January  2023, 19(1): 645-674. doi: 10.3934/jimo.2021200

Information sharing in two-tier supply chains considering cost reduction effort and information leakage

1. 

School of Management, Xi'an Jiaotong University, Xi'an, 710049, China

2. 

School of Economics and Management, Xidian University, Xi'an, 710126, China

3. 

Business School, Central South University, Changsha, 410083, China

* Corresponding author: Xiaomei Li

Received  January 2021 Revised  September 2021 Published  January 2023 Early access  November 2021

Fund Project: This resarch is supported by the Major Program of National Fund of Philosophy and Social Science of China (18ZDA104), National Natural Science Foundation of China (72102174), Ministry of Education of Humanities and Social Science Project (21XJC630004, 19YJA630068), Natural Science Basic Research Program of Shaanxi(2021JM-144), Fundamental Research Funds for the Central Universities (XJS200601)

This study investigates information sharing in two-tier supply chai-ns considering cost reduction effort and information leakage, with either upstream competition (system SC) or downstream competition (system RC). Results show that in system SC without information leakage, the retailer shares information with one supplier when suppliers are efficient in cost reduction, shares information with neither supplier when suppliers are inefficient in cost reduction, and shares information with two suppliers when suppliers are intermediate in cost reduction efficiency. nformation leakage won't affect the information sharing decisions of the retailer. In system RC with or without information leakage, both retailers share information with the supplier when the supplier is efficient in cost reduction and neither retailer shares information with the supplier when the supplier is inefficient in cost reduction. However, the threshold of cost reduction efficiency without information leakage is always lower than that with information leakage, which demonstrates that it is less likely for retailers to share information with information leakage. What's more, the two retailers choose the same information sharing strategies without information leakage but the opposite information sharing strategies with information leakage when the cost reduction efficiency is intermediate.

Citation: Xiaomei Li, Renjing Liu, Zhongquan Hu, Jiamin Dong. Information sharing in two-tier supply chains considering cost reduction effort and information leakage. Journal of Industrial and Management Optimization, 2023, 19 (1) : 645-674. doi: 10.3934/jimo.2021200
References:
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K. S. Anand and M. Goyal, Strategic information management under leakage in a supply chain, Management Science, 55 (2009), 438-452.  doi: 10.1287/mnsc.1080.0930.

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G. P. Cachon and M. Fisher, Supply chain inventory management and the value of shared information, Management Science, 46 (2000), 1032-1048.  doi: 10.1287/mnsc.46.8.1032.12029.

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K. CaiS. He and Z. He, Information sharing under different warranty policies with cost sharing in supply chains, Int. Trans. Oper. Res., 27 (2020), 1550-1572.  doi: 10.1111/itor.12597.

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E. Cao and G. Chen, Information sharing motivated by production cost reduction in a supply chain with downstream competition, Naval Research Logistics, 68 (2021), 898-907.  doi: 10.1002/nav.21977.

[5]

Y. Chen and Ö. Özer, Supply chain contracts that prevent information leakage, Management Science, 65 (2019), 5619-5650.  doi: 10.1287/mnsc.2018.3200.

[6]

D. Fang and Q. Ren, Optimal decision in a dual-channel supply chain under potential information leakage, Symmetry, 11 (2019), 308.  doi: 10.3390/sym11030308.

[7]

Z. GuanX. ZhangM. Zhou and Y. Dan, Demand information sharing in competing supply chains with manufacturer-provided service, Int. J. Production Economics, 220 (2020), 107450.  doi: 10.1016/j.ijpe.2019.07.023.

[8]

A. Y. HaQ. Tian and S. Tong, Information sharing in competing supply chains with production cost reduction, Manufacturing & Service Operations Management, 19 (2017), 246-262.  doi: 10.1287/msom.2016.0607.

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A. Y. HaS. Tong and H. Zhang, Sharing demand information in competing supply chains with production diseconomies, Management Science, 57 (2011), 566-581.  doi: 10.1287/mnsc.1100.1295.

[10]

J. HuQ. Hu and Y. Xia, Who should invest in cost reduction in supply chains?, Int. J. Production Economics, 207 (2019), 1-18.  doi: 10.1016/j.ijpe.2018.10.002.

[11]

L. Jiang and Z. Hao, Incentive-driven information dissemination in two-tier supply chains, Manufacturing & Service Operations Management, 18 (2016), 393-413.  doi: 10.1287/msom.2016.0575.

[12]

S. H. Kim and S. Netessine, Collaborative cost reduction and component procurement under information asymmetry, Management Science, 59 (2013), 189-206.  doi: 10.1287/mnsc.1120.1573.

[13]

G. KongS. Rajagopalan and H. Zhang, Revenue sharing and information leakage in a supply chain, Management Science, 59 (2013), 556-572.  doi: 10.1287/mnsc.1120.1627.

[14]

H. L. LeeK. C. So and C. S. Tang, The value of information sharing in a two-level supply chain, Management Science, 46 (2016), 626-643.  doi: 10.1287/mnsc.46.5.626.12047.

[15]

H. L. Lee and S. Whang, Information sharing in a supply chain, International Journal of Manufacturing Technology and Management, 1 (2000), 79-93.  doi: 10.1504/IJMTM.2000.001329.

[16]

H. Lei, J. Wang, H. Yang and H. Wan, The impact of ex-post information sharing on a two-echelon supply chain with horizontal competition and capacity constraint, Annals of Operations Research, (2020), 1–27.

[17]

L. Li, Cournot oligopoly with information sharing, The Rand Journal of Economics, (1985), 521–536.

[18]

L. Li, Information sharing in a supply chain with horizontal competition, Management Science, 48 (2002), 1196-1212. 

[19]

X. LiJ. Chen and X. Ai, Contract design in a cross-sales supply chain with demand information asymmetry, European J. Oper. Res., 275 (2019), 939-956.  doi: 10.1016/j.ejor.2018.12.023.

[20]

L. Li and H. Zhang, Confidentiality and information sharing in supply chain coordination, Management Science, 54 (2008), 1467-1481.  doi: 10.1287/mnsc.1070.0851.

[21]

T. Lisa, Retailer/Supplier Shared Data Study, 2015. Available from: https://consumergoods.com/2015-retailersupplier-shared-data-study.

[22]

H. LiuW. JiangG. Feng and K. S. Chin, Information leakage and supply chain contracts, Omega, 90 (2020), 101994.  doi: 10.1016/j.omega.2018.11.003.

[23]

M. Freedman, How Businesses Are Collecting Data, 2020. Available from: https://www.businessnewsdaily.com/10625-businesses-collecting-data.html.

[24]

S. K. MukhopadhyayD. Q. Yao and X. Yue, Information sharing of value-adding retailer in a mixed channel hi-tech supply chain, J. Business research, 61 (2008), 950-958.  doi: 10.1016/j.jbusres.2006.10.027.

[25]

W. ShangA. Y. Ha and S. Tong, Information sharing in a supply chain with a common retailer. Management Science, Management Science, 62 (2016), 245-263.  doi: 10.1287/mnsc.2014.2127.

[26]

4 Big Benefits of Retailers Sharing POS Data with Supply Chain Partners, Spring Global News, Spring Global, 2019, Available from: https://www.springglobal.com/blog/4-big-benefits-of-retailers-sharing-pos-data-with-supply-chain-partners.

[27]

X. SunW. TangJ. ChenS. Li and J. Zhang, Manufacturer encroachment with production cost reduction under asymmetric information, Transportation Research Part E: Logistics and Transportation Review, 128 (2019), 191-211.  doi: 10.1016/j.tre.2019.05.018.

[28]

K. H. TanW. P. Wong and L. Chung, Information and knowledge leakage in supply chain, Information Systems Frontiers, 18 (2016), 621-638.  doi: 10.1007/s10796-015-9553-6.

[29]

J. Wang, Z. Zhen and Q. Yan, Information sharing and leakage in the two-echelon supply chain, RAIRO-Oper. Res., 55 (2021), s307–s325. doi: 10.1051/ro/2019066.

[30]

A. Weinbaum, 9 Ways to Encourage Distributors to Submit Channel POS Data, 2017. Available from: https://computermarketresearch.com/inspire-channel-pos-data-submission/.

[31]

Z. Yu, H. Yan and T. E. Cheng, Benefits of information sharing with supply chain partnerships, Industrial Management & Data Systems, 101 (2001). doi: 10.1108/02635570110386625.

[32]

D. Y. ZhangX. CaoL. Wang and Y. Zeng, Mitigating the risk of information leakage in a two-level supply chain through optimal supplier selection, J. Intelligent Manufacturing, 23 (2019), 1351-1364.  doi: 10.1007/s10845-011-0527-3.

[33]

D. Y. ZhangY. ZengL. WangH. Li and Y. Geng, Modeling and evaluating information leakage caused by inferences in supply chains, Computers in Industry, 62 (2011), 351-363.  doi: 10.1016/j.compind.2010.10.002.

[34]

H. Zhang, Vertical information exchange in a supply chain with duopoly retailers, Production and Operations Management, 11 (2002), 531-546.  doi: 10.1111/j.1937-5956.2002.tb00476.x.

show all references

References:
[1]

K. S. Anand and M. Goyal, Strategic information management under leakage in a supply chain, Management Science, 55 (2009), 438-452.  doi: 10.1287/mnsc.1080.0930.

[2]

G. P. Cachon and M. Fisher, Supply chain inventory management and the value of shared information, Management Science, 46 (2000), 1032-1048.  doi: 10.1287/mnsc.46.8.1032.12029.

[3]

K. CaiS. He and Z. He, Information sharing under different warranty policies with cost sharing in supply chains, Int. Trans. Oper. Res., 27 (2020), 1550-1572.  doi: 10.1111/itor.12597.

[4]

E. Cao and G. Chen, Information sharing motivated by production cost reduction in a supply chain with downstream competition, Naval Research Logistics, 68 (2021), 898-907.  doi: 10.1002/nav.21977.

[5]

Y. Chen and Ö. Özer, Supply chain contracts that prevent information leakage, Management Science, 65 (2019), 5619-5650.  doi: 10.1287/mnsc.2018.3200.

[6]

D. Fang and Q. Ren, Optimal decision in a dual-channel supply chain under potential information leakage, Symmetry, 11 (2019), 308.  doi: 10.3390/sym11030308.

[7]

Z. GuanX. ZhangM. Zhou and Y. Dan, Demand information sharing in competing supply chains with manufacturer-provided service, Int. J. Production Economics, 220 (2020), 107450.  doi: 10.1016/j.ijpe.2019.07.023.

[8]

A. Y. HaQ. Tian and S. Tong, Information sharing in competing supply chains with production cost reduction, Manufacturing & Service Operations Management, 19 (2017), 246-262.  doi: 10.1287/msom.2016.0607.

[9]

A. Y. HaS. Tong and H. Zhang, Sharing demand information in competing supply chains with production diseconomies, Management Science, 57 (2011), 566-581.  doi: 10.1287/mnsc.1100.1295.

[10]

J. HuQ. Hu and Y. Xia, Who should invest in cost reduction in supply chains?, Int. J. Production Economics, 207 (2019), 1-18.  doi: 10.1016/j.ijpe.2018.10.002.

[11]

L. Jiang and Z. Hao, Incentive-driven information dissemination in two-tier supply chains, Manufacturing & Service Operations Management, 18 (2016), 393-413.  doi: 10.1287/msom.2016.0575.

[12]

S. H. Kim and S. Netessine, Collaborative cost reduction and component procurement under information asymmetry, Management Science, 59 (2013), 189-206.  doi: 10.1287/mnsc.1120.1573.

[13]

G. KongS. Rajagopalan and H. Zhang, Revenue sharing and information leakage in a supply chain, Management Science, 59 (2013), 556-572.  doi: 10.1287/mnsc.1120.1627.

[14]

H. L. LeeK. C. So and C. S. Tang, The value of information sharing in a two-level supply chain, Management Science, 46 (2016), 626-643.  doi: 10.1287/mnsc.46.5.626.12047.

[15]

H. L. Lee and S. Whang, Information sharing in a supply chain, International Journal of Manufacturing Technology and Management, 1 (2000), 79-93.  doi: 10.1504/IJMTM.2000.001329.

[16]

H. Lei, J. Wang, H. Yang and H. Wan, The impact of ex-post information sharing on a two-echelon supply chain with horizontal competition and capacity constraint, Annals of Operations Research, (2020), 1–27.

[17]

L. Li, Cournot oligopoly with information sharing, The Rand Journal of Economics, (1985), 521–536.

[18]

L. Li, Information sharing in a supply chain with horizontal competition, Management Science, 48 (2002), 1196-1212. 

[19]

X. LiJ. Chen and X. Ai, Contract design in a cross-sales supply chain with demand information asymmetry, European J. Oper. Res., 275 (2019), 939-956.  doi: 10.1016/j.ejor.2018.12.023.

[20]

L. Li and H. Zhang, Confidentiality and information sharing in supply chain coordination, Management Science, 54 (2008), 1467-1481.  doi: 10.1287/mnsc.1070.0851.

[21]

T. Lisa, Retailer/Supplier Shared Data Study, 2015. Available from: https://consumergoods.com/2015-retailersupplier-shared-data-study.

[22]

H. LiuW. JiangG. Feng and K. S. Chin, Information leakage and supply chain contracts, Omega, 90 (2020), 101994.  doi: 10.1016/j.omega.2018.11.003.

[23]

M. Freedman, How Businesses Are Collecting Data, 2020. Available from: https://www.businessnewsdaily.com/10625-businesses-collecting-data.html.

[24]

S. K. MukhopadhyayD. Q. Yao and X. Yue, Information sharing of value-adding retailer in a mixed channel hi-tech supply chain, J. Business research, 61 (2008), 950-958.  doi: 10.1016/j.jbusres.2006.10.027.

[25]

W. ShangA. Y. Ha and S. Tong, Information sharing in a supply chain with a common retailer. Management Science, Management Science, 62 (2016), 245-263.  doi: 10.1287/mnsc.2014.2127.

[26]

4 Big Benefits of Retailers Sharing POS Data with Supply Chain Partners, Spring Global News, Spring Global, 2019, Available from: https://www.springglobal.com/blog/4-big-benefits-of-retailers-sharing-pos-data-with-supply-chain-partners.

[27]

X. SunW. TangJ. ChenS. Li and J. Zhang, Manufacturer encroachment with production cost reduction under asymmetric information, Transportation Research Part E: Logistics and Transportation Review, 128 (2019), 191-211.  doi: 10.1016/j.tre.2019.05.018.

[28]

K. H. TanW. P. Wong and L. Chung, Information and knowledge leakage in supply chain, Information Systems Frontiers, 18 (2016), 621-638.  doi: 10.1007/s10796-015-9553-6.

[29]

J. Wang, Z. Zhen and Q. Yan, Information sharing and leakage in the two-echelon supply chain, RAIRO-Oper. Res., 55 (2021), s307–s325. doi: 10.1051/ro/2019066.

[30]

A. Weinbaum, 9 Ways to Encourage Distributors to Submit Channel POS Data, 2017. Available from: https://computermarketresearch.com/inspire-channel-pos-data-submission/.

[31]

Z. Yu, H. Yan and T. E. Cheng, Benefits of information sharing with supply chain partnerships, Industrial Management & Data Systems, 101 (2001). doi: 10.1108/02635570110386625.

[32]

D. Y. ZhangX. CaoL. Wang and Y. Zeng, Mitigating the risk of information leakage in a two-level supply chain through optimal supplier selection, J. Intelligent Manufacturing, 23 (2019), 1351-1364.  doi: 10.1007/s10845-011-0527-3.

[33]

D. Y. ZhangY. ZengL. WangH. Li and Y. Geng, Modeling and evaluating information leakage caused by inferences in supply chains, Computers in Industry, 62 (2011), 351-363.  doi: 10.1016/j.compind.2010.10.002.

[34]

H. Zhang, Vertical information exchange in a supply chain with duopoly retailers, Production and Operations Management, 11 (2002), 531-546.  doi: 10.1111/j.1937-5956.2002.tb00476.x.

Figure 1.  Supply chain members' expected profit in system SC without information leakage
Figure 2.  The retailer's information sharing strategies in system SC without information leakage
Figure 3.  Supply chain members' expected profits in system RC without information leakage
Figure 4.  Retailers' information sharing strategies in system RC without information leakage
Figure 5.  Supply chain members' expected profits in system RC with information leakage
Figure 6.  Retailers' information sharing strategies in system RC without information leakage
Table 1.  Summary table of literature review
Papers Influence factors Supply chain structures
Information leakage Cost reduction System SC System RC
Fang and Ren[6] $ \checkmark $ $ \times $ $ \times $ $ \times $
Wang et al.[29] $ \checkmark $ $ \times $ $ \times $ $ \checkmark $
Chen and Özer[5] $ \checkmark $ $ \times $ $ \times $ $ \checkmark $
Ha et al.[8] $ \times $ $ \checkmark $ $ \times $ $ \times $
Sun et al.[27] $ \times $ $ \checkmark $ $ \times $ $ \times $
Cao and Chen[4] $ \checkmark $ $ \checkmark $ $ \times $ $ \checkmark $
Our paper $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $
Papers Influence factors Supply chain structures
Information leakage Cost reduction System SC System RC
Fang and Ren[6] $ \checkmark $ $ \times $ $ \times $ $ \times $
Wang et al.[29] $ \checkmark $ $ \times $ $ \times $ $ \checkmark $
Chen and Özer[5] $ \checkmark $ $ \times $ $ \times $ $ \checkmark $
Ha et al.[8] $ \times $ $ \checkmark $ $ \times $ $ \times $
Sun et al.[27] $ \times $ $ \checkmark $ $ \times $ $ \times $
Cao and Chen[4] $ \checkmark $ $ \checkmark $ $ \times $ $ \checkmark $
Our paper $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $
Table 2.  Parameters and notations
Parameters and notations Description
$ p_{i} $ Selling price of product $ i $
$ q_{i} $ Order quantity of product $ i $
$ \alpha $ Potential market size
$ \theta $ Demand uncertainty with the mean of $ 0 $ and variance of $ \theta^2 $
$ \gamma $ Competition intensity and a larger $ \gamma $ implies more intense competition
$ c_i $ Production cost of product $ i $
$ x_i $ Production cost reduction level of product $ i $
$ k_i $ Cost reduction efficiency of product $ i $ and a lower $ k_i $ indicates a higher efficiency
$ Y_{(i)} $ Demand signal of retailer $ (i) $
$ s $ Accuracy of demand signal and a larger $ s $ indicates a less accurate information
$ n=(n_1,n_2) $ Information sharing decisions
$ \omega_i $ Wholesale price of product $ i $
$ \pi_{R_{(i)}}^{(n_1,n_2)} $ Profit of the retailer $ (i) $ under information sharing arrangement $ n=(n_1,n_2) $
$ \pi_{S_{(i)}}^{(n_1,n_2)} $ Profit of the supplier $ (i) $ under information sharing arrangement $ n=(n_1,n_2) $
Parameters and notations Description
$ p_{i} $ Selling price of product $ i $
$ q_{i} $ Order quantity of product $ i $
$ \alpha $ Potential market size
$ \theta $ Demand uncertainty with the mean of $ 0 $ and variance of $ \theta^2 $
$ \gamma $ Competition intensity and a larger $ \gamma $ implies more intense competition
$ c_i $ Production cost of product $ i $
$ x_i $ Production cost reduction level of product $ i $
$ k_i $ Cost reduction efficiency of product $ i $ and a lower $ k_i $ indicates a higher efficiency
$ Y_{(i)} $ Demand signal of retailer $ (i) $
$ s $ Accuracy of demand signal and a larger $ s $ indicates a less accurate information
$ n=(n_1,n_2) $ Information sharing decisions
$ \omega_i $ Wholesale price of product $ i $
$ \pi_{R_{(i)}}^{(n_1,n_2)} $ Profit of the retailer $ (i) $ under information sharing arrangement $ n=(n_1,n_2) $
$ \pi_{S_{(i)}}^{(n_1,n_2)} $ Profit of the supplier $ (i) $ under information sharing arrangement $ n=(n_1,n_2) $
Table 3.  Decisions and ex-ante profits in system SC without information leakage
$ n $ Decisions Ex-ante profits
$ (0,0) $ $ \omega_i^{(0,0)}=\omega^0 $ $ \pi_{R}^{(0,0)}=\pi_{R}^{0}+\frac{\sigma ^2}{2(1+s)(1+\gamma )} $
$ x_i^{(0,0)}=x^0 $
$ q_i^{(0,0)}=q^0+\frac{Y}{2(1+s)(1+\gamma )} $ $ \pi_{S_i}^{(0,0)}=\pi_{S}^{0} $
$ (1,1) $ $ \omega_i^{(1,1)}=\omega^0+\frac{(2k(1-\gamma^2)-1)Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)} $ $ \pi_{R}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2(1+\gamma)\sigma ^2}{(1+s)(2k(2-\gamma)(1+\gamma)-1)^2} $
$ x_i^{(1,1)}=x^0+\frac{Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)} $
$ q_i^{(1,1)}=q^0+\frac{kY}{(1+s)(2k(2-\gamma)(1+\gamma)-1)} $ $ \pi_{S_i}^{(1,1)}=\pi_{S}^{0}+\frac{k(4k(1-\gamma^2)-1)\sigma ^2}{2(1+s)(2k(2-\gamma)(1+\gamma)-1)^2} $
$ (1,0) $ $ \omega_1^{(1,0)}=\omega^0+\frac{(1-\gamma)(2k(1-\gamma^2)-1)Y}{(1+s)(4k(1+\gamma^2)-1)} $ $ \pi_{R}^{(1,0)}=\pi_{R}^{0} $
$ x_1^{(1,0)}=x^0+\frac{(1-\gamma)Y}{(1+s)(4k(1+\gamma^2)-1)} $ $ +\frac{(4k^2(1-\gamma)^2(1+\gamma)(5+3\gamma)-8k(1-\gamma^2)+1)\sigma ^2}{4(1+s)(4k(1-\gamma^2)-1)^2} $
$ \omega_2^{(1,0)}=\omega^0 $ $ \pi_{S_1}^{(1,0)}=\pi_{S}^{0}+\frac{(k(1-\gamma)^2\sigma ^2}{2(1+s)(4k(1-\gamma^2)-1)^2} $
$ x_2^{(1,0)}=x^0 $
$ q_1^{(1,0)}=q^0+\frac{(1-\gamma)kY}{(1+s)(4k(1+\gamma^2)-1)} $ $ \pi_{S_2}^{(1,0)}=\pi_{S}^{0} $
$ q_2^{(1,0)}=q^0+\frac{2k(2+\gamma)(1-\gamma)Y}{2(1+s)(4k(1+\gamma^2)-1)} $
Notes. $\omega^0=\frac{(2k(1-\gamma^2)-1)\alpha+2k(1+\gamma)c}{(2k(2-\gamma)(1+\gamma)-1}, x^0=\frac{\alpha-c}{(2k(2-\gamma)(1+\gamma)-1)}, q^0=\frac{k(\alpha-c)}{(2k(2-\gamma)(1+\gamma)-1},$
$\pi_{R}^{0}=\frac{2k^2(1+\gamma)(\alpha-c)^2}{(2k(2-\gamma)(1+\gamma)-1)^2},\pi_{S}^{0}=\frac{k(4k(1-\gamma^2)-1)(\alpha-c)^2}{2(2k(2-\gamma)(1+\gamma)-1)^2}.$
$ n $ Decisions Ex-ante profits
$ (0,0) $ $ \omega_i^{(0,0)}=\omega^0 $ $ \pi_{R}^{(0,0)}=\pi_{R}^{0}+\frac{\sigma ^2}{2(1+s)(1+\gamma )} $
$ x_i^{(0,0)}=x^0 $
$ q_i^{(0,0)}=q^0+\frac{Y}{2(1+s)(1+\gamma )} $ $ \pi_{S_i}^{(0,0)}=\pi_{S}^{0} $
$ (1,1) $ $ \omega_i^{(1,1)}=\omega^0+\frac{(2k(1-\gamma^2)-1)Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)} $ $ \pi_{R}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2(1+\gamma)\sigma ^2}{(1+s)(2k(2-\gamma)(1+\gamma)-1)^2} $
$ x_i^{(1,1)}=x^0+\frac{Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)} $
$ q_i^{(1,1)}=q^0+\frac{kY}{(1+s)(2k(2-\gamma)(1+\gamma)-1)} $ $ \pi_{S_i}^{(1,1)}=\pi_{S}^{0}+\frac{k(4k(1-\gamma^2)-1)\sigma ^2}{2(1+s)(2k(2-\gamma)(1+\gamma)-1)^2} $
$ (1,0) $ $ \omega_1^{(1,0)}=\omega^0+\frac{(1-\gamma)(2k(1-\gamma^2)-1)Y}{(1+s)(4k(1+\gamma^2)-1)} $ $ \pi_{R}^{(1,0)}=\pi_{R}^{0} $
$ x_1^{(1,0)}=x^0+\frac{(1-\gamma)Y}{(1+s)(4k(1+\gamma^2)-1)} $ $ +\frac{(4k^2(1-\gamma)^2(1+\gamma)(5+3\gamma)-8k(1-\gamma^2)+1)\sigma ^2}{4(1+s)(4k(1-\gamma^2)-1)^2} $
$ \omega_2^{(1,0)}=\omega^0 $ $ \pi_{S_1}^{(1,0)}=\pi_{S}^{0}+\frac{(k(1-\gamma)^2\sigma ^2}{2(1+s)(4k(1-\gamma^2)-1)^2} $
$ x_2^{(1,0)}=x^0 $
$ q_1^{(1,0)}=q^0+\frac{(1-\gamma)kY}{(1+s)(4k(1+\gamma^2)-1)} $ $ \pi_{S_2}^{(1,0)}=\pi_{S}^{0} $
$ q_2^{(1,0)}=q^0+\frac{2k(2+\gamma)(1-\gamma)Y}{2(1+s)(4k(1+\gamma^2)-1)} $
Notes. $\omega^0=\frac{(2k(1-\gamma^2)-1)\alpha+2k(1+\gamma)c}{(2k(2-\gamma)(1+\gamma)-1}, x^0=\frac{\alpha-c}{(2k(2-\gamma)(1+\gamma)-1)}, q^0=\frac{k(\alpha-c)}{(2k(2-\gamma)(1+\gamma)-1},$
$\pi_{R}^{0}=\frac{2k^2(1+\gamma)(\alpha-c)^2}{(2k(2-\gamma)(1+\gamma)-1)^2},\pi_{S}^{0}=\frac{k(4k(1-\gamma^2)-1)(\alpha-c)^2}{2(2k(2-\gamma)(1+\gamma)-1)^2}.$
Table 4.  Decisions and ex-ante profits in system SC without information leakage
$ n $ Decisions Ex-ante profits
$ (0,0) $ $ \omega_i^{(0,0)}=\omega^0 $ $ \pi_{R_i}^{(0,0)}=\pi_{R_i}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2} $
$ x_i^{(0,0)}=x^0 $
$ q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma} $ $ \pi_{S}^{(0,0)}=\pi_{S}^{0} $
$ (1,1) $ $ \omega_i^{(1,1)}=\omega^0+\alpha_i^{(1,1)} $ $ \pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\zeta_i^{(1,1)} $
$ x_i^{(1,1)}=x^0+\beta_i^{(1,1)} $
$ q_i^{(1,1)}=q^0+\delta_i^{(1,1)} $ $ \pi_{S}^{(1,1)}=\pi_{S}^{0}+\eta_i^{(1,1)} $
$ (1,0) $ $ \omega_1^{(1,0)}=\omega^0+\alpha_1^{(1,0)} $ $ \pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\zeta_1^{(1,0)} $
$ x_1^{(1,0)}=x^0+\beta_1^{(1,0)} $
$ \omega_2^{(1,0)}=\omega^0+\alpha_2^{(1,0)} $ $ \pi_{R_2}^{(1,0)}=\pi_{R}^{0}+\zeta_2^{(1,0)} $
$ x_2^{(1,0)}=x^0+\beta_2^{(1,0)} $
$ q_1^{(1,0)}=q^0+\delta_1^{(1,0)} $ $ \pi_{S}^{(1,0)}=\pi_{S}^{0}+\eta^{(1,0)} $
$ q_2^{(1,0)}=q^0+\delta_2^{(1,0)} $
Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1},x^0=\frac{\alpha-c}{2k(2+\gamma)-1},q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1},\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2},$,
$ \pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2} \qquad \left\{\begin{array}{l} \alpha _1^{(1,0)}=\frac{(2k^2(2+2s+\gamma)(4-\gamma ^2)+k((1+s)\gamma ^2-4(3+3s+\gamma ))+2+2s+\gamma )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \alpha _2^{(1,0)}=\frac{(2k^2(2+\gamma+s\gamma)(4-\gamma ^2)-k(12+(4+4s-\gamma )\gamma )+2+\gamma+s\gamma)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \alpha _i^{(1,1)}=\frac{(4k-1)(k(4-\gamma^2)-2)Y_i+(2k^2(4-\gamma^2)-4k+1)\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $ \left\{\begin{array}{l} \beta _1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _2^{(1,0)}=\frac{(2k(4-\gamma ^2)-(2+\gamma+s\gamma))Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _i^{(1,1)}=\frac{2(k(4-\gamma^2)-1)Y_i-\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$
$\left\{\begin{array}{l} \delta _1^{(1,0)}=\frac{k(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \delta _2^{(1,0)}=\frac{(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}+\frac{Y_2}{2+2s+\gamma},\\ \delta _i^{(1,1)}=\frac{2k(k(4-\gamma^2)-1)Y_i-k\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$
$ \left\{\begin{array}{l} \eta^{(1,0)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \eta^{(0,1)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \eta^{(1,1)}=\frac{(2k(2+2s+\gamma)(4-\gamma^2)-s(4+\gamma^2)-(2+\gamma)^2)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$
$ \left\{\begin{array}{l} \zeta_1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )^2k^2\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}, \\ \zeta_2^{(1,0)}=\frac{(2k(2+\gamma )-1)(2k(2-\gamma )-1)(2k^2(1+2s(2+s))(4-\gamma^2)-k(2+\gamma+s(16+8s+\gamma))+s(2+s))\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}+\\ \frac{(2k^2(1+2s(2-\gamma))(4-\gamma^2)-k(2+\gamma+s(17-7\gamma))+s(2-\gamma)(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2},\\ \zeta_i^{(1,1)}=\frac{(4k^3(1+s)(4-\gamma^2)^2+4k^2(4-\gamma^2)(s\gamma^2-(1+2s)\gamma-2(1+s)))k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}\\+\frac{(k((1-7s)\gamma^2+4(1+4s)\gamma+4(1+s))-s(2-\gamma)\gamma)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}. \end{array}\right.$
$ n $ Decisions Ex-ante profits
$ (0,0) $ $ \omega_i^{(0,0)}=\omega^0 $ $ \pi_{R_i}^{(0,0)}=\pi_{R_i}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2} $
$ x_i^{(0,0)}=x^0 $
$ q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma} $ $ \pi_{S}^{(0,0)}=\pi_{S}^{0} $
$ (1,1) $ $ \omega_i^{(1,1)}=\omega^0+\alpha_i^{(1,1)} $ $ \pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\zeta_i^{(1,1)} $
$ x_i^{(1,1)}=x^0+\beta_i^{(1,1)} $
$ q_i^{(1,1)}=q^0+\delta_i^{(1,1)} $ $ \pi_{S}^{(1,1)}=\pi_{S}^{0}+\eta_i^{(1,1)} $
$ (1,0) $ $ \omega_1^{(1,0)}=\omega^0+\alpha_1^{(1,0)} $ $ \pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\zeta_1^{(1,0)} $
$ x_1^{(1,0)}=x^0+\beta_1^{(1,0)} $
$ \omega_2^{(1,0)}=\omega^0+\alpha_2^{(1,0)} $ $ \pi_{R_2}^{(1,0)}=\pi_{R}^{0}+\zeta_2^{(1,0)} $
$ x_2^{(1,0)}=x^0+\beta_2^{(1,0)} $
$ q_1^{(1,0)}=q^0+\delta_1^{(1,0)} $ $ \pi_{S}^{(1,0)}=\pi_{S}^{0}+\eta^{(1,0)} $
$ q_2^{(1,0)}=q^0+\delta_2^{(1,0)} $
Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1},x^0=\frac{\alpha-c}{2k(2+\gamma)-1},q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1},\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2},$,
$ \pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2} \qquad \left\{\begin{array}{l} \alpha _1^{(1,0)}=\frac{(2k^2(2+2s+\gamma)(4-\gamma ^2)+k((1+s)\gamma ^2-4(3+3s+\gamma ))+2+2s+\gamma )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \alpha _2^{(1,0)}=\frac{(2k^2(2+\gamma+s\gamma)(4-\gamma ^2)-k(12+(4+4s-\gamma )\gamma )+2+\gamma+s\gamma)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \alpha _i^{(1,1)}=\frac{(4k-1)(k(4-\gamma^2)-2)Y_i+(2k^2(4-\gamma^2)-4k+1)\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $ \left\{\begin{array}{l} \beta _1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _2^{(1,0)}=\frac{(2k(4-\gamma ^2)-(2+\gamma+s\gamma))Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _i^{(1,1)}=\frac{2(k(4-\gamma^2)-1)Y_i-\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$
$\left\{\begin{array}{l} \delta _1^{(1,0)}=\frac{k(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \delta _2^{(1,0)}=\frac{(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}+\frac{Y_2}{2+2s+\gamma},\\ \delta _i^{(1,1)}=\frac{2k(k(4-\gamma^2)-1)Y_i-k\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$
$ \left\{\begin{array}{l} \eta^{(1,0)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \eta^{(0,1)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \eta^{(1,1)}=\frac{(2k(2+2s+\gamma)(4-\gamma^2)-s(4+\gamma^2)-(2+\gamma)^2)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$
$ \left\{\begin{array}{l} \zeta_1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )^2k^2\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}, \\ \zeta_2^{(1,0)}=\frac{(2k(2+\gamma )-1)(2k(2-\gamma )-1)(2k^2(1+2s(2+s))(4-\gamma^2)-k(2+\gamma+s(16+8s+\gamma))+s(2+s))\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}+\\ \frac{(2k^2(1+2s(2-\gamma))(4-\gamma^2)-k(2+\gamma+s(17-7\gamma))+s(2-\gamma)(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2},\\ \zeta_i^{(1,1)}=\frac{(4k^3(1+s)(4-\gamma^2)^2+4k^2(4-\gamma^2)(s\gamma^2-(1+2s)\gamma-2(1+s)))k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}\\+\frac{(k((1-7s)\gamma^2+4(1+4s)\gamma+4(1+s))-s(2-\gamma)\gamma)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}. \end{array}\right.$
Table 5.  Decisions and ex-ante profits in system SC without information leakage
$ n $ Decisions Ex-ante profits
$ (0,0) $ $ \omega^{(0,0)}=\omega^0 $ $ \pi_{R_i}^{(0,0)}=\pi_{R}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2} $
$ x^{(0,0)}=x^0 $
$ q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma} $ $ \pi_{S}^{(0,0)}=\pi_{S}^{0} $
$ (1,1) $ $ \omega^{(1,1)}=\omega^0+\frac{(2k+k\gamma-1)(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)} $ $ \pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2\sigma^2}{(2+s)(2k(2+\gamma)-1)^2} $
$ x^{(1,1)}=x^0+\frac{Y_1+Y_2}{(2+s)(2k(2+\gamma)-1)} $
$ q_i^{(1,1)}=q^0+\frac{k(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)} $ $ \pi_{S}^{(1,1)}=\pi_{S}^{0}+\frac{2k\sigma^2}{(2+s)(2k(2+\gamma)-1)} $
$ (1,0) $ $ \omega^{(1,0)}=\omega^0+\frac{(2k+k\gamma-1)Y_1}{(1+s)(2k(2+\gamma)-1)} $ $ \pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\frac{k^2\sigma^2}{(1+s)(2k(2+\gamma)-1)^2} $
$ x^{(1,0)}=x^0+\frac{Y_1}{(1+s)(2k(2+\gamma)-1)} $ $ \pi_{R_2}^{(1,0)}=\pi_{R}^{0} $
$ q_1^{(1,0)}=q^0+\frac{kY_1}{(1+s)(2k(2+\gamma)-1)} $ $ +\frac{(4k^2(2+5s)+4ks\gamma(4k+k\gamma-1)-(8k-1)s)\sigma^2}{4(1+s)(2+s)(2k(2+\gamma)-1)^2} $
$ q_2^{(1,0)}=q^0+\frac{(1+2ks-2k\gamma)Y_1}{2(1+s)(2+s)(2k(2+\gamma)-1)}+\frac{Y_2}{2(2+s)} $ $ \pi_{S}^{(1,0)}=\pi_{S}^{0}+\frac{k\sigma^2}{(1+s)(2k(2+\gamma)-1)} $
Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1}, \quad x^0=\frac{\alpha-c}{2k(2+\gamma)-1}, \quad q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1}, $
$\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2}, \qquad \pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2}$
$ n $ Decisions Ex-ante profits
$ (0,0) $ $ \omega^{(0,0)}=\omega^0 $ $ \pi_{R_i}^{(0,0)}=\pi_{R}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2} $
$ x^{(0,0)}=x^0 $
$ q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma} $ $ \pi_{S}^{(0,0)}=\pi_{S}^{0} $
$ (1,1) $ $ \omega^{(1,1)}=\omega^0+\frac{(2k+k\gamma-1)(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)} $ $ \pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2\sigma^2}{(2+s)(2k(2+\gamma)-1)^2} $
$ x^{(1,1)}=x^0+\frac{Y_1+Y_2}{(2+s)(2k(2+\gamma)-1)} $
$ q_i^{(1,1)}=q^0+\frac{k(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)} $ $ \pi_{S}^{(1,1)}=\pi_{S}^{0}+\frac{2k\sigma^2}{(2+s)(2k(2+\gamma)-1)} $
$ (1,0) $ $ \omega^{(1,0)}=\omega^0+\frac{(2k+k\gamma-1)Y_1}{(1+s)(2k(2+\gamma)-1)} $ $ \pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\frac{k^2\sigma^2}{(1+s)(2k(2+\gamma)-1)^2} $
$ x^{(1,0)}=x^0+\frac{Y_1}{(1+s)(2k(2+\gamma)-1)} $ $ \pi_{R_2}^{(1,0)}=\pi_{R}^{0} $
$ q_1^{(1,0)}=q^0+\frac{kY_1}{(1+s)(2k(2+\gamma)-1)} $ $ +\frac{(4k^2(2+5s)+4ks\gamma(4k+k\gamma-1)-(8k-1)s)\sigma^2}{4(1+s)(2+s)(2k(2+\gamma)-1)^2} $
$ q_2^{(1,0)}=q^0+\frac{(1+2ks-2k\gamma)Y_1}{2(1+s)(2+s)(2k(2+\gamma)-1)}+\frac{Y_2}{2(2+s)} $ $ \pi_{S}^{(1,0)}=\pi_{S}^{0}+\frac{k\sigma^2}{(1+s)(2k(2+\gamma)-1)} $
Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1}, \quad x^0=\frac{\alpha-c}{2k(2+\gamma)-1}, \quad q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1}, $
$\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2}, \qquad \pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2}$
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