doi: 10.3934/jimo.2022102
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Producing two substitutable products under a supply chain including two manufacturers and one retailer: A game-theoretic approach

1. 

Assistant Professor, Industrial Engineering Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan 87717-67498, Iran

2. 

Assistant Professor, Department of Industrial Engineering, Quchan University of Technology, Quchan, Iran

*Corresponding author: Hamed Jafari

Received  January 2022 Revised  April 2022 Early access June 2022

In the current study, a supply chain structure is considered consisting of duopolistic manufacturers and monopolistic retailer. In the considered structure, the manufacturers product two substitutable products and deliver them to end customers. One of the manufacturers sells his/her products only through the common retailer whereas another one sells directly to customers as well as through the common retailer. Obviously, a competitive behavior is established in order to price the products under the considered supply chain. In this point of view, a game-theoretic approach is applied including Nash, Manufacturers-Stackelberg, and Retailer-Stackelberg games to analyze the pricing decisions. Then, the equilibrium strategies obtained from the investigated games are discussed and some insights are presented. The results indicate that Manufacturers-Stackelberg and Nash games respectively lead to the highest and the lowest profits for the manufacturers and the whole system. Furthermore, the highest profit for the common retailer is obtained from Retailer-Stackelberg game.

Citation: Hamed Jafari, Soroush Safarzadeh. Producing two substitutable products under a supply chain including two manufacturers and one retailer: A game-theoretic approach. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022102
References:
[1]

X. AiJ. ChenH. Zhao and X. Tang, Competition among supply chains: Implications of full returns policy, Internat. J. Prod. Econ., 139 (2012), 257-265.  doi: 10.1016/j.ijpe.2012.04.018.

[2]

C. Amountzias, Pricing decisions and competitive conduct across manufacturing sectors: Evidence from 19 European Union manufacturing industries, J. Industry Competition Trade, 19 (2019), 413-440.  doi: 10.1007/s10842-019-00293-5.

[3]

J. Asirvatham and S. Bhuyan, Incentives and impacts of vertical coordination in a food production-marketing chain: A non-cooperative multi-stage, multi-player analysis, J. Industry Competition Trade, 18 (2018), 59-95.  doi: 10.1007/s10842-017-0247-2.

[4]

Y.-H. ChenX.-W. Wen and M.-Z. Luo, Corporate social responsibility spillover and competition effects on the food industry, Aus. Econ. Papers, 55 (2016), 1-13.  doi: 10.1111/1467-8454.12058.

[5]

M. Christopher, Logistics and Supply Chain Management,

[6]

S. Delpachitra, Price rigidity in the downstream petroleum industry in New Zealand: Where does it happen?, Energy Econ., 24 (2002), 597-613.  doi: 10.1016/S0140-9883(02)00021-X.

[7]

S. Delpachitra and D. Beal, Petrol prices disparity: Did the removal of price surveillance create price competition?, Econ. Papers, 21 (2002), 56-65.  doi: 10.1111/j.1759-3441.2002.tb00310.x.

[8]

N. C. P. EdirisingheB. Bichescu and X. Shi, Equilibrium analysis of supply chain structures under power imbalance, European J. Oper. Res., 214 (2011), 568-578.  doi: 10.1016/j.ejor.2011.05.008.

[9]

A. E. E. EltoukhyZ. X. WangF. T. S. Chan and S. H. Chung, Joint optimization using a leader-follower Stackelberg game for coordinated configuration of stochastic operational aircraft maintenance routing and maintenance staffing, Comp. Indust. Engrg., 125 (2018), 46-68.  doi: 10.1016/j.cie.2018.08.012.

[10]

H. FangL. Xu and X. Wang, Coordinated multiple-relays based physical-layer security improvement: A single-leader multiple-followers Stackelberg game scheme, IEEE Trans. Info. Forensics Security, 13 (2018), 197-209.  doi: 10.1109/TIFS.2017.2746001.

[11]

H. Jafari, Investigating environmental and economic aspects of sustainability by recycling PET plastic bottles: A game-theoretic approach, Clean Tech. Environ. Policy, 24 (2022), 829-842.  doi: 10.1007/s10098-021-02216-7.

[12]

H. Jafari, Nurse scheduling problem by considering total number of required nurses as well as nurses' preferences for working shifts: An algorithmic game-theoretic approach, in press, Scientia Iranica.

[13]

H. Jafari, Sustainable development by reusing of recyclables in a textile industry including two collectors and three firms: A game-theoretic approach for pricing decisions, J. Cleaner Prod., 229 (2019), 598-610.  doi: 10.1016/j.jclepro.2019.04.222.

[14]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Game theoretical approach to price a product under two-echelon supply chain containing e-tail selling channel, Internat. J. Services Oper. Mgmt., 36 (2020), 131-160.  doi: 10.1504/IJSOM.2020.10028974.

[15]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Pricing decisions in dual-channel supply chain including monopolistic manufacturer and duopolistic retailers: A game-theoretic approach, J. Industry Competition Trade, 16 (2016), 323-343.  doi: 10.1007/s10842-016-0224-1.

[16]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Pricing decisions in dual-channel supply chain with one manufacturer and multiple retailers: A game-theoretic approach, RAIRO Oper. Res., 51 (2017), 1269-1287.  doi: 10.1051/ro/2017003.

[17]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Sustainable development by waste recycling under a three-echelon supply chain: A game-theoretic approach, J. Cleaner Prod., 142 (2017), 2252-2261.  doi: 10.1016/j.jclepro.2016.11.051.

[18]

H. Jafari, S. Safarzadeh and E. Azad-Farsani, Effects of governmental policies on energy-efficiency improvement of hydrogen fuel cell cars: A game-theoretic approach, Energy, 254 (2022). doi: 10.1016/j. energy. 2022.124394.

[19]

C. K. JaggiM. GuptaA. Kausar and S. Tiwari, Inventory and credit decisions for deteriorating items with displayed stock dependent demand in two-echelon supply chain using Stackelberg and Nash equilibrium solution, Ann. Oper. Res., 274 (2019), 309-329.  doi: 10.1007/s10479-018-2925-9.

[20]

J. Jiang and X. Liu, Multi-objective Stackelberg game model for water supply networks against interdictions with incomplete information, European J. Oper. Res., 266 (2018), 920-933.  doi: 10.1016/j.ejor.2017.10.034.

[21]

D. Kesavayuth and V. Zikos, Price or quantity? A new irrelevance result for mixed markets, Aus. Econ. Papers, 55 (2016), 29-42.  doi: 10.1111/1467-8454.12059.

[22]

B. X. LiY.-W. ZhouJ.-Z. Li and S.-P. Zhou, Contract choice game of supply chain competition at both manufacturer and retailer levels, Internat. J. Prod. Econ., 143 (2013), 188-197.  doi: 10.1016/j.ijpe.2013.01.007.

[23]

J. LiS. Wang and T. C. E. Cheng, Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption, Internat. J. Prod. Econ., 124 (2010), 137-150.  doi: 10.1016/j.ijpe.2009.10.017.

[24]

P.-C. Liao, Strategic delegation under unionised duopoly: Who will bargain with unions?, Aus. Econ. Papers, 49 (2010), 276-288.  doi: 10.1111/j.1467-8454.2010.00402.x.

[25]

Y. LiuH. XuS.-J. S. Yang and J. Zhang, Distributionally robust equilibrium for continuous games: Nash and Stackelberg models, European J. Oper. Res., 265 (2018), 631-643.  doi: 10.1016/j.ejor.2017.07.050.

[26]

J.-C. LuY.-C. Tsao and C. Charoensiriwath, Competition under manufacturer service and retail price, Econ. Modell., 28 (2011), 1256-1264.  doi: 10.1016/j.econmod.2011.01.008.

[27]

A. LuciaP. A. DiMaggio and D. Alonso-Martinez, Metabolic pathway analysis using a Nash equilibrium approach, J. Global Optim., 71 (2018), 537-550.  doi: 10.1007/s10898-018-0605-6.

[28]

T. Matsumura and A. Ogawa, Endogenous timing in mixed duopolies with externality, Aus. Econ. Papers, 56 (2017), 304-327.  doi: 10.1111/1467-8454.12104.

[29]

J. F. Nash Jr., Equilibrium points in $n$-person games, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.

[30]

M. A. NchakeL. Edwards and A. Sundaram, Price-setting behavior and competition in developing countries: An analysis of retail outlets in Lesotho, J. Industry Competition Trade, 18 (2018), 529-547.  doi: 10.1007/s10842-018-0275-6.

[31]

K. PanK. K. LaiS. C. H. Leung and D. Xiao, Revenue-sharing versus wholesale price mechanisms under different channel power structures, European J. Oper. Res., 203 (2010), 532-538.  doi: 10.1016/j.ejor.2009.08.010.

[32]

M. Rasti-BarzokiH. Jafari and S. R. Hejazi, Game-theoretic approach for pricing decisions in dual-channel supply chain, Internat. J. Industrial Engrg. Prod. Res., 28 (2017), 1-8.  doi: 10.22068/ijiepr.28.1.1.

[33]

S. Sinha and S. P. Sarmah, Coordination and price competition in a duopoly common retailer supply chain, Comp. Indust. Engrg., 59 (2010), 280-295.  doi: 10.1016/j.cie.2010.04.010.

[34]

H.-L. Sun and X.-J. Chen, Two-stage stochastic variational inequalities: Theory, algorithms and applications, J. Oper. Res. Soc. China, 9 (2021), 1-32.  doi: 10.1007/s40305-019-00267-8.

[35]

C. TangS. Zhu and E. Cao, Equilibrium analysis of a dual-channel supply chain under different market powers, Aus. Econ. Papers, 57 (2018), 19-34.  doi: 10.1111/1467-8454.12107.

[36]

C. WangP.-Y. Nie and Y. Meng, Duopoly competition with corporate social responsibility, Aus. Econ. Papers, 57 (2018), 327-345.  doi: 10.1111/1467-8454.12119.

[37]

F.-M. WangJ.-J. WangN. LiY.-J. Jiang and S.-C. Li, A cost-sharing scheme for the $k$-level facility location game with penalties, J. Oper. Res. Soc. China, 10 (2022), 173-182.  doi: 10.1007/s40305-021-00345-w.

[38]

J. WeiJ. Zhao and Y. Li, Pricing decisions for complementary products with firms' different market powers, European J. Oper. Res., 224 (2013), 507-519.  doi: 10.1016/j.ejor.2012.09.011.

[39]

W. WeiX. FanH. SongX. Fan and J. Yang, Imperfect information dynamic Stackelberg game based resource allocation using hidden Markov for cloud computing, IEEE Trans. Services Comput., 11 (2018), 78-89.  doi: 10.1109/TSC.2016.2528246.

[40]

C.-H. Wu, Price and service competition between new and remanufactured products in a two-echelon supply chain, Internat. J. Prod. Econ., 140 (2012), 496-507.  doi: 10.1016/j.ijpe.2012.06.034.

[41]

D. D. Wu, Bargaining in supply chain with price and promotional effort dependent demand, Math. Comput. Modelling, 58 (2013), 1659-1669.  doi: 10.1016/j.mcm.2010.12.035.

[42]

T. Xiao and T. M. Choi, Purchasing choices and channel structure strategies for a two-echelon system with risk-averse players, Internat. J. Prod. Econ., 120 (2009), 54-65.  doi: 10.1016/j.ijpe.2008.07.028.

[43]

T. Xiao and D. Yang, Price and service competition of supply chains with risk-averse retailers under demand uncertainty, Internat. J. Prod. Econ., 114 (2008), 187-200.  doi: 10.1016/j.ijpe.2008.01.006.

[44]

G. XieS. Wang and K. K. Lai, Quality improvement in competing supply chains, Internat. J. Prod. Econ., 134 (2011), 262-270.  doi: 10.1016/j.ijpe.2011.07.007.

[45]

H. XuN. Yao and S. Tong, Sourcing under cost information asymmetry when facing time-sensitive customers, Internat. J. Prod. Econ., 144 (2013), 599-609.  doi: 10.1016/j.ijpe.2013.04.023.

[46]

Z. Yang and X. Zhang, Competitive equilibria and benefit distributions of population production economies with external increasing returns, J. Oper. Res. Soc. China, 9 (2021), 723-740.  doi: 10.1007/s40305-021-00340-1.

[47]

G. Ye, Leadership and privatisation in a mixed multi-product oligopoly: An endogenous timing model, Aus. Econ. Papers, 55 (2016), 170-180.  doi: 10.1111/1467-8454.12062.

[48]

B.-Y. Zhang and S. Pei, Game theory and the evolution of cooperation, J. Oper. Res. Soc. China, 10 (2022), 379-399.  doi: 10.1007/s40305-021-00350-z.

[49]

Y. ZhangY. XuY. XuY. YangY. LuoQ. Wu and X. Liu, A multi-leader one-follower Stackelberg game approach for cooperative anti-jamming: No pains, no Gains, IEEE Comm. Lett., 22 (2018), 1680-1683.  doi: 10.1109/LCOMM.2018.2843374.

[50]

J. ZhaoW. TangR. Zhao and J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European J. Oper. Res., 216 (2012), 409-419.  doi: 10.1016/j.ejor.2011.07.026.

show all references

References:
[1]

X. AiJ. ChenH. Zhao and X. Tang, Competition among supply chains: Implications of full returns policy, Internat. J. Prod. Econ., 139 (2012), 257-265.  doi: 10.1016/j.ijpe.2012.04.018.

[2]

C. Amountzias, Pricing decisions and competitive conduct across manufacturing sectors: Evidence from 19 European Union manufacturing industries, J. Industry Competition Trade, 19 (2019), 413-440.  doi: 10.1007/s10842-019-00293-5.

[3]

J. Asirvatham and S. Bhuyan, Incentives and impacts of vertical coordination in a food production-marketing chain: A non-cooperative multi-stage, multi-player analysis, J. Industry Competition Trade, 18 (2018), 59-95.  doi: 10.1007/s10842-017-0247-2.

[4]

Y.-H. ChenX.-W. Wen and M.-Z. Luo, Corporate social responsibility spillover and competition effects on the food industry, Aus. Econ. Papers, 55 (2016), 1-13.  doi: 10.1111/1467-8454.12058.

[5]

M. Christopher, Logistics and Supply Chain Management,

[6]

S. Delpachitra, Price rigidity in the downstream petroleum industry in New Zealand: Where does it happen?, Energy Econ., 24 (2002), 597-613.  doi: 10.1016/S0140-9883(02)00021-X.

[7]

S. Delpachitra and D. Beal, Petrol prices disparity: Did the removal of price surveillance create price competition?, Econ. Papers, 21 (2002), 56-65.  doi: 10.1111/j.1759-3441.2002.tb00310.x.

[8]

N. C. P. EdirisingheB. Bichescu and X. Shi, Equilibrium analysis of supply chain structures under power imbalance, European J. Oper. Res., 214 (2011), 568-578.  doi: 10.1016/j.ejor.2011.05.008.

[9]

A. E. E. EltoukhyZ. X. WangF. T. S. Chan and S. H. Chung, Joint optimization using a leader-follower Stackelberg game for coordinated configuration of stochastic operational aircraft maintenance routing and maintenance staffing, Comp. Indust. Engrg., 125 (2018), 46-68.  doi: 10.1016/j.cie.2018.08.012.

[10]

H. FangL. Xu and X. Wang, Coordinated multiple-relays based physical-layer security improvement: A single-leader multiple-followers Stackelberg game scheme, IEEE Trans. Info. Forensics Security, 13 (2018), 197-209.  doi: 10.1109/TIFS.2017.2746001.

[11]

H. Jafari, Investigating environmental and economic aspects of sustainability by recycling PET plastic bottles: A game-theoretic approach, Clean Tech. Environ. Policy, 24 (2022), 829-842.  doi: 10.1007/s10098-021-02216-7.

[12]

H. Jafari, Nurse scheduling problem by considering total number of required nurses as well as nurses' preferences for working shifts: An algorithmic game-theoretic approach, in press, Scientia Iranica.

[13]

H. Jafari, Sustainable development by reusing of recyclables in a textile industry including two collectors and three firms: A game-theoretic approach for pricing decisions, J. Cleaner Prod., 229 (2019), 598-610.  doi: 10.1016/j.jclepro.2019.04.222.

[14]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Game theoretical approach to price a product under two-echelon supply chain containing e-tail selling channel, Internat. J. Services Oper. Mgmt., 36 (2020), 131-160.  doi: 10.1504/IJSOM.2020.10028974.

[15]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Pricing decisions in dual-channel supply chain including monopolistic manufacturer and duopolistic retailers: A game-theoretic approach, J. Industry Competition Trade, 16 (2016), 323-343.  doi: 10.1007/s10842-016-0224-1.

[16]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Pricing decisions in dual-channel supply chain with one manufacturer and multiple retailers: A game-theoretic approach, RAIRO Oper. Res., 51 (2017), 1269-1287.  doi: 10.1051/ro/2017003.

[17]

H. JafariS. R. Hejazi and M. Rasti-Barzoki, Sustainable development by waste recycling under a three-echelon supply chain: A game-theoretic approach, J. Cleaner Prod., 142 (2017), 2252-2261.  doi: 10.1016/j.jclepro.2016.11.051.

[18]

H. Jafari, S. Safarzadeh and E. Azad-Farsani, Effects of governmental policies on energy-efficiency improvement of hydrogen fuel cell cars: A game-theoretic approach, Energy, 254 (2022). doi: 10.1016/j. energy. 2022.124394.

[19]

C. K. JaggiM. GuptaA. Kausar and S. Tiwari, Inventory and credit decisions for deteriorating items with displayed stock dependent demand in two-echelon supply chain using Stackelberg and Nash equilibrium solution, Ann. Oper. Res., 274 (2019), 309-329.  doi: 10.1007/s10479-018-2925-9.

[20]

J. Jiang and X. Liu, Multi-objective Stackelberg game model for water supply networks against interdictions with incomplete information, European J. Oper. Res., 266 (2018), 920-933.  doi: 10.1016/j.ejor.2017.10.034.

[21]

D. Kesavayuth and V. Zikos, Price or quantity? A new irrelevance result for mixed markets, Aus. Econ. Papers, 55 (2016), 29-42.  doi: 10.1111/1467-8454.12059.

[22]

B. X. LiY.-W. ZhouJ.-Z. Li and S.-P. Zhou, Contract choice game of supply chain competition at both manufacturer and retailer levels, Internat. J. Prod. Econ., 143 (2013), 188-197.  doi: 10.1016/j.ijpe.2013.01.007.

[23]

J. LiS. Wang and T. C. E. Cheng, Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption, Internat. J. Prod. Econ., 124 (2010), 137-150.  doi: 10.1016/j.ijpe.2009.10.017.

[24]

P.-C. Liao, Strategic delegation under unionised duopoly: Who will bargain with unions?, Aus. Econ. Papers, 49 (2010), 276-288.  doi: 10.1111/j.1467-8454.2010.00402.x.

[25]

Y. LiuH. XuS.-J. S. Yang and J. Zhang, Distributionally robust equilibrium for continuous games: Nash and Stackelberg models, European J. Oper. Res., 265 (2018), 631-643.  doi: 10.1016/j.ejor.2017.07.050.

[26]

J.-C. LuY.-C. Tsao and C. Charoensiriwath, Competition under manufacturer service and retail price, Econ. Modell., 28 (2011), 1256-1264.  doi: 10.1016/j.econmod.2011.01.008.

[27]

A. LuciaP. A. DiMaggio and D. Alonso-Martinez, Metabolic pathway analysis using a Nash equilibrium approach, J. Global Optim., 71 (2018), 537-550.  doi: 10.1007/s10898-018-0605-6.

[28]

T. Matsumura and A. Ogawa, Endogenous timing in mixed duopolies with externality, Aus. Econ. Papers, 56 (2017), 304-327.  doi: 10.1111/1467-8454.12104.

[29]

J. F. Nash Jr., Equilibrium points in $n$-person games, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.

[30]

M. A. NchakeL. Edwards and A. Sundaram, Price-setting behavior and competition in developing countries: An analysis of retail outlets in Lesotho, J. Industry Competition Trade, 18 (2018), 529-547.  doi: 10.1007/s10842-018-0275-6.

[31]

K. PanK. K. LaiS. C. H. Leung and D. Xiao, Revenue-sharing versus wholesale price mechanisms under different channel power structures, European J. Oper. Res., 203 (2010), 532-538.  doi: 10.1016/j.ejor.2009.08.010.

[32]

M. Rasti-BarzokiH. Jafari and S. R. Hejazi, Game-theoretic approach for pricing decisions in dual-channel supply chain, Internat. J. Industrial Engrg. Prod. Res., 28 (2017), 1-8.  doi: 10.22068/ijiepr.28.1.1.

[33]

S. Sinha and S. P. Sarmah, Coordination and price competition in a duopoly common retailer supply chain, Comp. Indust. Engrg., 59 (2010), 280-295.  doi: 10.1016/j.cie.2010.04.010.

[34]

H.-L. Sun and X.-J. Chen, Two-stage stochastic variational inequalities: Theory, algorithms and applications, J. Oper. Res. Soc. China, 9 (2021), 1-32.  doi: 10.1007/s40305-019-00267-8.

[35]

C. TangS. Zhu and E. Cao, Equilibrium analysis of a dual-channel supply chain under different market powers, Aus. Econ. Papers, 57 (2018), 19-34.  doi: 10.1111/1467-8454.12107.

[36]

C. WangP.-Y. Nie and Y. Meng, Duopoly competition with corporate social responsibility, Aus. Econ. Papers, 57 (2018), 327-345.  doi: 10.1111/1467-8454.12119.

[37]

F.-M. WangJ.-J. WangN. LiY.-J. Jiang and S.-C. Li, A cost-sharing scheme for the $k$-level facility location game with penalties, J. Oper. Res. Soc. China, 10 (2022), 173-182.  doi: 10.1007/s40305-021-00345-w.

[38]

J. WeiJ. Zhao and Y. Li, Pricing decisions for complementary products with firms' different market powers, European J. Oper. Res., 224 (2013), 507-519.  doi: 10.1016/j.ejor.2012.09.011.

[39]

W. WeiX. FanH. SongX. Fan and J. Yang, Imperfect information dynamic Stackelberg game based resource allocation using hidden Markov for cloud computing, IEEE Trans. Services Comput., 11 (2018), 78-89.  doi: 10.1109/TSC.2016.2528246.

[40]

C.-H. Wu, Price and service competition between new and remanufactured products in a two-echelon supply chain, Internat. J. Prod. Econ., 140 (2012), 496-507.  doi: 10.1016/j.ijpe.2012.06.034.

[41]

D. D. Wu, Bargaining in supply chain with price and promotional effort dependent demand, Math. Comput. Modelling, 58 (2013), 1659-1669.  doi: 10.1016/j.mcm.2010.12.035.

[42]

T. Xiao and T. M. Choi, Purchasing choices and channel structure strategies for a two-echelon system with risk-averse players, Internat. J. Prod. Econ., 120 (2009), 54-65.  doi: 10.1016/j.ijpe.2008.07.028.

[43]

T. Xiao and D. Yang, Price and service competition of supply chains with risk-averse retailers under demand uncertainty, Internat. J. Prod. Econ., 114 (2008), 187-200.  doi: 10.1016/j.ijpe.2008.01.006.

[44]

G. XieS. Wang and K. K. Lai, Quality improvement in competing supply chains, Internat. J. Prod. Econ., 134 (2011), 262-270.  doi: 10.1016/j.ijpe.2011.07.007.

[45]

H. XuN. Yao and S. Tong, Sourcing under cost information asymmetry when facing time-sensitive customers, Internat. J. Prod. Econ., 144 (2013), 599-609.  doi: 10.1016/j.ijpe.2013.04.023.

[46]

Z. Yang and X. Zhang, Competitive equilibria and benefit distributions of population production economies with external increasing returns, J. Oper. Res. Soc. China, 9 (2021), 723-740.  doi: 10.1007/s40305-021-00340-1.

[47]

G. Ye, Leadership and privatisation in a mixed multi-product oligopoly: An endogenous timing model, Aus. Econ. Papers, 55 (2016), 170-180.  doi: 10.1111/1467-8454.12062.

[48]

B.-Y. Zhang and S. Pei, Game theory and the evolution of cooperation, J. Oper. Res. Soc. China, 10 (2022), 379-399.  doi: 10.1007/s40305-021-00350-z.

[49]

Y. ZhangY. XuY. XuY. YangY. LuoQ. Wu and X. Liu, A multi-leader one-follower Stackelberg game approach for cooperative anti-jamming: No pains, no Gains, IEEE Comm. Lett., 22 (2018), 1680-1683.  doi: 10.1109/LCOMM.2018.2843374.

[50]

J. ZhaoW. TangR. Zhao and J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European J. Oper. Res., 216 (2012), 409-419.  doi: 10.1016/j.ejor.2011.07.026.

Figure 1.  Changes of the profits in the manufacturing costs
Figure 2.  Changes of the profits in the market scale for the first product in direct channel
Figure 3.  Changes of the profits in the market scales for the products in the retail channel
Figure 4.  Changes of the profits in the self-price sensitivity of the demands
Figure 5.  Changes of the profits in the cross-price sensitivity of the demands
Table 1.  The values of the parameters in the numerical example
Parameters $ K_1=K_2 $ $ \varphi $ $ {\vartheta }_1={\vartheta }_2 $ $ \lambda $ $ \zeta $
Values 100 200 300 3.0 1.0
Parameters $ K_1=K_2 $ $ \varphi $ $ {\vartheta }_1={\vartheta }_2 $ $ \lambda $ $ \zeta $
Values 100 200 300 3.0 1.0
Table 2.  The results obtained from the investigated models for the numerical example
Games Prices
$ W_1 $ $ W_2 $ $ P $ $ P_1 $ $ P_2 $
Nash 130.23 130.23 141.86 160.47 160.47
Manufacturers-Stackelberg 161.39 153.16 157.59 195.09 190.98
Retailer-Stackelberg 134.58 133.86 145.24 169.15 167.73
Demands
$ Q $ $ Q_1 $ $ Q_2 $
Nash 95.35 120.93 120.93
Manufacturers-Stackelberg 113.29 63.29 79.75
Retailer-Stackelberg 101.15 105.51 111.21
Profits
$ M_1 $ $ M_2 $ $ R $ $ SC $
Nash 7647.38 3656.03 7312.06 18615.47
Manufacturers-Stackelberg 9534.33 3627.72 6637.06 19799.11
Retailer-Stackelberg 8224.57 3766.11 7414.31 19405.00
Games Prices
$ W_1 $ $ W_2 $ $ P $ $ P_1 $ $ P_2 $
Nash 130.23 130.23 141.86 160.47 160.47
Manufacturers-Stackelberg 161.39 153.16 157.59 195.09 190.98
Retailer-Stackelberg 134.58 133.86 145.24 169.15 167.73
Demands
$ Q $ $ Q_1 $ $ Q_2 $
Nash 95.35 120.93 120.93
Manufacturers-Stackelberg 113.29 63.29 79.75
Retailer-Stackelberg 101.15 105.51 111.21
Profits
$ M_1 $ $ M_2 $ $ R $ $ SC $
Nash 7647.38 3656.03 7312.06 18615.47
Manufacturers-Stackelberg 9534.33 3627.72 6637.06 19799.11
Retailer-Stackelberg 8224.57 3766.11 7414.31 19405.00
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