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March  2022, 18(2): 1365-1391. doi: 10.3934/jimo.2021024

Effects of disruption risk on a supply chain with a risk-averse retailer

1. 

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

2. 

Department of Electronic Business, South China University of Technology, Guangzhou, Guangdong 510006, China

3. 

School of Management, Fudan University, Shanghai 200433, China

* Corresponding author: Wei Wang

Received  February 2020 Revised  December 2020 Published  March 2022 Early access  February 2021

Fund Project: The authors contributed equally. This work was partially supported by National Nature Science Foundation of China (No. 71531005, No. 72072036), the Fundamental Research Funds for the Central Universities, SCUT (No. D2193040), Guangdong University Characteristic Innovation Project (No. 2017WTSCX002) and Guangdong Natural Science Foundation Doctoral Research Project (No. B6180990)

This paper studies a supply chain consisting of two unreliable suppliers and a retailer, where the two suppliers' default risks are correlated. We use a mean-variance function to characterize the retailer's risk aversion. In the case of exogenous wholesale prices, we find that the retailer's risk aversion has a non-monotonic effect on its total ordering quantity. We also show that when the suppliers' default correlation increases, the retailer's total ordering quantity is non-increasing. In the case of endogenous wholesale prices, we find that the profits of the suppliers and the retailer are non-monotonic in retailer's risk aversion level or suppliers' default correlation. As risk aversion level increases, the retailer becomes less sensitive to wholesale prices. Finally, the numerical results indicate that when the suppliers' delivery rates are different, the supplier with a low delivery rate can benefit from the retailer's risk aversion under certain conditions.

Citation: Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1365-1391. doi: 10.3934/jimo.2021024
References:
[1]

D. Adelman and S. Wang, Supply disruption with a risk-averse buyer, Chicago Booth Research Paper, working paper, (2013). doi: 10.2139/ssrn.2847279.

[2]

R. Anupindi and R. Akella, Diversification under supply uncertainty, Management Science, 39 (1993), 944-963.  doi: 10.1287/mnsc.39.8.944.

[3]

A. AryaB. Mittendorf and D. E. M. Sappington, The bright side of supplier encroachment, Marketing Science, 26 (2007), 651-659.  doi: 10.1287/mksc.1070.0280.

[4]

V. BabichA. N. Burnetas and P. H. Ritchken, Competition and diversification effects in supply chains with supplier default risk, Manufacturing and Service Operations Management, 9 (2007), 123-146. 

[5]

P. D. BergerA. Gerstenfeld and A. Z. Zeng, How many suppliers are best? A decision-analysis approach, Omega, 32 (2004), 9-15.  doi: 10.1016/j.omega.2003.09.001.

[6]

W. Chen and Y. Zhe, Using backup supply with responsive pricing to mitigate disruption risk for a risk-averse firm, International Journal of Production Research, 56 (2018), 1-17. 

[7]

K. ChoiR. Narasimhan and S. W. Kim, Postponement strategy for international transfer of products in a global supply chain: A system dynamics examination, Journal of Operations Management, 30 (2012), 167-179. 

[8]

T. M. Choi and C. H. Chiu, Mean-downside-risk and mean-variance newsvendor models: Implications for sustainable fashion retailing, International Journal of Production Economics, 135 (2012), 552-560.  doi: 10.1016/j.ijpe.2010.10.004.

[9]

S. DasD. DuffieN. Kapadia and L. Saita, Common failings: How corporate defaults are correlated, The Journal of Finance, 62 (2007), 93-117.  doi: 10.3386/w11961.

[10]

A. de Servigny and O. Renault, Default correlation: Empirical evidence, Standard and Poors Risk Solutions.

[11]

A. Federgruen and N. Yang, Procurement strategies with unreliable suppliers, Operations Research, 59 (2011), 1033-1039.  doi: 10.1287/opre.1110.0935.

[12]

B. C. Giri and S. Bardhan, Coordinating a supply chain under uncertain demand and random yield in presence of supply disruption, International Journal of Production Research, 53 (2015), 5070-5084.  doi: 10.1080/00207543.2015.1030469.

[13]

X. GongX. Chao and S. Zheng, Dynamic pricing and inventory management with dual suppliers of different leadtimes and disruption risks, Production and Operations Management, 23 (2015), 2058-2074. 

[14]

V. GuptaB. He and S. P. Sethi, Contingent sourcing under supply disruption and competition, International Journal of Production Research, 53 (2015), 3006-3027.  doi: 10.1080/00207543.2014.965351.

[15]

V. Gupta, D. Ivanov and T. Choi, Competitive pricing of substitute products under supply disruption, Omega, 102279.

[16]

K. Hendricks and V. R. Singhal, Supply chain glitches and operating performance, Management Science, 51 (2005), 695-711. 

[17]

Hongo and Kubota, Disaster strands thousands in Japan, The Wall Street Journal.

[18]

T. H. P. U. N. RaghavanS. Kumara and R. Albert, Survivability of multiagent-based supply networks: A topological perspect, IEEE Intelligent Systems, 19 (2004), 24-31. 

[19]

X. HuH. Gurnani and W. Ling, Managing risk of supply disruptions: Incentives for capacity restoration, Production and Operations Management, 22 (2013), 137-150.  doi: 10.1111/j.1937-5956.2012.01342.x.

[20]

S. G. KhokharQ. Min and C. Su, Bird flu (h7n9) outbreak and its implications on the supply chain of poultry meat in China, Journal of Applied Poultry Research, 24 (2015), 215-221.  doi: 10.3382/japr/pfv007.

[21]

M. Lij. ZhangY. Xu and W. Wang, Effect of disruption risk on a supply chain with price-dependent demand, Journal of Industrial and Management Optimization, 16 (2020), 3083-3103.  doi: 10.3934/jimo.2019095.

[22]

Z. LiS. M. Gilbert and G. Lai, Supplier encroachment under asymmetric information, Management Science, 60 (2013), 449-462. 

[23]

D. J. Lucas, Default correlation and credit analysis, Journal of Fixed Income, 4 (1995), 76-87.  doi: 10.3905/jfi.1995.408124.

[24]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. 

[25]

H. Markowitz, Meancvariance approximations to expected utility, European Journal of Operational Research, 234 (2014), 346-355.  doi: 10.1016/j.ejor.2012.08.023.

[26]

Y. Merzifonluoglu and Y. Feng, Newsvendor problem with multiple unreliable suppliers, International Journal of Production Research, 52 (2014), 221-242.  doi: 10.1080/00207543.2013.835497.

[27]

Z. Pi, W. Fang and B. Zhang, Service and pricing strategies with competition and cooperation in a dual-channel supply chain with demand disruption, Computers and Industrial Engineering, 138 (2019), 106130. doi: 10.1016/j.cie.2019.106130.

[28]

M. R. PowersA. BandyopadhyayT. Chherawala and A. Saha, Calibrating asset correlation for indian corporate exposures, Journal of Risk Finance, 8 (2007), 330-348. 

[29]

L. Qi and Z. J. M. Shen, A supply chain design model with unreliable supply, Naval Research Logistics, 54 (2010), 829-844.  doi: 10.1002/nav.20255.

[30]

P. Ray and M. Jenamani, Sourcing decision with correlated supplier disruption: An mv framework, IEEE International Conference on Industrial Engineering and Engineering Management, 2015 (2015), 868-871.  doi: 10.1109/IEEM.2014.7058762.

[31]

J. Rubio-Herrero and M. Baykal-Grsoy, Mean-variance analysis of the newsvendor problem with price-dependent, isoelastic demand, European Journal of Operational Research, 283 (2020), 942-953.  doi: 10.1016/j.ejor.2019.11.064.

[32]

A. J. Ruiz-Torres and F. Mahmoodi, The optimal number of suppliers considering the costs of individual supplier failures, Omega, 35 (2007), 104-115. 

[33]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 404-420.  doi: 10.1287/mnsc.46.3.404.12070.

[34]

B. Shen and Q. Li, Market disruptions in supply chains: A review of operational models, International Transactions in Operational Research, 24 (2017), 697-711.  doi: 10.1111/itor.12333.

[35]

D. P. SongJ. X. Dong and J. Xu, Integrated inventory management and supplier base reduction in a supply chain with multiple uncertainties, European Journal of Operational Research, 232 (2014), 522-536.  doi: 10.1016/j.ejor.2013.07.044.

[36]

R. Swinney and S. Netessine, Long-term contracts under the threat of supplier default, Manufacturing and Service Operations Management, 11 (2009), 109-127.  doi: 10.1287/msom.1070.0199.

[37]

C. Tang and B. Tomlin, The power of flexibility for mitigating supply chain risks, International Journal of Production Economics, 116 (2016), 12-27.  doi: 10.1007/s10479-018-2840-0.

[38]

C. S. Tang, Perspectives in supply chain risk management, International Journal of Production Economics, 103 (2006), 451-488.  doi: 10.1016/j.ijpe.2005.12.006.

[39]

B. Tomlin, On the value of mitigation and contingency strategies for managing supply chain disruption risks, Management Science, 52 (2006), 639-657.  doi: 10.1287/mnsc.1060.0515.

[40]

S. M. Wagner and J. L. Johnson, Configuring and managing strategic supplier portfolios, 33 (2004), 717–730. doi: 10.1016/j.indmarman.2004.01.005.

[41]

Y. WangW. Gilland and B. Tomlin, Mitigating supply risk: Dual sourcing or process improvement?, Manufacturing and Service Operations Management, 12 (2010), 489-510.  doi: 10.1287/msom.1090.0279.

[42]

J. WuJ. LiS. Wang and T. C. E. Cheng, Mean-variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009), 724-730.  doi: 10.1016/j.omega.2008.02.005.

[43]

T. Xiao and X. Qi, Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers, Omega, 36 (2008), 741-753.  doi: 10.1016/j.omega.2006.02.008.

[44]

Z. YangG. AydinV. Babich and D. Beil, Using a dual-sourcing option in the presence of asymmetric information about supplier reliability: Competition vs. diversification, Manufacturing and Service Operations Management, 14 (2012), 202-217. 

[45]

C. Zhou, An analysis of default correlations and multiple defaults, Review of Financial Studies, 14 (2001), 555-576.  doi: 10.1093/rfs/14.2.555.

[46]

W. ZhuoL. Shao and H. Yang, Meancvariance analysis of option contracts in a two-echelon supply chain, European Journal of Operational Research, 271 (2018), 535-547.  doi: 10.1016/j.ejor.2018.05.033.

show all references

References:
[1]

D. Adelman and S. Wang, Supply disruption with a risk-averse buyer, Chicago Booth Research Paper, working paper, (2013). doi: 10.2139/ssrn.2847279.

[2]

R. Anupindi and R. Akella, Diversification under supply uncertainty, Management Science, 39 (1993), 944-963.  doi: 10.1287/mnsc.39.8.944.

[3]

A. AryaB. Mittendorf and D. E. M. Sappington, The bright side of supplier encroachment, Marketing Science, 26 (2007), 651-659.  doi: 10.1287/mksc.1070.0280.

[4]

V. BabichA. N. Burnetas and P. H. Ritchken, Competition and diversification effects in supply chains with supplier default risk, Manufacturing and Service Operations Management, 9 (2007), 123-146. 

[5]

P. D. BergerA. Gerstenfeld and A. Z. Zeng, How many suppliers are best? A decision-analysis approach, Omega, 32 (2004), 9-15.  doi: 10.1016/j.omega.2003.09.001.

[6]

W. Chen and Y. Zhe, Using backup supply with responsive pricing to mitigate disruption risk for a risk-averse firm, International Journal of Production Research, 56 (2018), 1-17. 

[7]

K. ChoiR. Narasimhan and S. W. Kim, Postponement strategy for international transfer of products in a global supply chain: A system dynamics examination, Journal of Operations Management, 30 (2012), 167-179. 

[8]

T. M. Choi and C. H. Chiu, Mean-downside-risk and mean-variance newsvendor models: Implications for sustainable fashion retailing, International Journal of Production Economics, 135 (2012), 552-560.  doi: 10.1016/j.ijpe.2010.10.004.

[9]

S. DasD. DuffieN. Kapadia and L. Saita, Common failings: How corporate defaults are correlated, The Journal of Finance, 62 (2007), 93-117.  doi: 10.3386/w11961.

[10]

A. de Servigny and O. Renault, Default correlation: Empirical evidence, Standard and Poors Risk Solutions.

[11]

A. Federgruen and N. Yang, Procurement strategies with unreliable suppliers, Operations Research, 59 (2011), 1033-1039.  doi: 10.1287/opre.1110.0935.

[12]

B. C. Giri and S. Bardhan, Coordinating a supply chain under uncertain demand and random yield in presence of supply disruption, International Journal of Production Research, 53 (2015), 5070-5084.  doi: 10.1080/00207543.2015.1030469.

[13]

X. GongX. Chao and S. Zheng, Dynamic pricing and inventory management with dual suppliers of different leadtimes and disruption risks, Production and Operations Management, 23 (2015), 2058-2074. 

[14]

V. GuptaB. He and S. P. Sethi, Contingent sourcing under supply disruption and competition, International Journal of Production Research, 53 (2015), 3006-3027.  doi: 10.1080/00207543.2014.965351.

[15]

V. Gupta, D. Ivanov and T. Choi, Competitive pricing of substitute products under supply disruption, Omega, 102279.

[16]

K. Hendricks and V. R. Singhal, Supply chain glitches and operating performance, Management Science, 51 (2005), 695-711. 

[17]

Hongo and Kubota, Disaster strands thousands in Japan, The Wall Street Journal.

[18]

T. H. P. U. N. RaghavanS. Kumara and R. Albert, Survivability of multiagent-based supply networks: A topological perspect, IEEE Intelligent Systems, 19 (2004), 24-31. 

[19]

X. HuH. Gurnani and W. Ling, Managing risk of supply disruptions: Incentives for capacity restoration, Production and Operations Management, 22 (2013), 137-150.  doi: 10.1111/j.1937-5956.2012.01342.x.

[20]

S. G. KhokharQ. Min and C. Su, Bird flu (h7n9) outbreak and its implications on the supply chain of poultry meat in China, Journal of Applied Poultry Research, 24 (2015), 215-221.  doi: 10.3382/japr/pfv007.

[21]

M. Lij. ZhangY. Xu and W. Wang, Effect of disruption risk on a supply chain with price-dependent demand, Journal of Industrial and Management Optimization, 16 (2020), 3083-3103.  doi: 10.3934/jimo.2019095.

[22]

Z. LiS. M. Gilbert and G. Lai, Supplier encroachment under asymmetric information, Management Science, 60 (2013), 449-462. 

[23]

D. J. Lucas, Default correlation and credit analysis, Journal of Fixed Income, 4 (1995), 76-87.  doi: 10.3905/jfi.1995.408124.

[24]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. 

[25]

H. Markowitz, Meancvariance approximations to expected utility, European Journal of Operational Research, 234 (2014), 346-355.  doi: 10.1016/j.ejor.2012.08.023.

[26]

Y. Merzifonluoglu and Y. Feng, Newsvendor problem with multiple unreliable suppliers, International Journal of Production Research, 52 (2014), 221-242.  doi: 10.1080/00207543.2013.835497.

[27]

Z. Pi, W. Fang and B. Zhang, Service and pricing strategies with competition and cooperation in a dual-channel supply chain with demand disruption, Computers and Industrial Engineering, 138 (2019), 106130. doi: 10.1016/j.cie.2019.106130.

[28]

M. R. PowersA. BandyopadhyayT. Chherawala and A. Saha, Calibrating asset correlation for indian corporate exposures, Journal of Risk Finance, 8 (2007), 330-348. 

[29]

L. Qi and Z. J. M. Shen, A supply chain design model with unreliable supply, Naval Research Logistics, 54 (2010), 829-844.  doi: 10.1002/nav.20255.

[30]

P. Ray and M. Jenamani, Sourcing decision with correlated supplier disruption: An mv framework, IEEE International Conference on Industrial Engineering and Engineering Management, 2015 (2015), 868-871.  doi: 10.1109/IEEM.2014.7058762.

[31]

J. Rubio-Herrero and M. Baykal-Grsoy, Mean-variance analysis of the newsvendor problem with price-dependent, isoelastic demand, European Journal of Operational Research, 283 (2020), 942-953.  doi: 10.1016/j.ejor.2019.11.064.

[32]

A. J. Ruiz-Torres and F. Mahmoodi, The optimal number of suppliers considering the costs of individual supplier failures, Omega, 35 (2007), 104-115. 

[33]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 404-420.  doi: 10.1287/mnsc.46.3.404.12070.

[34]

B. Shen and Q. Li, Market disruptions in supply chains: A review of operational models, International Transactions in Operational Research, 24 (2017), 697-711.  doi: 10.1111/itor.12333.

[35]

D. P. SongJ. X. Dong and J. Xu, Integrated inventory management and supplier base reduction in a supply chain with multiple uncertainties, European Journal of Operational Research, 232 (2014), 522-536.  doi: 10.1016/j.ejor.2013.07.044.

[36]

R. Swinney and S. Netessine, Long-term contracts under the threat of supplier default, Manufacturing and Service Operations Management, 11 (2009), 109-127.  doi: 10.1287/msom.1070.0199.

[37]

C. Tang and B. Tomlin, The power of flexibility for mitigating supply chain risks, International Journal of Production Economics, 116 (2016), 12-27.  doi: 10.1007/s10479-018-2840-0.

[38]

C. S. Tang, Perspectives in supply chain risk management, International Journal of Production Economics, 103 (2006), 451-488.  doi: 10.1016/j.ijpe.2005.12.006.

[39]

B. Tomlin, On the value of mitigation and contingency strategies for managing supply chain disruption risks, Management Science, 52 (2006), 639-657.  doi: 10.1287/mnsc.1060.0515.

[40]

S. M. Wagner and J. L. Johnson, Configuring and managing strategic supplier portfolios, 33 (2004), 717–730. doi: 10.1016/j.indmarman.2004.01.005.

[41]

Y. WangW. Gilland and B. Tomlin, Mitigating supply risk: Dual sourcing or process improvement?, Manufacturing and Service Operations Management, 12 (2010), 489-510.  doi: 10.1287/msom.1090.0279.

[42]

J. WuJ. LiS. Wang and T. C. E. Cheng, Mean-variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009), 724-730.  doi: 10.1016/j.omega.2008.02.005.

[43]

T. Xiao and X. Qi, Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers, Omega, 36 (2008), 741-753.  doi: 10.1016/j.omega.2006.02.008.

[44]

Z. YangG. AydinV. Babich and D. Beil, Using a dual-sourcing option in the presence of asymmetric information about supplier reliability: Competition vs. diversification, Manufacturing and Service Operations Management, 14 (2012), 202-217. 

[45]

C. Zhou, An analysis of default correlations and multiple defaults, Review of Financial Studies, 14 (2001), 555-576.  doi: 10.1093/rfs/14.2.555.

[46]

W. ZhuoL. Shao and H. Yang, Meancvariance analysis of option contracts in a two-echelon supply chain, European Journal of Operational Research, 271 (2018), 535-547.  doi: 10.1016/j.ejor.2018.05.033.

Figure 1.  Retailer's optimal order quantities given $ (w_1, w_2) $
Figure 2.  Retailer's optimal order quantities given $ (w_1, w_2) $ with different risk aversion levels
Figure 3.  Retailer's optimal order quantities given $ (w_1, w_2) $ with different default correlations
Figure 4.  Retailer's optimal order quantities with the increase of the default correlation or the risk aversion level
Figure 5.  Suppliers' profits with the increase of the default correlation for different risk aversion levels
Figure 6.  Suppliers' profits with the increase of the risk aversion level for different default correlations
Figure 7.  Examples for the optimal wholesale prices in $ \Omega_2 $ and $ \Omega_3 $
Figure 8.  Retailer's optimal order quantities given $ (w_1, w_2) $ when $ \alpha>\beta $
Figure 9.  Suppliers' profits with the increase of the risk aversion level
Figure 10.  Suppliers' profits with the increase of the default correlation
Table 1.  Equilibrium results when $ \alpha = 0.5 $ and $ p_{00} = \beta $
$ k $ $ w_1^* $ $ w_2* $ $ Q_1^* $ $ Q_2^* $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.00031 1.38 0 100 100 138 0
0.00041 1.268 0.044 96.6463 3.3537 122.5476 0.1476
0.00061 1.5613 0.3907 79.9863 20.0137 124.8853 7.8187
0.00071 1.708 0.564 75.1761 24.8239 128.4007 14.0007
0.00081 1.6 0.48 69.1358 24.6914 110.6173 11.8519
$ k $ $ w_1^* $ $ w_2* $ $ Q_1^* $ $ Q_2^* $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.00031 1.38 0 100 100 138 0
0.00041 1.268 0.044 96.6463 3.3537 122.5476 0.1476
0.00061 1.5613 0.3907 79.9863 20.0137 124.8853 7.8187
0.00071 1.708 0.564 75.1761 24.8239 128.4007 14.0007
0.00081 1.6 0.48 69.1358 24.6914 110.6173 11.8519
Table 2.  Equilibrium results when $ \alpha = 0.9 $ and $ k = 0.0034 $
$ p_{00} $ $ w_1^* $ $ w_2* $ $ Q_1^* $ $ Q_2^* $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.271 4.4964 1.4750 73.4740 10.3295 330.3661 15.2359
0.272 4.4926 1.4499 73.4243 10.1556 329.8676 14.7248
0.273 8.9749 0.7090 59.1954 41.6294 531.2713 29.5148
0.274 8.9815 0.683 59.0538 41.5197 530.3940 28.3595
0.275 8.9881 0.6571 58.9137 41.4115 529.5238 27.2100
$ p_{00} $ $ w_1^* $ $ w_2* $ $ Q_1^* $ $ Q_2^* $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.271 4.4964 1.4750 73.4740 10.3295 330.3661 15.2359
0.272 4.4926 1.4499 73.4243 10.1556 329.8676 14.7248
0.273 8.9749 0.7090 59.1954 41.6294 531.2713 29.5148
0.274 8.9815 0.683 59.0538 41.5197 530.3940 28.3595
0.275 8.9881 0.6571 58.9137 41.4115 529.5238 27.2100
Table 3.  Equilibrium results when $ \alpha = 0.7 $ and $ p_{00} = 0.27 $
$ k $ $ w_1^* $ $ w_2* $ $ Q_1^* $ $ Q_2^* $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.0009 3.2376 0.3073 74.3118 59.4039 240.5899 18.2533
0.0010 3.3548 0.3184 69.3784 55.4602 232.7506 17.6585
0.0011 3.2083 0.9583 74.9360 22.3835 240.4198 21.4508
0.0012 3.2083 0.9583 68.7430 20.5336 220.5504 19.6781
0.0013 3.2083 0.9583 63.4954 18.9662 203.7145 18.1759
$ k $ $ w_1^* $ $ w_2* $ $ Q_1^* $ $ Q_2^* $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.0009 3.2376 0.3073 74.3118 59.4039 240.5899 18.2533
0.0010 3.3548 0.3184 69.3784 55.4602 232.7506 17.6585
0.0011 3.2083 0.9583 74.9360 22.3835 240.4198 21.4508
0.0012 3.2083 0.9583 68.7430 20.5336 220.5504 19.6781
0.0013 3.2083 0.9583 63.4954 18.9662 203.7145 18.1759
Table 4.  Equilibrium results when $ \alpha = 0.5 $ and $ p_{00} = \beta $
$ k $ $ g_1 $ $ g_2 $ $ c $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.0004 0.5 0.5 0.15 130 0
0.0005 0.5 0.5 0.15 128.4028 1.7361
0.0006 0.5 0.5 0.15 130.0208 6.0208
0.0007 0.5 0.5 0.15 133.0972 11.7639
0.0008 0.5 0.5 0.15 118.2447 11.1747
$ k $ $ g_1 $ $ g_2 $ $ c $ $ \pi_{S_1}^* $ $ \pi_{S_2}^* $
0.0004 0.5 0.5 0.15 130 0
0.0005 0.5 0.5 0.15 128.4028 1.7361
0.0006 0.5 0.5 0.15 130.0208 6.0208
0.0007 0.5 0.5 0.15 133.0972 11.7639
0.0008 0.5 0.5 0.15 118.2447 11.1747
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