doi: 10.3934/jimo.2022098
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Evolutionary game theory analysis of supply chain with fairness concerns of retailers

1. 

Institute for Logistics and Emergency Management, School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, Sichuan, China

2. 

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, Guizhou, China

*Corresponding author: Ying Dai

Received  November 2021 Revised  April 2022 Early access June 2022

This study focuses on evolutionary game analysis of supply chain (SC) strategies with fairness concerns. First, the equilibrium solutions under the eight combined strategies are obtained through the Stackelberg game. Subsequently, evolutionary game theory is applied to analyze the stability of these equilibrium solutions and to discuss the evolutionary stability strategy of the tripartite game. The study suggests that the effect of distributional fairness concerns on SC pricing differs from that of peer-induced fairness concerns in the short term. Implanting fairness concerns into SC pricing decisions is always detrimental to the supplier and not always beneficial to retailers. From a long-term perspective, only one evolutionarily stable equilibrium point exists. Retailers do not pay attention to the fairness of the channel profit distribution, and the supplier does not consider the fairness of retailers. The results help enterprises understand the influence of fairness preferences on pricing decisions and provide a reference for the efficient operation of SCs.

Citation: Yadong Shu, Ying Dai, Zujun Ma. Evolutionary game theory analysis of supply chain with fairness concerns of retailers. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022098
References:
[1]

M. Alinaghian and A. Goli, Allocation and routing of temporary health centers in rural areas in crisis, solved by improved harmony search algorithm, International Journal of Computational Intelligence Systems, 10 (2017), 894-913. 

[2]

J. J. Angel and D. McCabe, Fairness in financial markets: The case of high frequency trading, Journal of Business Ethics, 112 (2013), 585-595. 

[3]

P. AssarzadeganS. R. Hejazi and G. A. Raissi, An evolutionary game theoretic model for analyzing retailers' behavior when introducing economy and premium private labels, Journal of Retailing and Consumer Services, 57 (2020), 102227. 

[4]

S. BarariG. AgarwalW. J. ZhangB. Mahanty and M. K. Tiwari, A decision framework for the analysis of green supply chain contracts: An evolutionary game approach, Expert Systems with Applications, 39 (2012), 2965-2976. 

[5]

S. Choi and P. R. Messinger, The role of fairness in competitive supply chain relationships: An experimental study, European Journal of Operational Research, 251 (2016), 798-813. 

[6]

T. H. CuiJ. S. Raju and Z. J. Zhang, Fairness and channel coordination, Management Science, 53 (2007), 1303-1314. 

[7]

G. B. DahlK. V. Løken and M. Mogstad, Peer effects in program participation, American Economic Review, 104 (2014), 2049-2074. 

[8]

S. F. DuL. WeiY. G. Zhu and T. F. Nie, Peer-regarding fairness in supply chain, International Journal of Production Research, 56 (2018), 3384-3396. 

[9]

E. Fehr and and K. M. Schmidt, A theory of fairness, competition and cooperation, Quarterly Journal of Economics, 114 (1999), 817-868. 

[10]

D. Friedman, On economic applications of evolutionary game theory, Journal of Evolutionary Economics, 8 (1998), 15-54. 

[11]

A. Goli, and T. Keshavarz, Just-in-time scheduling in identical parallel machine sequence-dependent group scheduling problem, Journal of Industrial and Management Optimization, 17 (2021).http://dx.doi.org/10.3934/jimo.2021124. doi: 10.3934/jimo.2021124.

[12]

A. GoliH. Khademi-ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers and Industrial Engineering, 137 (2019), 106090. 

[13]

A. Goli, H. Khademi-Zare, R. Tavakkoli-Moghaddam, et al., An integrated approach based on artificial intelligence and novel meta-heuristic algorithms to predict demand for dairy products: A case study, Network: Computation in Neural Systems, 32 (2021), 1-35.

[14]

A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, International Journal of Intelligent Transportation Systems Research, 18 (2018), 140-152. 

[15]

A. Goli and H. Mohammadi, Developing a sustainable operational management system using hybrid Shapley value and Multimoora method: Case study petrochemical supply chain, Environment, Development and Sustainability, 23 (2021), 1-30. 

[16]

T. H. Ho and X. M. Su, Peer-induced fairness in games, American Economic Review, 99 (2009), 2022-2049. 

[17]

T. H. HoX. M. Su and Y. Z. Wu, Distributional and peer-induced fairness in supply chain contract design, Production and Operations Management, 23 (2014), 161-175. 

[18]

T. H. Ho and J. J. Zhang, Designing pricing contracts for boundedly rational customers: Does the framing of the fixed fee matter?, Management Science, 54 (2008), 686-700. 

[19]

P. JiX. Ma and G. Li, Developing green purchasing relationships for the manufacturing industry: An evolutionary game theory perspective, International Journal of Production Economics, 166 (2015), 155-162. 

[20]

D. A. KahnemanJ. L. Knetsch and R. H. Thaler, The journal of business, International Journal of Production Economics, 59 (1986), 285-300. 

[21]

E. KatokT. L. Olsen and V. Pavlov, Wholesale pricing under mild and privately known concerns for fairness, Production and Operations Management, 23 (2014), 285-302. 

[22]

E. Katok and V. Pavlov, Fairness in supply chain contracts: A laboratory study, Journal of Operations Management, 31 (2013), 129-137. 

[23]

K. T. KimJ. S. Lee and S. Y. Lee, The effects of supply chain fairness and the buyer's power sources on the innovation performance of the supplier: A mediating role of social capital accumulation, Journal of Business Industrial Marketing, 31 (2017), 987-997. 

[24]

B. LiP. Hou and Q. Li, Cooperative advertising in a dual-channel supply chain with a fairness concern of the manufacturer, IMA Journal of Management Mathematics, 28 (2017), 259-277.  doi: 10.1093/imaman/dpv025.

[25]

C. F. LiX. Q. Guo and D. L. Du, Pricing decisions in dual-channel closed-loop supply chain under retailer's risk aversion and fairness concerns, Journal of the Operations Research Society of China, 9 (2021), 641-657.  doi: 10.1007/s40305-020-00324-7.

[26]

P. Lii and F. I. Kuo, Innovation-oriented supply chain integration for combined competitiveness and ?rm performance, International Journal of Production Economics, 174 (2016), 142-155. 

[27]

K. J. Li and S. Jain, Behavior-based pricing: An analysis of the impact of peer-induced fairness, Management Science, 62 (2016), 2705-2721. 

[28]

X. LiX. Q. CuiY. J. LiD. Q. Xu and F. C. Xu, Optimisation of reverse supply chain with used-product collection effort under collector's fairness concerns, International Journal of Production Research, 59 (2019), 652-663. 

[29]

T. T. LiJ. X. XieX. B. Zhao and J. F. Tang, On supplier encroachment with retailer's fairness concerns, Computers and Industrial Engineering, 98 (2016), 499-512. 

[30]

W. H. LiuD. WangX. R. ShenX. Y. Yan and W. Y. Wei, The impacts of distributional and peer-induced fairness concerns on the decision-making of order allocation in logistics service supply chain, Transportation Research Part E: Logistics and Transportation Review, 116 (2018), 102-122. 

[31]

C. H. Loch and Y. Z. Wu, Social preferences and supply chain performance: An experimental study, Management Science, 54 (2008), 1835-1849. 

[32]

R. LotfiB. KargarS. H. HoseiniS. NazariS. Safavi and G. W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, International Journal of Energy Research, 45 (2021), 17749-17766. 

[33]

R. Lotfi, B. Kargar, A. Gharehbaghi and G. W. Weber, Viable medical waste chain network design by considering risk and robustness, Environmental Science and Pollution Research, (2021), 1–16.

[34]

R. Lotfi, S. Safavi, A. Gharehbaghi, S. G. Zare, R. Hazrati and G. W. Weber, Viable supply chain network design by considering blockchain technology and cryptocurrency, Mathematical Problems in Engineering, (2021).http://dx.doi.org/10.1155/2021/7347389. doi: RHazrati2021.

[35]

R. LotfiG. W. WeberS. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial Management Optimization, 16 (2020), 117-140.  doi: 10.3934/jimo.2018143.

[36]

B. Nguyen and P. P. Klaus, Retail fairness: Exploring consumer perceptions of fairness towards retailers' marketing tactics, Journal of Retailing and Consumer Services, 20 (2013), 311-324. 

[37]

T. F. Nie and S. F. Du, Dual-fairness supply chain with quantity discount contracts, European Journal of Operational Research, 258 (2017), 491-500.  doi: 10.1016/j.ejor.2016.08.051.

[38]

B. Z. NiuQ. Q. Cui and J. Zhang, Impact of channel power and fairness concern on supplier's market entry decision, Journal of the Operational Research Society, 68 (2017), 1570-1581. 

[39]

K. W. PanZ. B. CuiA. X. Xing and Q. H. Lu, Impact of fairness concern on retailer-dominated supply chain, Computers and Industrial Engineering, 139 (2020), 106209. 

[40]

F. QinF. MaiM. J. Fry and A. S. Raturi, Supply-chain performance anomalies: Fairness concerns under private cost information, European Journal of Operational Research, 252 (2016), 170-182. 

[41]

Y. H. QinG. X. Wei and J. X. Dong, The signaling game model under asymmetric fairness-concern information, European Journal of Operational Research, 22 (2019), 5547-5562. 

[42]

A. Sharma, Game-Theoretic analysis of pricing models in a Dyadic supply chain with fairness concerns, International Journal of Strategic Decision Sciences, 10 (2019), 1-24.  doi: 10.1142/S0219198920500176.

[43]

Y. C. Shi and J. A. Zhu, Game-theoretic analysis for supply chain with distributional and peer-induced fairness concerned retailers, Management Science and Engineering, 8 (2014), 78-84. 

[44]

J. TaoL. S. ShaoZ. M. GuanW. Ho and S. Talluri, Incorporating risk aversion and fairness considerations into procurement and distribution decisions in a supply chain, International Journal of Production Research, 58 (2020), 1950-1967.  doi: 10.1016/j.cor.2020.105105.

[45]

E. B. TirkolaeeA. GoliP. Ghasemi and F. Goodarzian, Designing a sustainable closed-loop supply chain network of face masks during the COVID-19 pandemic: Pareto-based algorithms, Journal of Cleaner Production, 333 (2022), 130056. 

[46]

N. N. WangZ. P. Fan and X. Chen, Effect of fairness on channel choice of the mobile phone supply chain, International Transactions in Operational Research, 28 (2019), 2110-2138.  doi: 10.1111/itor.12660.

[47]

Y. Y. WangZ. Q. Yu and L. Shen, Study on the decision-making and coordination of an e-commerce supply chain with manufacturer fairness concerns, International Journal of Production Research, 57 (2019), 2788-2808. 

[48]

Y. Y. WangR. J. FanL. Shen and M. Z. Jin, Decisions and coordination of green e-commerce supply chain considering green manufacturer's fairness concerns, International Journal of Production Research, 58 (2020), 7471-7489. 

[49]

Y. Y. WangM. SuL. Shen and R. Y. Tang, Decision-making of closed-loop supply chain under corporate social responsibility and fairness concerns, Journal of Cleaner Production, 284 (2021), 125373. 

[50]

B. WuP. F. Liu and X. F. Xu, An evolutionary analysis of low-carbon strategies based on the government-enterprise game in the complex network context, Journal of Cleaner Production, 141 (2017), 168-179. 

[51]

T. Xiao and G. H. Chen, Wholesale pricing and evolutionarily stable strategies of retailers with imperfectly observable objective, European Journal of Operational Research, 196 (2009), 1190-1201.  doi: 10.1016/j.ejor.2008.04.009.

[52]

Y. Y. Yi and H. S. Yang, Wholesale pricing and evolutionary stable strategies of retailers under network externality, European Journal of Operational Research, 259 (2017), 37-47.  doi: 10.1016/j.ejor.2016.09.014.

[53]

R. Yoshihara and N. Matsubayashi, Channel coordination between manufacturer and competing retailers with fairness concerns, European Journal of Operational Research, 290 (2021), 546-555.  doi: 10.1016/j.ejor.2020.08.023.

[54]

H. S. YuA. Z. Zeng and L. D. Zhao, Analyzing the evolutionary stability of the vendor-managed inventory supply chains, Computers and industrial engineering, 56 (2009), 274-282. 

[55]

L. H. ZhangH. ZhouY. Y. Liu and R. Lu, Optimal environmental quality and price with consumer environmental awareness and retailer's fairness concerns in supply chain, Journal of Cleaner Production, 213 (2019), 1063-1079.  doi: 10.1007/s10651-018-0412-8.

[56]

N. Zhang and B. Li, Pricing and coordination of green closed-loop supply chain with fairness concerns, IEEE Access, 8 (2020), 224178. 

[57]

R. R. ZhangW. M. MaH. Y. SiJ. J. Liu and L. Liao, Cooperative game analysis of coordination mechanisms under fairness concerns of a green retailer, Journal of Retailing and Consumer Services, 59 (2021), 102361. 

[58]

T. Zhang and X. C. Wang, The impact of fairness concern on the three-party supply chain coordination, Industrial Marketing Management, 73 (2018), 99-115. 

[59]

Q. ZhouQ. H. LiX. Q. HuW. Yang and W. X. Chen, Optimal contract design problem considering the retailer's fairness concern with asymmetric demand information, Journal of Cleaner Production, 287 (2021), 125407. 

[60]

H. ZouJ. Qin and B. Dai, Optimal pricing decisions for a low-carbon supply chain considering fairness concern under carbon quota policy, International Journal of Environmental Research and Public Health, 18 (2021), 556. 

show all references

References:
[1]

M. Alinaghian and A. Goli, Allocation and routing of temporary health centers in rural areas in crisis, solved by improved harmony search algorithm, International Journal of Computational Intelligence Systems, 10 (2017), 894-913. 

[2]

J. J. Angel and D. McCabe, Fairness in financial markets: The case of high frequency trading, Journal of Business Ethics, 112 (2013), 585-595. 

[3]

P. AssarzadeganS. R. Hejazi and G. A. Raissi, An evolutionary game theoretic model for analyzing retailers' behavior when introducing economy and premium private labels, Journal of Retailing and Consumer Services, 57 (2020), 102227. 

[4]

S. BarariG. AgarwalW. J. ZhangB. Mahanty and M. K. Tiwari, A decision framework for the analysis of green supply chain contracts: An evolutionary game approach, Expert Systems with Applications, 39 (2012), 2965-2976. 

[5]

S. Choi and P. R. Messinger, The role of fairness in competitive supply chain relationships: An experimental study, European Journal of Operational Research, 251 (2016), 798-813. 

[6]

T. H. CuiJ. S. Raju and Z. J. Zhang, Fairness and channel coordination, Management Science, 53 (2007), 1303-1314. 

[7]

G. B. DahlK. V. Løken and M. Mogstad, Peer effects in program participation, American Economic Review, 104 (2014), 2049-2074. 

[8]

S. F. DuL. WeiY. G. Zhu and T. F. Nie, Peer-regarding fairness in supply chain, International Journal of Production Research, 56 (2018), 3384-3396. 

[9]

E. Fehr and and K. M. Schmidt, A theory of fairness, competition and cooperation, Quarterly Journal of Economics, 114 (1999), 817-868. 

[10]

D. Friedman, On economic applications of evolutionary game theory, Journal of Evolutionary Economics, 8 (1998), 15-54. 

[11]

A. Goli, and T. Keshavarz, Just-in-time scheduling in identical parallel machine sequence-dependent group scheduling problem, Journal of Industrial and Management Optimization, 17 (2021).http://dx.doi.org/10.3934/jimo.2021124. doi: 10.3934/jimo.2021124.

[12]

A. GoliH. Khademi-ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers and Industrial Engineering, 137 (2019), 106090. 

[13]

A. Goli, H. Khademi-Zare, R. Tavakkoli-Moghaddam, et al., An integrated approach based on artificial intelligence and novel meta-heuristic algorithms to predict demand for dairy products: A case study, Network: Computation in Neural Systems, 32 (2021), 1-35.

[14]

A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, International Journal of Intelligent Transportation Systems Research, 18 (2018), 140-152. 

[15]

A. Goli and H. Mohammadi, Developing a sustainable operational management system using hybrid Shapley value and Multimoora method: Case study petrochemical supply chain, Environment, Development and Sustainability, 23 (2021), 1-30. 

[16]

T. H. Ho and X. M. Su, Peer-induced fairness in games, American Economic Review, 99 (2009), 2022-2049. 

[17]

T. H. HoX. M. Su and Y. Z. Wu, Distributional and peer-induced fairness in supply chain contract design, Production and Operations Management, 23 (2014), 161-175. 

[18]

T. H. Ho and J. J. Zhang, Designing pricing contracts for boundedly rational customers: Does the framing of the fixed fee matter?, Management Science, 54 (2008), 686-700. 

[19]

P. JiX. Ma and G. Li, Developing green purchasing relationships for the manufacturing industry: An evolutionary game theory perspective, International Journal of Production Economics, 166 (2015), 155-162. 

[20]

D. A. KahnemanJ. L. Knetsch and R. H. Thaler, The journal of business, International Journal of Production Economics, 59 (1986), 285-300. 

[21]

E. KatokT. L. Olsen and V. Pavlov, Wholesale pricing under mild and privately known concerns for fairness, Production and Operations Management, 23 (2014), 285-302. 

[22]

E. Katok and V. Pavlov, Fairness in supply chain contracts: A laboratory study, Journal of Operations Management, 31 (2013), 129-137. 

[23]

K. T. KimJ. S. Lee and S. Y. Lee, The effects of supply chain fairness and the buyer's power sources on the innovation performance of the supplier: A mediating role of social capital accumulation, Journal of Business Industrial Marketing, 31 (2017), 987-997. 

[24]

B. LiP. Hou and Q. Li, Cooperative advertising in a dual-channel supply chain with a fairness concern of the manufacturer, IMA Journal of Management Mathematics, 28 (2017), 259-277.  doi: 10.1093/imaman/dpv025.

[25]

C. F. LiX. Q. Guo and D. L. Du, Pricing decisions in dual-channel closed-loop supply chain under retailer's risk aversion and fairness concerns, Journal of the Operations Research Society of China, 9 (2021), 641-657.  doi: 10.1007/s40305-020-00324-7.

[26]

P. Lii and F. I. Kuo, Innovation-oriented supply chain integration for combined competitiveness and ?rm performance, International Journal of Production Economics, 174 (2016), 142-155. 

[27]

K. J. Li and S. Jain, Behavior-based pricing: An analysis of the impact of peer-induced fairness, Management Science, 62 (2016), 2705-2721. 

[28]

X. LiX. Q. CuiY. J. LiD. Q. Xu and F. C. Xu, Optimisation of reverse supply chain with used-product collection effort under collector's fairness concerns, International Journal of Production Research, 59 (2019), 652-663. 

[29]

T. T. LiJ. X. XieX. B. Zhao and J. F. Tang, On supplier encroachment with retailer's fairness concerns, Computers and Industrial Engineering, 98 (2016), 499-512. 

[30]

W. H. LiuD. WangX. R. ShenX. Y. Yan and W. Y. Wei, The impacts of distributional and peer-induced fairness concerns on the decision-making of order allocation in logistics service supply chain, Transportation Research Part E: Logistics and Transportation Review, 116 (2018), 102-122. 

[31]

C. H. Loch and Y. Z. Wu, Social preferences and supply chain performance: An experimental study, Management Science, 54 (2008), 1835-1849. 

[32]

R. LotfiB. KargarS. H. HoseiniS. NazariS. Safavi and G. W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, International Journal of Energy Research, 45 (2021), 17749-17766. 

[33]

R. Lotfi, B. Kargar, A. Gharehbaghi and G. W. Weber, Viable medical waste chain network design by considering risk and robustness, Environmental Science and Pollution Research, (2021), 1–16.

[34]

R. Lotfi, S. Safavi, A. Gharehbaghi, S. G. Zare, R. Hazrati and G. W. Weber, Viable supply chain network design by considering blockchain technology and cryptocurrency, Mathematical Problems in Engineering, (2021).http://dx.doi.org/10.1155/2021/7347389. doi: RHazrati2021.

[35]

R. LotfiG. W. WeberS. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial Management Optimization, 16 (2020), 117-140.  doi: 10.3934/jimo.2018143.

[36]

B. Nguyen and P. P. Klaus, Retail fairness: Exploring consumer perceptions of fairness towards retailers' marketing tactics, Journal of Retailing and Consumer Services, 20 (2013), 311-324. 

[37]

T. F. Nie and S. F. Du, Dual-fairness supply chain with quantity discount contracts, European Journal of Operational Research, 258 (2017), 491-500.  doi: 10.1016/j.ejor.2016.08.051.

[38]

B. Z. NiuQ. Q. Cui and J. Zhang, Impact of channel power and fairness concern on supplier's market entry decision, Journal of the Operational Research Society, 68 (2017), 1570-1581. 

[39]

K. W. PanZ. B. CuiA. X. Xing and Q. H. Lu, Impact of fairness concern on retailer-dominated supply chain, Computers and Industrial Engineering, 139 (2020), 106209. 

[40]

F. QinF. MaiM. J. Fry and A. S. Raturi, Supply-chain performance anomalies: Fairness concerns under private cost information, European Journal of Operational Research, 252 (2016), 170-182. 

[41]

Y. H. QinG. X. Wei and J. X. Dong, The signaling game model under asymmetric fairness-concern information, European Journal of Operational Research, 22 (2019), 5547-5562. 

[42]

A. Sharma, Game-Theoretic analysis of pricing models in a Dyadic supply chain with fairness concerns, International Journal of Strategic Decision Sciences, 10 (2019), 1-24.  doi: 10.1142/S0219198920500176.

[43]

Y. C. Shi and J. A. Zhu, Game-theoretic analysis for supply chain with distributional and peer-induced fairness concerned retailers, Management Science and Engineering, 8 (2014), 78-84. 

[44]

J. TaoL. S. ShaoZ. M. GuanW. Ho and S. Talluri, Incorporating risk aversion and fairness considerations into procurement and distribution decisions in a supply chain, International Journal of Production Research, 58 (2020), 1950-1967.  doi: 10.1016/j.cor.2020.105105.

[45]

E. B. TirkolaeeA. GoliP. Ghasemi and F. Goodarzian, Designing a sustainable closed-loop supply chain network of face masks during the COVID-19 pandemic: Pareto-based algorithms, Journal of Cleaner Production, 333 (2022), 130056. 

[46]

N. N. WangZ. P. Fan and X. Chen, Effect of fairness on channel choice of the mobile phone supply chain, International Transactions in Operational Research, 28 (2019), 2110-2138.  doi: 10.1111/itor.12660.

[47]

Y. Y. WangZ. Q. Yu and L. Shen, Study on the decision-making and coordination of an e-commerce supply chain with manufacturer fairness concerns, International Journal of Production Research, 57 (2019), 2788-2808. 

[48]

Y. Y. WangR. J. FanL. Shen and M. Z. Jin, Decisions and coordination of green e-commerce supply chain considering green manufacturer's fairness concerns, International Journal of Production Research, 58 (2020), 7471-7489. 

[49]

Y. Y. WangM. SuL. Shen and R. Y. Tang, Decision-making of closed-loop supply chain under corporate social responsibility and fairness concerns, Journal of Cleaner Production, 284 (2021), 125373. 

[50]

B. WuP. F. Liu and X. F. Xu, An evolutionary analysis of low-carbon strategies based on the government-enterprise game in the complex network context, Journal of Cleaner Production, 141 (2017), 168-179. 

[51]

T. Xiao and G. H. Chen, Wholesale pricing and evolutionarily stable strategies of retailers with imperfectly observable objective, European Journal of Operational Research, 196 (2009), 1190-1201.  doi: 10.1016/j.ejor.2008.04.009.

[52]

Y. Y. Yi and H. S. Yang, Wholesale pricing and evolutionary stable strategies of retailers under network externality, European Journal of Operational Research, 259 (2017), 37-47.  doi: 10.1016/j.ejor.2016.09.014.

[53]

R. Yoshihara and N. Matsubayashi, Channel coordination between manufacturer and competing retailers with fairness concerns, European Journal of Operational Research, 290 (2021), 546-555.  doi: 10.1016/j.ejor.2020.08.023.

[54]

H. S. YuA. Z. Zeng and L. D. Zhao, Analyzing the evolutionary stability of the vendor-managed inventory supply chains, Computers and industrial engineering, 56 (2009), 274-282. 

[55]

L. H. ZhangH. ZhouY. Y. Liu and R. Lu, Optimal environmental quality and price with consumer environmental awareness and retailer's fairness concerns in supply chain, Journal of Cleaner Production, 213 (2019), 1063-1079.  doi: 10.1007/s10651-018-0412-8.

[56]

N. Zhang and B. Li, Pricing and coordination of green closed-loop supply chain with fairness concerns, IEEE Access, 8 (2020), 224178. 

[57]

R. R. ZhangW. M. MaH. Y. SiJ. J. Liu and L. Liao, Cooperative game analysis of coordination mechanisms under fairness concerns of a green retailer, Journal of Retailing and Consumer Services, 59 (2021), 102361. 

[58]

T. Zhang and X. C. Wang, The impact of fairness concern on the three-party supply chain coordination, Industrial Marketing Management, 73 (2018), 99-115. 

[59]

Q. ZhouQ. H. LiX. Q. HuW. Yang and W. X. Chen, Optimal contract design problem considering the retailer's fairness concern with asymmetric demand information, Journal of Cleaner Production, 287 (2021), 125407. 

[60]

H. ZouJ. Qin and B. Dai, Optimal pricing decisions for a low-carbon supply chain considering fairness concern under carbon quota policy, International Journal of Environmental Research and Public Health, 18 (2021), 556. 

Figure 1.  SC Structure diagram with fairness concerns implanted
Figure 2.  Tripartite Stackelberg game
Figure 3.  The relation of $ \pi_S^{l*} $ to $ \lambda $
Figure 4.  The relation of $ \pi_S^{l*} $ to $ \theta $
Figure 5.  The relation of $ \pi _{{R_1}}^{l*} $ to $ \lambda $
Figure 6.  The relation of $ \pi _{{R_2}}^{l*} $ to $ \lambda $
Figure 7.  The relation of $ \pi _{{R_2}}^{l*} $ to $ \theta $
Figure 8.  The relation of $ \ u_{{R_i}}^{l*} $ to $ \lambda $
Table 1.  The literature positioning of this paper
Reference Number of participants Consider multiple fairness concerns Leader "actively" or "fairness" considers the fairness Stability analysis of equilibrium concerns of followers points
Assarzadegan et al.[3] 2 $ \times $ Passively $ \surd $
Ho et al. [17] 3 $ \surd $ Passively $ \times $
Li et al. [29] 2 $ \times $ Passively $ \times $
Liu et al. [30] 3 $ \surd $ Passively $ \times $
Nie and Du[37] 3 $ \surd $ Passively $ \times $
Sharma[42] 2 $ \times $ Passively $ \times $
Shi and Zhu[43] 3 $ \surd $ Passively $ \times $
Yoshihara et al.[53] 3 $ \times $ Passively $ \times $
Zhang et al. [55] 2 $ \times $ Passively $ \times $
Zhang et al. [57] 3 $ \times $ Passively $ \times $
Zhang and Wang[58] 3 $ \surd $ Passively $ \times $
This paper 3 $ \surd $ Passively and actively $ \surd $
Reference Number of participants Consider multiple fairness concerns Leader "actively" or "fairness" considers the fairness Stability analysis of equilibrium concerns of followers points
Assarzadegan et al.[3] 2 $ \times $ Passively $ \surd $
Ho et al. [17] 3 $ \surd $ Passively $ \times $
Li et al. [29] 2 $ \times $ Passively $ \times $
Liu et al. [30] 3 $ \surd $ Passively $ \times $
Nie and Du[37] 3 $ \surd $ Passively $ \times $
Sharma[42] 2 $ \times $ Passively $ \times $
Shi and Zhu[43] 3 $ \surd $ Passively $ \times $
Yoshihara et al.[53] 3 $ \times $ Passively $ \times $
Zhang et al. [55] 2 $ \times $ Passively $ \times $
Zhang et al. [57] 3 $ \times $ Passively $ \times $
Zhang and Wang[58] 3 $ \surd $ Passively $ \times $
This paper 3 $ \surd $ Passively and actively $ \surd $
Table 2.  Notations description
Parameter Meaning
$ S $ Supplier
$ R_{i} $ Retailer $ i $, $ i=1, 2 $
$ D_{i} $ Demand function of the retailer $ R_{i} $, $ D=D_{1}+D_{2} $
$ Superscript $ "$ \ast $" The optimal solution
$ \lambda_{R_{i}} $ Distributional fairness concerns coefficient of the retailer $ R_{i} $, let $ \lambda_{R_{1}}=\lambda_{R_{2}} $, and $ 0\leq \lambda \leq 1 $
$ \theta $ Peer-induced fairness concerns coefficient, and $ 0\leq \theta \leq 1 $
Decision variables
$ p_{i} $ Selling price, and it is the retailer $ R_{i}^{'} $s decision variable
$ \omega_{i} $ Wholesale price, and it is the supplier's decision variable
Performance
$ \pi_{S}^{l} $ The profit of the supplier under model $ l $,
$ l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\} $
$ \pi_{R_{i}}^{l} $ The profit of the retailer $ \pi_{R_{i}} $ under model $ l $,
$ l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\} $
$ \pi_{SJ} $ The supplier's profit in the $ j $-th round of the game, $ j=1, 2 $
$ \pi_{J} $ The profit of the SC in the $ j $-th round of the game, $ j=1, 2 $
$ \bar{\pi}_{R_{i}} $ Nash bargaining solution of the retailer $ R_{i} $
$ \bar{\pi}_{S} $ Nash bargaining solution of the supplier
$ u_{R_{i}}^{l} $ The utility of the retailer $ R_{i} $ under model $ l $,
$ l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\} $
Parameter Meaning
$ S $ Supplier
$ R_{i} $ Retailer $ i $, $ i=1, 2 $
$ D_{i} $ Demand function of the retailer $ R_{i} $, $ D=D_{1}+D_{2} $
$ Superscript $ "$ \ast $" The optimal solution
$ \lambda_{R_{i}} $ Distributional fairness concerns coefficient of the retailer $ R_{i} $, let $ \lambda_{R_{1}}=\lambda_{R_{2}} $, and $ 0\leq \lambda \leq 1 $
$ \theta $ Peer-induced fairness concerns coefficient, and $ 0\leq \theta \leq 1 $
Decision variables
$ p_{i} $ Selling price, and it is the retailer $ R_{i}^{'} $s decision variable
$ \omega_{i} $ Wholesale price, and it is the supplier's decision variable
Performance
$ \pi_{S}^{l} $ The profit of the supplier under model $ l $,
$ l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\} $
$ \pi_{R_{i}}^{l} $ The profit of the retailer $ \pi_{R_{i}} $ under model $ l $,
$ l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\} $
$ \pi_{SJ} $ The supplier's profit in the $ j $-th round of the game, $ j=1, 2 $
$ \pi_{J} $ The profit of the SC in the $ j $-th round of the game, $ j=1, 2 $
$ \bar{\pi}_{R_{i}} $ Nash bargaining solution of the retailer $ R_{i} $
$ \bar{\pi}_{S} $ Nash bargaining solution of the supplier
$ u_{R_{i}}^{l} $ The utility of the retailer $ R_{i} $ under model $ l $,
$ l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\} $
Table 3.  The relation between the supplier's profits and distributional fairness concerns
$ \frac{\partial \pi_{S}^{CCC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{CCN\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{CNC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{CNN\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NCC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NCN\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NNC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NNN\ast}}{\partial \lambda} $
$ - $ $ - $ $ - $ $ - $ $ - $ $ - $ $ - $ $ /$
$ \frac{\partial \pi_{S}^{CCC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{CCN\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{CNC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{CNN\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NCC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NCN\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NNC\ast}}{\partial \lambda} $ $ \frac{\partial \pi_{S}^{NNN\ast}}{\partial \lambda} $
$ - $ $ - $ $ - $ $ - $ $ - $ $ - $ $ - $ $ /$
Table 4.  The relation between the supplier's profits and peer-induced fairness concerns
$ \frac{{\partial\pi_S^{CCC*}}}{{\partial\theta}} $ $ \frac{\partial \pi_{S}^{CCN\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{CNC\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{CNN\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NCC\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NCN\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NNC\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NNN\ast}}{\partial \theta} $
$ + $ $ + $ $ + $ $ + $ $ + $ $ /$ $ + $ $ /$
$ \frac{{\partial\pi_S^{CCC*}}}{{\partial\theta}} $ $ \frac{\partial \pi_{S}^{CCN\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{CNC\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{CNN\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NCC\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NCN\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NNC\ast}}{\partial \theta} $ $ \frac{\partial \pi_{S}^{NNN\ast}}{\partial \theta} $
$ + $ $ + $ $ + $ $ + $ $ + $ $ /$ $ + $ $ /$
Table 5.  The relation between the retailer $ {R_1} $'s profits and distributional fairness concerns
$ \frac{{\partial \pi _{{R_1}}^{CCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{CCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{CNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{CNN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NNN*}}}{{\partial \lambda }} $
$ + $ $ + $ $ + $ $ + $ $ - $ $ - $ $ /$ $ /$
$ \frac{{\partial \pi _{{R_1}}^{CCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{CCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{CNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{CNN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_1}}^{NNN*}}}{{\partial \lambda }} $
$ + $ $ + $ $ + $ $ + $ $ - $ $ - $ $ /$ $ /$
Table 6.  The relation between the retailer $ {R_2} $'s profits and distributional fairness concerns
$ \frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \lambda }} $
$ + $ $ + $ $ + $ $ + $ $ - $ $ /$ $ - $ $ /$
$ \frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \lambda }} $ $ \frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \lambda }} $
$ + $ $ + $ $ + $ $ + $ $ - $ $ /$ $ - $ $ /$
Table 7.  The relation between the retailer $ {R_2} $'s profits and peer-induced fairness concerns
$ \frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \theta}} $ $ \frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \theta }} $
$ + $ $ + $ $ + $ $ + $ $ - $ $ /$ $ - $ $ /$
$ \frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \theta }} $ $ \frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \theta}} $ $ \frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \theta }} $
$ + $ $ + $ $ + $ $ + $ $ - $ $ /$ $ - $ $ /$
Table 8.  Equilibrium solutions under different strategy combination
$ w_1^{CCC*} $ $ w_1^{CCN*} $ $ w_1^{CNC*} $ $ w_1^{CNN*} $ $ w_1^{NCC*} $ $ w_1^{NCN*} $ $ w_1^{NNC*} $ $ w_1^{NNN*} $
$ 416.7 $ $ 416.7 $ $ 416.7 $ $ 416.7 $ $ 500 $ $ 500 $ $ 500 $ $ 500 $
$ w_2^{CCC*} $ $ w_2^{CCN*} $ $ w_2^{CNC*} $ $ w_2^{CNN*} $ $ w_2^{NCC*} $ $ w_2^{NCN*} $ $ w_2^{NNC*} $ $ w_2^{NNN*} $
$ 437.5 $ $ 437.5 $ $ 437.5 $ $ 437.5 $ $ 500 $ $ 500 $ $ 500 $ $ 500 $
$ p_1^{CCC*} $ $ p_1^{CCN*} $ $ p_1^{CNC*} $ $ p_1^{CNN*} $ $ p_1^{NCC*} $ $ p_1^{NCN*} $ $ p_1^{NNC*} $ $ p_1^{NNN*} $
$ 750 $ $ 750 $ $ 708.3 $ $ 708.3 $ $ 800 $ $ 800 $ $ 750 $ $ 750 $
$ p_2^{CCC*} $ $ p_2^{CCN*} $ $ p_2^{CNC*} $ $ p_2^{CNN*} $ $ p_2^{NCC*} $ $ p_2^{NCN*} $ $ p_2^{NNC*} $ $ p_2^{NNN*} $
$ 750 $ $ 718.7 $ $ 750 $ $ 718.7 $ $ 785.7 $ $ 750 $ $ 785.7 $ $ 750 $
$ w_1^{CCC*} $ $ w_1^{CCN*} $ $ w_1^{CNC*} $ $ w_1^{CNN*} $ $ w_1^{NCC*} $ $ w_1^{NCN*} $ $ w_1^{NNC*} $ $ w_1^{NNN*} $
$ 416.7 $ $ 416.7 $ $ 416.7 $ $ 416.7 $ $ 500 $ $ 500 $ $ 500 $ $ 500 $
$ w_2^{CCC*} $ $ w_2^{CCN*} $ $ w_2^{CNC*} $ $ w_2^{CNN*} $ $ w_2^{NCC*} $ $ w_2^{NCN*} $ $ w_2^{NNC*} $ $ w_2^{NNN*} $
$ 437.5 $ $ 437.5 $ $ 437.5 $ $ 437.5 $ $ 500 $ $ 500 $ $ 500 $ $ 500 $
$ p_1^{CCC*} $ $ p_1^{CCN*} $ $ p_1^{CNC*} $ $ p_1^{CNN*} $ $ p_1^{NCC*} $ $ p_1^{NCN*} $ $ p_1^{NNC*} $ $ p_1^{NNN*} $
$ 750 $ $ 750 $ $ 708.3 $ $ 708.3 $ $ 800 $ $ 800 $ $ 750 $ $ 750 $
$ p_2^{CCC*} $ $ p_2^{CCN*} $ $ p_2^{CNC*} $ $ p_2^{CNN*} $ $ p_2^{NCC*} $ $ p_2^{NCN*} $ $ p_2^{NNC*} $ $ p_2^{NNN*} $
$ 750 $ $ 718.7 $ $ 750 $ $ 718.7 $ $ 785.7 $ $ 750 $ $ 785.7 $ $ 750 $
Table 9.  Tripartite game benefit matrix
Supplier
Supplier $ R_{1} $ Concerns ($ y $) No concerns ($ 1-y $) Retailer $ R_{2} $
Consideration ($ x $) $ A, I, q $ $ b, T, r $ Concerns ($ z $)
No consideration ($ 1-x $) $ c, k, s $ $ d, I, B $
Consideration ($ x $) $ c, m, u $ $ g, n, v $ (No concerns $ 1-z $)
No consideration ($ 1-x $) $ f, o, W $ $ h, p, \varphi $
Supplier
Supplier $ R_{1} $ Concerns ($ y $) No concerns ($ 1-y $) Retailer $ R_{2} $
Consideration ($ x $) $ A, I, q $ $ b, T, r $ Concerns ($ z $)
No consideration ($ 1-x $) $ c, k, s $ $ d, I, B $
Consideration ($ x $) $ c, m, u $ $ g, n, v $ (No concerns $ 1-z $)
No consideration ($ 1-x $) $ f, o, W $ $ h, p, \varphi $
Table 10.  ESS stability analysis
Equilibrium Points Eigenvalues Asymptotic Stability
$ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{3} $
$ E_{1}(0, 0, 0) $ $ \eta_{1} $ $ \eta_{1} $ $ \eta_{1} $ ESS
$ - $ $ - $ $ - $
$ E_{2}(1, 0, 0) $ $ -\eta_{1} $ $ \alpha_{2}+\eta_{2} $ $ \alpha_{3}+\eta_{3} $ Saddle-point
$ + $ $ - $ $ - $
$ E_{3}(1, 1, 1) $ $ -(\alpha_{1}+\theta_{1}+\eta_{1}) $ $ -(\alpha_{2}+\eta_{2}) $ $ -(\alpha_{3}+\eta_{3}) $ Saddle-point
$ - $ $ + $ $ + $
$ E_{4}(0, 1, 0) $ $ \alpha_{1}+\eta_{1} $ $ -\eta_{2} $ $ \eta_{3} $ Saddle-point
$ + $ $ + $ $ - $
$ E_{5}(0, 0, 1) $ $ \theta_{1}+\eta_{1} $ $ \eta_{2} $ $ -\eta_{3} $ Saddle-point
$ Uncertain $ $ - $ $ + $
$ E_{6}(1, 1, 0) $ $ -(\alpha_{1}+\eta_{1}) $ $ -(\alpha_{2}+\eta_{2}) $ $ \alpha_{3}+\eta_{3} $ Saddle-point
$ - $ $ + $ $ - $
$ E_{7}(1, 0, 1) $ $ -(\theta_{1}+\eta_{1}) $ $ \alpha_{2}+\eta_{2} $ $ -(\alpha_{3}+\eta_{3}) $ Saddle-point
$ Uncertain $ $ - $ $ + $
$ E_{8}(0, 1, 1) $ $ \alpha_{1}+\theta_{1}+\eta_{1} $ $ -\eta_{2} $ $ -\eta_{3} $ Saddle-point
$ + $ $ + $ $ + $
Equilibrium Points Eigenvalues Asymptotic Stability
$ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{3} $
$ E_{1}(0, 0, 0) $ $ \eta_{1} $ $ \eta_{1} $ $ \eta_{1} $ ESS
$ - $ $ - $ $ - $
$ E_{2}(1, 0, 0) $ $ -\eta_{1} $ $ \alpha_{2}+\eta_{2} $ $ \alpha_{3}+\eta_{3} $ Saddle-point
$ + $ $ - $ $ - $
$ E_{3}(1, 1, 1) $ $ -(\alpha_{1}+\theta_{1}+\eta_{1}) $ $ -(\alpha_{2}+\eta_{2}) $ $ -(\alpha_{3}+\eta_{3}) $ Saddle-point
$ - $ $ + $ $ + $
$ E_{4}(0, 1, 0) $ $ \alpha_{1}+\eta_{1} $ $ -\eta_{2} $ $ \eta_{3} $ Saddle-point
$ + $ $ + $ $ - $
$ E_{5}(0, 0, 1) $ $ \theta_{1}+\eta_{1} $ $ \eta_{2} $ $ -\eta_{3} $ Saddle-point
$ Uncertain $ $ - $ $ + $
$ E_{6}(1, 1, 0) $ $ -(\alpha_{1}+\eta_{1}) $ $ -(\alpha_{2}+\eta_{2}) $ $ \alpha_{3}+\eta_{3} $ Saddle-point
$ - $ $ + $ $ - $
$ E_{7}(1, 0, 1) $ $ -(\theta_{1}+\eta_{1}) $ $ \alpha_{2}+\eta_{2} $ $ -(\alpha_{3}+\eta_{3}) $ Saddle-point
$ Uncertain $ $ - $ $ + $
$ E_{8}(0, 1, 1) $ $ \alpha_{1}+\theta_{1}+\eta_{1} $ $ -\eta_{2} $ $ -\eta_{3} $ Saddle-point
$ + $ $ + $ $ + $
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