi, S. P., & Zhang, X. (2021a). Contracting green product supply chains considering marketing efforts in the circular economy era. *Int. J. Prod. Econ.*, **234**, Article 108041.

Li, G., Li, L., Choi, T. M., & Sethi, S. P. (2020a). Green supply chain management in Chinese firms: Innovative measures and the moderating role of quick response technology. *J. Oper. Manag.*, **66(7–8)**, 958–988.

Li, X., Li, Y., 2014. Chain-to-chain competition on product sustainability 1–8.

Li, G., Li, X., & Liu, M. (2023). Inducing supplier backup via manufacturer information sharing under supply disruption risk. *Comput. Ind. Eng.*, **176**, Article 108914.

Liu, B., Li, T., & Tsai, S. (2017). Low Carbon Strategy Analysis of Competing Supply Chains with Different Power Structures. *Sustainability*, **9**, 835. https://doi.org/10.3390/su9050835
Liu, K., Li, W., Jia, F., & Lan, Y. (2022). Optimal strategies of green product supply chains based on behaviour-based pricing. *J. Clean. Prod.*, **130288**.

Liu, Z., Xu, L., Ren, X., Lu, Q., Wang, X., & Arisian, S. (2023). Contagion effect in the adoption of environmental corporate social responsibility. *Ann. Oper. Res.*, 1-38.

Madani, S. R., & Rasti-Barzoki, M. (2017). Sustainable supply chain management with pricing, greening and governmental tariffs determining strategies: A game-theoretic approach. *Comput. Ind. Eng.*, https://doi.org/10.1016/j.cie.2017.01.017

Mahmoudi, R., & Rasti-Barzoki, M. (2018). Sustainable supply chains under government intervention with a real-world case study: An evolutionary game theoretic approach. *Comput. Ind. Eng.*, **116**, 130-143.

Matinfard, S., Yaghoubi, S., & Kharaji Manouchehrabadi, M. (2022). Impact of government policies on photovoltaic supply chain considering quality in the power distribution system: A case study. *Environ. Sci. Pollut. Res.*, **29**(39), 58810-58827.

Mu, D., Ren, H., & Wang, C. (2021). A Literature Review of Taxes in Cross-Border Supply Chain Modeling: Themes, Tax Types and New Trade-Offs. *J. Theor. Appl. Electron. Commer. Res.*, **17**(1), 20-46.

Ni, D., & Li, K. W. (2012). A game-theoretic analysis of social responsibility conduct in two-echelon supply chains. *Int. J. Prod. Econ.*, **138**, 303-313. https://doi.org/10.1016/j.ijepr.2012.04.002

Ni, D., Li, K. W., & Tang, X. (2010). Social responsibility allocation in two-echelon supply chains: Insights from wholesale price contracts. *Eur. J. Oper. Res.*, **207**, 1269-1279. https://doi.org/10.1016/j.ejor.2010.06.026

Nersesian, L., Hafezalkotob, A., & Reza-Gharehbagh, R. (2022). Alternative governmental carbon policies on populations of green and non-green supply chains in a competitive market. *Environment, Development and Sustainability*, 1-34.

Pakseresht, M., Shirazi, B., Mahdavi, I., & Mahdavi-Amiri, N. (2020). Toward sustainable optimization with stackelberg game between green product family and downstream supply chain. *Sustainable Prod. Consumpt.*, **23**, 198-211.

Panda, S., & Modak, N. M. (2016). Exploring the effects of social responsibility on coordination and profit division in a supply chain. *J. Clean. Prod.*, **139**, 25-40. https://doi.org/10.1016/j.jclepro.2016.07.118

Qin Zhang, Lei Yang, 2016. Chain-to-chain competition under the cap-and-trade scheme, in: 2016 13th International Conference on Service Systems and Service Management (ICSSSM). IEEE, pp. 1-6. doi:10.1109/ICSSSM.2016.7538481.

Quarshie, A. M., Salmi, A., & Leuschner, R. (2016). Sustainability and corporate social responsibility in supply chains: The state of research in supply chain management and business ethics journals. *J. Purch. Supply Manag.*, **22**, 82-97. https://doi.org/10.1016/j.pursup.2015.11.001

Rahimi, M., Hafezalkotob, A., Asian, S., & Martínez, L. (2021). Environmental policy making in supply chains under ambiguity and competition: A fuzzy Stackelberg game approach. *Sustainability*, **13**(4), 2367.

Rasti-Barzoki, M., & Moon, I. (2021). A game theoretic approach for analyzing electric and gasoline-based vehicles' competition in a supply chain under government sustainable strategies: A case study of South Korea. *Renew. Sustain. Energy Rev.*, **146**, Article 111139.

Reza-Gharehbagh, R., Arisian, S., Hafezalkotob, A., & Makui, A. (2022). Sustainable supply chain finance through digital platforms: A pathway to green entrepreneurship. *Ann. Oper. Res.*, 1-35.

Reza-Gharehbagh, R., Asian, S., Hafezalkotob, A., & Wei, C. (2021). Reframing supply chain finance in an era of reglocalization: On the value of multi-sided crowdfunding platforms. *Transp. Res. E: Logist. Transp. Rev.*, **149**, Article 102298.

Rodríguez-González, R. M., Maldonado-Guzman, G., & Madrid-Guijarro, A. (2022). The effect of green strategies and eco-innovation on Mexican automotive industry sustainable and financial performance: Sustainable supply chains as a mediating variable. *Corp. Soc. Respon. Environ. Manag.*

Ruan, X., Ding, N., & Yang, J. (2022). Dual-level eco-efficiency analysis in sustainable management: An empirical study on textile manufacturers and supply chains in China. *J. Environ. Manage.*, **304**, Article 114227.

Rotter, J. P., Airike, P. E., & Mark-Herbert, C. (2013). Exploring Political Corporate Social Responsibility in Global Supply Chains. *J. Bus. Ethics*, **125**, 581-599. https://doi.org/10.1007/s10551-013-1927-4

Seuring, S., & Müller, M. (2008). From a literature review to a conceptual framework for sustainable supply chain management. *J. Clean. Prod.*, **16**, 1699-1710. https://doi.org/10.1016/j.jclepro.2008.04.020

Sheu, J. (2011). Bargaining framework for competitive green supply chains under governmental financial intervention. *Transp. Res. Part E Logist. Transp. Rev.*, **47**, 573-592. https://doi.org/10.1016/j.tre.2010.12.006

Sheu, J., & Chen, Y. J. (2012). Impact of government financial intervention on competition among green supply chains. *Int. J. Prod. Econ.*, **138**, 201-213. https://doi.org/10.1016/j.ijepr.2012.03.024

Sudusinghe, J. I., & Seuring, S. (2022). Supply chain collaboration and sustainability performance in circular economy: A systematic literature review. *Int. J. Prod. Econ.*, **245**, Article 108402.

Taleizadeh, A. A., Ahmadzadeh, K., Sarker, B. R., & Ghavamifar, A. (2022). Designing an optimal sustainable supply chain system considering pricing decisions and resilience factors. *J. Clean. Prod.*, **332**, Article 129895.

Tamás, M. M., Bade Shrestha, S. O., & Zhou, H. (2010). Feed-in tariff and tradable green certificate in oligopoly. *Energy Policy*, **38**, 4040-4047. https://doi.org/10.1016/j.enpol.2010.03.028

Tiwari, M., Tiwari, T., Santhose, S. S., Mishra, L., Rejeesh, M. R., & Sundararaj, V. (2021). Corporate social responsibility and supply chain: A study for evaluating corporate hypocrisy with special focus on stakeholders. *Int. J. Financ. Econ.*

Waldman, K. B., & Kerr, J. M. (2014). Limitations of certification and supply chain standards for environmental protection in commodity crop production. *Annu. Rev. Resour. Econ.*, **6**(1), 429-449.

Xie, G. (2015). Modeling decision processes of a green supply chain with regulation on energy saving level. *Comput. Oper. Res.*, **54**, 266-273. https://doi.org/10.1016/j.cor.2013.11.020

Yang, D., & Xiao, T. (2017). Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties. *J. Clean. Prod.*, https://doi.org/10.1016/j.jclepro.2017.02.138

Yang, L., Zhang, Q., & Ji, J. (2017). Pricing and carbon emission reduction decisions in supply chains with vertical and horizontal cooperation. *Int. J. Prod. Econ.*, **191**, 286-297. https://doi.org/10.1016/j.ijepr.2017.06.021

Zand, F., & Yaghoubi, S. (2022). Effects of a dominant retailer on green supply chain activities with government cooperation. *Environ. Dev. Sustain.*, **24**(1), 1313-1334.

Zhang, L., Wang, J., & You, J. (2015). Consumer environmental awareness and channel coordination with two substitutable products. *Eur. J. Oper. Res.*, **241**, 63-73. https://doi.org/10.1016/j.ejor.2014.07.043

Zhang, Y. H., & Wang, Y. (2017). The impact of government incentive on the two competing supply chains under the perspective of Corporation Social Responsibility: A case study of Photovoltaic industry. *J. Clean. Prod.*, **154**, 102-113. https://doi.org/10.1016/j.jclepro.2017.03.127

Further reading

Li, G., Li, N., & Sethi, S. P. (2021b). Does CSR reduce idiosyncratic risk? Roles of operational efficiency and AI innovation. *Prod. Oper. Managt.*, **30**(7), 2027-2045.

Li, G., Tian, L., & Zheng, H. (2021c). Information sharing in an online marketplace with co-opetitive sellers. *Prod. Oper. Manag.*, **30**(10), 3713-3734.

Li, G., Lim, M. K., & Wang, Z. (2020b). Stakeholders, green manufacturing, and practice performance: Empirical evidence from Chinese fashion businesses. *Ann. Oper. Res.*, **290**, 961-982.
Liu, K., Li, W., Jia, F., & Lan, Y. (2022). Optimal strategies of green product supply chains based on behaviour-based pricing. *J. Clean. Prod.*, **130288**.

Liu, Z., Xu, L., Ren, X., Lu, Q., Wang, X., & Arisian, S. (2023). Contagion effect in the adoption of environmental corporate social responsibility. *Ann. Oper. Res.*, 1-38.

Madani, S. R., & Rasti-Barzoki, M. (2017). Sustainable supply chain management with pricing, greening and governmental tariffs determining strategies: A game-theoretic approach. *Comput. Ind. Eng.*, https://doi.org/10.1016/j.cie.2017.01.017

Mahmoudi, R., & Rasti-Barzoki, M. (2018). Sustainable supply chains under government intervention with a real-world case study: An evolutionary game theoretic approach. *Comput. Ind. Eng.*, **116**, 130-143.

Matinfard, S., Yaghoubi, S., & Kharaji Manouchehrabadi, M. (2022). Impact of government policies on photovoltaic supply chain considering quality in the power distribution system: A case study. *Environ. Sci. Pollut. Res.*, **29**(39), 58810-58827.

Mu, D., Ren, H., & Wang, C. (2021). A Literature Review of Taxes in Cross-Border Supply Chain Modeling: Themes, Tax Types and New Trade-Offs. *J. Theor. Appl. Electron. Commer. Res.*, **17**(1), 20-46.

Ni, D., & Li, K. W. (2012). A game-theoretic analysis of social responsibility conduct in two-echelon supply chains. *Int. J. Prod. Econ.*, **138**, 303-313. https://doi.org/10.1016/j.ijepr.2012.04.002

Ni, D., Li, K. W., & Tang, X. (2010). Social responsibility allocation in two-echelon supply chains: Insights from wholesale price contracts. *Eur. J. Oper. Res.*, **207**, 1269-1279. https://doi.org/10.1016/j.ejor.2010.06.026

Nersesian, L., Hafezalkotob, A., & Reza-Gharehbagh, R. (2022). Alternative governmental carbon policies on populations of green and non-green supply chains in a competitive market. *Environment, Development and Sustainability*, 1-34.

Pakseresht, M., Shirazi, B., Mahdavi, L., & Mahdavi-Amiri, N. (2020). Toward sustainable optimization with stackelberg game between green product family and downstream supply chain. *Sustainable Prod. Consumpt.*, **23**, 198-211.

Panda, S., & Modak, N. M. (2016). Exploring the effects of social responsibility on coordination and profit division in a supply chain. *J. Clean. Prod.*, **139**, 25-40. https://doi.org/10.1016/j.jclepro.2016.07.118

Qin Zhang, Lei Yang, 2016. Chain-to-chain competition under the cap-and-trade scheme, in: 2016 13th International Conference on Service Systems and Service Management (ICSSSM). IEEE, pp. 1-6. doi:10.1109/ICSSSM.2016.7538481.

Quarshie, A. M., Salmi, A., & Leuschner, R. (2016). Sustainability and corporate social responsibility in supply chains: The state of research in supply chain management and business ethics journals. *J. Purch. Supply Manag.*, **22**, 82-97. https://doi.org/10.1016/j.pursup.2015.11.001

Rahimi, M., Hafezalkotob, A., Asian, S., & Martínez, L. (2021). Environmental policy making in supply chains under ambiguity and competition: A fuzzy Stackelberg game approach. *Sustainability*, **13**(4), 2367.

Rasti-Barzoki, M., & Moon, I. (2021). A game theoretic approach for analyzing electric and gasoline-based vehicles' competition in a supply chain under government sustainable strategies: A case study of South Korea. *Renew. Sustain. Energy Rev.*, **146**, Article 111139.

Reza-Gharehbagh, R., Arisian, S., Hafezalkotob, A., & Makui, A. (2022). Sustainable supply chain finance through digital platforms: A pathway to green entrepreneurship. *Ann. Oper. Res.*, 1-35.

Reza-Garehbagh, R., Asian, S., Hafezalkotob, A., & Wei, C. (2021). Reframing supply chain finance in an era of reglocalization: On the value of multi-sided crowdfunding platforms. *Transp. Res. E: Logist. Transp. Rev.*, **149**, Article 102298.

Rodríguez-González, R. M., Maldonado-Guzman, G., & Madrid-Guijarro, A. (2022). The effect of green strategies and eco-innovation on Mexican automotive industry sustainable and financial performance: Sustainable supply chains as a mediating variable. *Corp. Soc. Respon. Environ. Manag.*

Ruan, X., Ding, N., & Yang, J. (2022). Dual-level eco-efficiency analysis in sustainable management: An empirical study on textile manufacturers and supply chains in China. *J. Environ. Manage.*, **304**, Article 114227.

Rotter, J. P., Airike, P. E., & Mark-Herbert, C. (2013). Exploring Political Corporate Social Responsibility in Global Supply Chains. *J. Bus. Ethics*, **125**, 581-599. https://doi.org/10.1007/s10551-013-1927-4

Seuring, S., & Müller, M. (2008). From a literature review to a conceptual framework for sustainable supply chain management. *J. Clean. Prod.*, **16**, 1699-1710. https://doi.org/10.1016/j.jclepro.2008.04.020

Sheu, J. (2011). Bargaining framework for competitive green supply chains under governmental financial intervention. *Transp. Res. Part E Logist. Transp. Rev.*, **47**, 573-592. https://doi.org/10.1016/j.tre.2010.12.006

Sheu, J., & Chen, Y. J. (2012). Impact of government financial intervention on competition among green supply chains. *Int. J. Prod. Econ.*, **138**, 201-213. https://doi.org/10.1016/j.ijepr.2012.03.024

Sudusinghe, J. I., & Seuring, S. (2022). Supply chain collaboration and sustainability performance in circular economy: A systematic literature review. *Int. J. Prod. Econ.*, **245**, Article 108402.

Taleizadeh, A. A., Ahmadzadeh, K., Sarker, B. R., & Ghavamifar, A. (2022). Designing an optimal sustainable supply chain system considering pricing decisions and resilience factors. *J. Clean. Prod.*, **332**, Article 129895.

Tamás, M. M., Bade Shrestha, S. O., & Zhou, H. (2010). Feed-in tariff and tradable green certificate in oligopoly. *Energy Policy*, **38**, 4040-4047. https://doi.org/10.1016/j.enpol.2010.03.028

Tiwari, M., Tiwari, T., Santhose, S. S., Mishra, L., Rejeesh, M. R., & Sundararaj, V. (2021). Corporate social responsibility and supply chain: A study for evaluating corporate hypocrisy with special focus on stakeholders. *Int. J. Financ. Econ.*

Waldman, K. B., & Kerr, J. M. (2014). Limitations of certification and supply chain standards for environmental protection in commodity crop production. *Annu. Rev. Resour. Econ.*, **6**(1), 429-449.

Xie, G. (2015). Modeling decision processes of a green supply chain with regulation on energy saving level. *Comput. Oper. Res.*, **54**, 266-273. https://doi.org/10.1016/j.cor.2013.11.020

Yang, D., & Xiao, T. (2017). Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties. *J. Clean. Prod.*, https://doi.org/10.1016/j.jclepro.2017.02.138

Yang, L., Zhang, Q., & Ji, J. (2017). Pricing and carbon emission reduction decisions in supply chains with vertical and horizontal cooperation. *Int. J. Prod. Econ.*, **191**, 286-297. https://doi.org/10.1016/j.ijepr.2017.06.021

Zand, F., & Yaghoubi, S. (2022). Effects of a dominant retailer on green supply chain activities with government cooperation. *Environ. Dev. Sustain.*, **24**(1), 1313-1334.

Zhang, L., Wang, J., & You, J. (2015). Consumer environmental awareness and channel coordination with two substitutable products. *Eur. J. Oper. Res.*, **241**, 63-73. https://doi.org/10.1016/j.ejor.2014.07.043

Zhang, Y. H., & Wang, Y. (2017). The impact of government incentive on the two competing supply chains under the perspective of Corporation Social Responsibility: A case study of Photovoltaic industry. *J. Clean. Prod.*, **154**, 102-113. https://doi.org/10.1016/j.jclepro.2017.03.127

Further reading

Li, G., Li, N., & Sethi, S. P. (2021b). Does CSR reduce idiosyncratic risk? Roles of operational efficiency and AI innovation. *Prod. Oper. Managt.*, **30**(7), 2027-2045.

Li, G., Tian, L., & Zheng, H. (2021c). Information sharing in an online marketplace with co-opetitive sellers. *Prod. Oper. Manag.*, **30**(10), 3713-3734.

Li, G., Lim, M. K., & Wang, Z. (2020b). Stakeholders, green manufacturing, and practice performance: Empirical evidence from Chinese fashion businesses. *Ann. Oper. Res.*, **290**, 961-982.