American Institute of Mathematical Sciences

Advanced
May  2021, 17(3): 1505-1529. doi: 10.3934/jimo.2020032

The comparison between selling and leasing for new and remanufactured products with quality level in the electric vehicle industry

Kai Li 1,2, , Tao Zhou 1,, and  Bohai Liu 1,3,

1. 

School of Management, Hefei University of Technology, Hefei, China
2. 

Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei, China
3. 

Research Center of Industrial Transfer and Innovation Development, Hefei University of Technology, Hefei, China

* Corresponding author: Tao Zhou

Received  May 2019 Revised  September 2019 Published  May 2021 Early access  February 2020

Fund Project: The first author is supported by National Natural Science Foundation of China under grants 71871076, 71521001, 71531008, 71690235

Process uncertainty makes remanufacturing operations management sophisticated. To reduce the uncertainty of the timing, quality and quantity of product returns in remanufacturing, motivated by the selling and leasing of electric vehicle batteries, we consider a monopolist vendor who markets her products by adopting two models: (1) a single leasing model, and (2) a single selling model. We first investigate the firm's marketing model with remanufacturing and analyze the impact of the quality level of the returned products on the firm's marketing and remarketing models. Then we compare selling and leasing models. We first find that only when the quality level of returned sold products is relatively high will the vendor choose to remanufacture under the single selling model. Conversely, only when the quality level of returned leased products is relatively low will the vendor decide to remanufacture under the single leasing model. Secondly, we show that the space of remanufacturable quality level under the single selling model is bigger than the space under the single leasing model. Thirdly, selling is more profitable than leasing when the quality level of returned sold products is sufficiently high. These results are further demonstrated by a numerical study. Our study provides firms with guidance on how to optimally adopt remanufacturing and marketing strategies that take into account the quality level of the returned products.

Keywords: Selling, leasing, remanufacturing, quality level, electric vehicle battery.
Mathematics Subject Classification: Primary: 90B60; Secondary: 91B16, 91B24.
Citation: Kai Li, Tao Zhou, Bohai Liu. The comparison between selling and leasing for new and remanufactured products with quality level in the electric vehicle industry. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1505-1529. doi: 10.3934/jimo.2020032
References:
[1]

J. D. Abbey, V. D. R. Guide Jr. and G. C. Souza, Delayed differentiation for multiple lifecycle products, Prod. Oper. Manag., 22 (2013), 588-602.  doi: 10.1111/j.1937-5956.2012.01370.x.

[2]

V. V. Agrawal, M. Ferguson, L. B. Toktay and V. M. Thomas, Is leasing greener than selling?, Manage. Sci., 58 (2012), 523-533.  doi: 10.1287/mnsc.1110.1428.

[3]

N. Aras, R. Güllü and S. Yürülmez, Optimal inventory and pricing policies for remanufacturable leased products, Int. J. Prod. Econ., 133 (2011), 262-271.  doi: 10.1016/j.ijpe.2010.01.024.

[4]

A. Atasu, V. D. R. Guide Jr. and L. N. Van Wassenhove, So what if remanufacturing cannibalizes my new product sales?, Calif. Manage. Rev., 52 (2010), 1-21.  doi: 10.1525/cmr.2010.52.2.56.

[5]

A. Atasu and G. C. Souza, How does product recovery affect quality choice?, Prod. Oper. Manag., 22 (2013), 991-1010.  doi: 10.1111/j.1937-5956.2011.01290.x.

[6]

B. Avci, K. Girotra and S. Netessine, Electric vehicles with a battery switching station: Adoption and environmental impact, Manage. Sci., 61 (2015), 772-794.  doi: 10.1287/mnsc.2014.1916.

[7]

S. R. Bhaskaran and S. M. Gilbert, Implications of channel structure and operational mode upon a manufacturer's durability choice, Prod. Oper. Manag., 24 (2015), 1071-1085.  doi: 10.1111/poms.12335.

[8]

S. R. Bhaskaran and S. M. Gilbert, Implications of channel structure for leasing or selling durable goods, Market. Sci., 28 (2009), 918-934.  doi: 10.1287/mksc.1080.0458.

[9]

J. I. Bulow, Durable-goods monopolists, J. Polit. Econ., 90 (1982), 314-332.  doi: 10.1086/261058.

[10]

P. Desai and D. Purohit, Competition in durable goods markets: The strategic consequences of leasing and selling, Market. Sci., 18 (1999), 42-58.  doi: 10.1287/mksc.18.1.42.

[11]

P. Desai and D. Purohit, Leasing and selling: Optimal marketing strategies for a durable goods firm, Manage. Sci., 44 (1998), S19–S34. doi: 10.1287/mnsc.44.11.S19.

[12]

M. Ferguson, V. D. R. Guide Jr., E. Koca and G. C. Souza, The value of quality grading in remanufacturing, Prod. Oper. Manag., 18 (2009), 300-314.  doi: 10.1111/j.1937-5956.2009.01033.x.

[13]

G. Ferrer and J. M. Swaminathan, Managing new and remanufactured products, Manage. Sci., 52 (2006), 15-26.  doi: 10.1287/mnsc.1050.0465.

[14]

V. D. R. Guide Jr., R. H. Teunter and L. N. Van Wassenhove, Matching demand and supply to maximize profits from remanufacturing, Manuf. Serv. Oper. Manag., 5 (2003), 303-316. 

[15]

E. Huang, China buys one out of every two electric vehicles sold globally, Centre for Solar Energy and Hydrogen Research Baden-Württemberg, Germany, 2019. Available from: https://qz.com/1552991/china-buys-one-out-of-every-two-electric-vehicles-sold-globally/.

[16]

K. J. Li and S. H. Xu, The comparison between trade-in and leasing of a product with technology innovations, Omega, 54 (2015), 134-146.  doi: 10.1016/j.omega.2015.01.018.

[17]

Z. Li and M. Ouyang, A win-win marginal rent analysis for operator and consumer under battery leasing mode in China electric vehicle market, Energ. Policy, 39 (2011), 3222-3237.  doi: 10.1016/j.enpol.2011.03.014.

[18]

M. K. Lim, H. Y. Mak and Y. Rong, Toward mass adoption of electric vehicles: Impacts of the range and resale anxieties, Manuf. Serv. Oper. Manag., 17 (2015), 101-119.  doi: 10.2139/ssrn.2236560.

[19]

P. V. Loon, C. Delagarde and L. N. Van Wassenhove, The role of second-hand markets in circular business: A simple model for leasing versus selling consumer products, Int. J. Prod. Res., 56 (2018), 960-973.  doi: 10.1080/00207543.2017.1398429.

[20]

P. Majumder and H. Groenevelt, Competition in remanufacturing, Prod. Oper. Manag., 10 (2001), 125-141. 

[21]

O. Mont, C. Dalhammar and N. Jacobsson, A new business model for baby prams based on leasing and product remanufacturing, J. Clean. Prod., 14 (2006), 1509-1518.  doi: 10.1016/j.jclepro.2006.01.024.

[22]

K. S. Moorthy, Product and price competition in a duopoly, Market. Sci., 7 (1988), 141-168.  doi: 10.1287/mksc.7.2.141.

[23]

A. Mutha, S. Bansal and V. D. R. Guide Jr., Managing demand uncertainty through core acquisition in remanufacturing, Prod. Oper. Manag., 25 (2016), 1449-1464.  doi: 10.1111/poms.12554.

[24]

J. Östlin, E. Sundin and M. Björkman, Importance of closed-loop supply chain relationships for product remanufacturing, Int. J. Prod. Econ., 115 (2008), 336-348.  doi: 10.1016/j.ijpe.2008.02.020.

[25]

A. Robotis, S. Bhattacharya and L. N. Van Wassenhove, Lifecycle pricing for installed base management with constrained capacity and remanufacturing, Prod. Oper. Manag., 21 (2012), 236-252.  doi: 10.1111/j.1937-5956.2011.001262.x.

[26]

G. C. Souza, Closed-loop supply chains: A critical review, and future research, Decision Sci., 44 (2013), 7-38.  doi: 10.1111/j.1540-5915.2012.00394.x.

[27]

W. R. Stahel, The Circular Economy, Routledge, London, 2019. doi: 10.4324/9780429259203.

[28]

C. R. Standridge and L. Corneal, Remanufacturing, repurposing, and recycling of post-vehicle-application lithium-ion batteries, Mineta Transportation Institute, San José, 2014. Available from: https://rosap.ntl.bts.gov/view/dot/27425.

[29]

C. R. Standridge, L. Corneal and N. Baine, Advances in repurposing and recycling of post-vehicle-application lithium-ion batteries, Mineta Transportation Institute, San José, 2016. Available from: http://transweb.sjsu.edu/project/1238.html.

[30]

D. W. Steeneck and S. C. Sarin, Product design for leased products under remanufacturing, Int. J. Prod. Econ., 202 (2018), 132-144.  doi: 10.1016/j.ijpe.2018.04.025.

[31]

M. Thierry, M. Salomon, J. Van Nunen and L. N. Van Wassenhove, Strategic issues in product recovery management, Calif. Manage. Rev., 37 (1995), 114-135.  doi: 10.2307/41165792.

[32]

B. K. Thorn and P. Rogerson, Take it back, IIE Solutions, 34 (2002), 34-40. 

[33]

M. W. Toffel, Strategic management of product recovery, Calif. Manage. Rev., 46 (2004), 120-141.  doi: 10.2307/41166214.

[34]

L. B. Toktay and D. Wei, Cost allocation in manufacturing-remanufacturing operations, Prod. Oper. Manag., 20 (2011), 841-847.  doi: 10.1111/j.1937-5956.2011.01236.x.

[35]

T. Tse, M. Esposito and K. Soufani, How businesses can support a circular economy, Harvard. Bus. Rev., 2 (2016), 1-6. 

[36]

M. Waldman, Eliminating the market for secondhand goods: An alternative explanation for leasing, J. Law. Econ., 40 (1997), 61-92.  doi: 10.1086/467366.

[37]

L. Wang, G. Cai, A. A. Tsay and A. J. Vakharia, Design of the reverse channel for remanufacturing: Must profit-maximization harm the environment?, Prod. Oper. Manag., 26 (2017), 1585-1603.  doi: 10.1111/poms.12709.

[38]

WEEE Forum, 2012 Annual Report, European Association of Electric and Electronic Waste Take-Back Systems, 2013. Available from: http://www.weee-forum.org/what-is-the-weee-forum/annual_report_2012_final.pdf.

[39]

B. Yalabik, D. Chhajed and N. C. Petruzzi, Product and sales contract design in remanufacturing, Int. J. Prod. Econ., 154 (2014), 299-312.  doi: 10.1016/j.ijpe.2013.09.008.

[40]

W. Zhang, Q. Zhang, K. J. Mizgier and Y. Zhang, Integrating the customers' perceived risks and benefits into the triple-channel retailing, Int. J. Prod. Res., 55 (2017), 6676-6690.  doi: 10.1080/00207543.2017.1336679.

[41]

Y. Zhang, C. Y. Wang and X. Tang, Cycling degradation of an automotive LiFePO_4, lithium-ion battery, J. Power Sources, 196 (2011), 1513-1520.  doi: 10.1016/j.jpowsour.2010.08.070.

show all references

References:
[1]

J. D. Abbey, V. D. R. Guide Jr. and G. C. Souza, Delayed differentiation for multiple lifecycle products, Prod. Oper. Manag., 22 (2013), 588-602.  doi: 10.1111/j.1937-5956.2012.01370.x.

[2]

V. V. Agrawal, M. Ferguson, L. B. Toktay and V. M. Thomas, Is leasing greener than selling?, Manage. Sci., 58 (2012), 523-533.  doi: 10.1287/mnsc.1110.1428.

[3]

N. Aras, R. Güllü and S. Yürülmez, Optimal inventory and pricing policies for remanufacturable leased products, Int. J. Prod. Econ., 133 (2011), 262-271.  doi: 10.1016/j.ijpe.2010.01.024.

[4]

A. Atasu, V. D. R. Guide Jr. and L. N. Van Wassenhove, So what if remanufacturing cannibalizes my new product sales?, Calif. Manage. Rev., 52 (2010), 1-21.  doi: 10.1525/cmr.2010.52.2.56.

[5]

A. Atasu and G. C. Souza, How does product recovery affect quality choice?, Prod. Oper. Manag., 22 (2013), 991-1010.  doi: 10.1111/j.1937-5956.2011.01290.x.

[6]

B. Avci, K. Girotra and S. Netessine, Electric vehicles with a battery switching station: Adoption and environmental impact, Manage. Sci., 61 (2015), 772-794.  doi: 10.1287/mnsc.2014.1916.

[7]

S. R. Bhaskaran and S. M. Gilbert, Implications of channel structure and operational mode upon a manufacturer's durability choice, Prod. Oper. Manag., 24 (2015), 1071-1085.  doi: 10.1111/poms.12335.

[8]

S. R. Bhaskaran and S. M. Gilbert, Implications of channel structure for leasing or selling durable goods, Market. Sci., 28 (2009), 918-934.  doi: 10.1287/mksc.1080.0458.

[9]

J. I. Bulow, Durable-goods monopolists, J. Polit. Econ., 90 (1982), 314-332.  doi: 10.1086/261058.

[10]

P. Desai and D. Purohit, Competition in durable goods markets: The strategic consequences of leasing and selling, Market. Sci., 18 (1999), 42-58.  doi: 10.1287/mksc.18.1.42.

[11]

P. Desai and D. Purohit, Leasing and selling: Optimal marketing strategies for a durable goods firm, Manage. Sci., 44 (1998), S19–S34. doi: 10.1287/mnsc.44.11.S19.

[12]

M. Ferguson, V. D. R. Guide Jr., E. Koca and G. C. Souza, The value of quality grading in remanufacturing, Prod. Oper. Manag., 18 (2009), 300-314.  doi: 10.1111/j.1937-5956.2009.01033.x.

[13]

G. Ferrer and J. M. Swaminathan, Managing new and remanufactured products, Manage. Sci., 52 (2006), 15-26.  doi: 10.1287/mnsc.1050.0465.

[14]

V. D. R. Guide Jr., R. H. Teunter and L. N. Van Wassenhove, Matching demand and supply to maximize profits from remanufacturing, Manuf. Serv. Oper. Manag., 5 (2003), 303-316. 

[15]

E. Huang, China buys one out of every two electric vehicles sold globally, Centre for Solar Energy and Hydrogen Research Baden-Württemberg, Germany, 2019. Available from: https://qz.com/1552991/china-buys-one-out-of-every-two-electric-vehicles-sold-globally/.

[16]

K. J. Li and S. H. Xu, The comparison between trade-in and leasing of a product with technology innovations, Omega, 54 (2015), 134-146.  doi: 10.1016/j.omega.2015.01.018.

[17]

Z. Li and M. Ouyang, A win-win marginal rent analysis for operator and consumer under battery leasing mode in China electric vehicle market, Energ. Policy, 39 (2011), 3222-3237.  doi: 10.1016/j.enpol.2011.03.014.

[18]

M. K. Lim, H. Y. Mak and Y. Rong, Toward mass adoption of electric vehicles: Impacts of the range and resale anxieties, Manuf. Serv. Oper. Manag., 17 (2015), 101-119.  doi: 10.2139/ssrn.2236560.

[19]

P. V. Loon, C. Delagarde and L. N. Van Wassenhove, The role of second-hand markets in circular business: A simple model for leasing versus selling consumer products, Int. J. Prod. Res., 56 (2018), 960-973.  doi: 10.1080/00207543.2017.1398429.

[20]

P. Majumder and H. Groenevelt, Competition in remanufacturing, Prod. Oper. Manag., 10 (2001), 125-141. 

[21]

O. Mont, C. Dalhammar and N. Jacobsson, A new business model for baby prams based on leasing and product remanufacturing, J. Clean. Prod., 14 (2006), 1509-1518.  doi: 10.1016/j.jclepro.2006.01.024.

[22]

K. S. Moorthy, Product and price competition in a duopoly, Market. Sci., 7 (1988), 141-168.  doi: 10.1287/mksc.7.2.141.

[23]

A. Mutha, S. Bansal and V. D. R. Guide Jr., Managing demand uncertainty through core acquisition in remanufacturing, Prod. Oper. Manag., 25 (2016), 1449-1464.  doi: 10.1111/poms.12554.

[24]

J. Östlin, E. Sundin and M. Björkman, Importance of closed-loop supply chain relationships for product remanufacturing, Int. J. Prod. Econ., 115 (2008), 336-348.  doi: 10.1016/j.ijpe.2008.02.020.

[25]

A. Robotis, S. Bhattacharya and L. N. Van Wassenhove, Lifecycle pricing for installed base management with constrained capacity and remanufacturing, Prod. Oper. Manag., 21 (2012), 236-252.  doi: 10.1111/j.1937-5956.2011.001262.x.

[26]

G. C. Souza, Closed-loop supply chains: A critical review, and future research, Decision Sci., 44 (2013), 7-38.  doi: 10.1111/j.1540-5915.2012.00394.x.

[27]

W. R. Stahel, The Circular Economy, Routledge, London, 2019. doi: 10.4324/9780429259203.

[28]

C. R. Standridge and L. Corneal, Remanufacturing, repurposing, and recycling of post-vehicle-application lithium-ion batteries, Mineta Transportation Institute, San José, 2014. Available from: https://rosap.ntl.bts.gov/view/dot/27425.

[29]

C. R. Standridge, L. Corneal and N. Baine, Advances in repurposing and recycling of post-vehicle-application lithium-ion batteries, Mineta Transportation Institute, San José, 2016. Available from: http://transweb.sjsu.edu/project/1238.html.

[30]

D. W. Steeneck and S. C. Sarin, Product design for leased products under remanufacturing, Int. J. Prod. Econ., 202 (2018), 132-144.  doi: 10.1016/j.ijpe.2018.04.025.

[31]

M. Thierry, M. Salomon, J. Van Nunen and L. N. Van Wassenhove, Strategic issues in product recovery management, Calif. Manage. Rev., 37 (1995), 114-135.  doi: 10.2307/41165792.

[32]

B. K. Thorn and P. Rogerson, Take it back, IIE Solutions, 34 (2002), 34-40. 

[33]

M. W. Toffel, Strategic management of product recovery, Calif. Manage. Rev., 46 (2004), 120-141.  doi: 10.2307/41166214.

[34]

L. B. Toktay and D. Wei, Cost allocation in manufacturing-remanufacturing operations, Prod. Oper. Manag., 20 (2011), 841-847.  doi: 10.1111/j.1937-5956.2011.01236.x.

[35]

T. Tse, M. Esposito and K. Soufani, How businesses can support a circular economy, Harvard. Bus. Rev., 2 (2016), 1-6. 

[36]

M. Waldman, Eliminating the market for secondhand goods: An alternative explanation for leasing, J. Law. Econ., 40 (1997), 61-92.  doi: 10.1086/467366.

[37]

L. Wang, G. Cai, A. A. Tsay and A. J. Vakharia, Design of the reverse channel for remanufacturing: Must profit-maximization harm the environment?, Prod. Oper. Manag., 26 (2017), 1585-1603.  doi: 10.1111/poms.12709.

[38]

WEEE Forum, 2012 Annual Report, European Association of Electric and Electronic Waste Take-Back Systems, 2013. Available from: http://www.weee-forum.org/what-is-the-weee-forum/annual_report_2012_final.pdf.

[39]

B. Yalabik, D. Chhajed and N. C. Petruzzi, Product and sales contract design in remanufacturing, Int. J. Prod. Econ., 154 (2014), 299-312.  doi: 10.1016/j.ijpe.2013.09.008.

[40]

W. Zhang, Q. Zhang, K. J. Mizgier and Y. Zhang, Integrating the customers' perceived risks and benefits into the triple-channel retailing, Int. J. Prod. Res., 55 (2017), 6676-6690.  doi: 10.1080/00207543.2017.1336679.

[41]

Y. Zhang, C. Y. Wang and X. Tang, Cycling degradation of an automotive LiFePO_4, lithium-ion battery, J. Power Sources, 196 (2011), 1513-1520.  doi: 10.1016/j.jpowsour.2010.08.070.

Figure 1.  Optimal quantity as a function of $ v_{ir} $, given that $ c $ and $ \gamma $
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Figure 2.  Optimal price as a function of $ v_{ir} $, given that $ c $ and $ \gamma $
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Figure 3.  Optimal profits as a function of $ v_{ir} $, given that $ c $ and $ \gamma $
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Figure 4.  Optimal quantity as a function of $ c $, given that $ v_{ij} $ and $ \gamma $
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Figure 5.  Optimal price as a function of $ c $, given that $ v_{ij} $ and $ \gamma $
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Figure 6.  Optimal profits as a function of $ c $, given that $ v_{ij} $ and $ \gamma $
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Table 1.  Some Key Literature on Remanufacturing and Marketing Strategies
Marketing strategy
Selling Leasing Combination of selling and Leasing
Remanufacturing strategy Remanufacturing Product performance (durability, depreciation, quality, etc.) Atasu and Souza [5] Mont et al. [21]
Robotis et al. [25]
Steeneck and Sarin [30]
Product returns (reverse logistics, acquisition policy, inventory, etc.) Guide et al. [14]
Mutha et al. [23]
Östlin et al. [24]
Wang et al. [37]		 Aras et al. [3]
Loon et al. [19]
Yalabik et al. [39]
No remanufacturing Product performance Agrawal et al. [2] Desai and Purohit [11]
Product returns (secondary market or discarding) Waldman [36]
Table Options

Download as excel

Marketing strategy
Selling Leasing Combination of selling and Leasing
Remanufacturing strategy Remanufacturing Product performance (durability, depreciation, quality, etc.) Atasu and Souza [5] Mont et al. [21]
Robotis et al. [25]
Steeneck and Sarin [30]
Product returns (reverse logistics, acquisition policy, inventory, etc.) Guide et al. [14]
Mutha et al. [23]
Östlin et al. [24]
Wang et al. [37]		 Aras et al. [3]
Loon et al. [19]
Yalabik et al. [39]
No remanufacturing Product performance Agrawal et al. [2] Desai and Purohit [11]
Product returns (secondary market or discarding) Waldman [36]
Table 2.  Summary of Notation
Symbol Definition
$ \gamma $ A discount factor signifying the value of future cash flows
$ v_{ij} $ The quality level for product $ j $ under decision $ i $
$ U^{m}_{kij} $ The utility of a customer for product $ j $ of decision $ i $ in period $ k $ under market model $ m $
$ c(v_{ij}) $ The marginal cost of product $ j $ of decision $ i $
$ p^{m}_{kij} $ The one-period price for the product $ j $ of decision $ i $ in period $ k $ under market model $ m $
$ q^{m}_{kij} $ The period $ k $ demand for product $ j $ of decision $ i $ under market model $ m $
$ \pi^{m}_{k} $ The profits under market model $ m $ in period $ k $
$ \pi^{m} $ The total profits under market model $ m $
Table Options

Download as excel

Symbol Definition
$ \gamma $ A discount factor signifying the value of future cash flows
$ v_{ij} $ The quality level for product $ j $ under decision $ i $
$ U^{m}_{kij} $ The utility of a customer for product $ j $ of decision $ i $ in period $ k $ under market model $ m $
$ c(v_{ij}) $ The marginal cost of product $ j $ of decision $ i $
$ p^{m}_{kij} $ The one-period price for the product $ j $ of decision $ i $ in period $ k $ under market model $ m $
$ q^{m}_{kij} $ The period $ k $ demand for product $ j $ of decision $ i $ under market model $ m $
$ \pi^{m}_{k} $ The profits under market model $ m $ in period $ k $
$ \pi^{m} $ The total profits under market model $ m $
Table 3.  Optimal Solutions in Each Model
Solutions of parties Model L Model S
$ q^{m\ast}_{1i n} $ $ \frac{1-c}{2} $ $ \frac{1-c}{2}+\frac{\gamma(v_{sr}+c(1-v^{2}_{sr}))}{4} $
$ q^{m\ast}_{2i n} $ $ \frac{1-v_{lr}-cv_{lr}-cv^{2}_{lr}}{2(1-v_{lr})} $ $ \frac{1-v_{sr}-cv^{2}_{sr}}{2(1-v_{sr})} $
$ q^{m\ast}_{2i r} $ $ \frac{c(v^{2}_{lr}+2v_{lr}-1)}{2v_{lr}(1-v_{lr})} $ $ \frac{c(v^{2}_{sr}+v_{sr}-1)}{2v_{sr}(1-v_{sr})} $
$ p^{m\ast}_{1i n} $ $ \frac{1+c}{2} $ $ \frac{1+c}{2}+\frac{\gamma(v_{sr}+c(1-v^{2}_{sr}))}{4} $
$ p^{m\ast}_{2i n} $ $ \frac{1+c}{2} $ $ \frac{1+c}{2} $
$ p^{m\ast}_{2i r} $ $ \frac{v+c(1-v_{lr}-v^{2}_{lr})}{2} $ $ \frac{v_{sr}+c(1-v^{2}_{sr})}{2} $
$ \pi^{m\ast}_{2} $ $ \frac{1-2c-6c^2-5c^2v_{lr}-c^2v^{2}_{lr}}{4}+\frac{c^2(1+3v_{lr})}{4v_{lr}(1-v_{lr})} $ $ \frac{1-2c-c^2-3c^2v_{sr}-c^2v^{2}_{sr}}{4}+\frac{c^2}{4v_{sr}(1-v_{sr})} $
$ \pi^{m\ast} $ $ \frac{(1-c)^2}{4}+\gamma(\frac{1-2c-6c^2-5c^2v_{lr}-c^2v^{2}_{lr}}{4}+\frac{c^2(1+3v_{lr})}{4v_{lr}(1-v_{lr})}) $ $ (\frac{1-c}{2}+\frac{r(v_{sr}+c(1-v^{2}_{sr}))}{4})^2+\gamma(\frac{1-2c-c^2-3c^2v_{sr}-c^2v^{2}_{sr}}{4}+\frac{c^2}{4v_{sr}(1-v_{sr})}) $
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Solutions of parties Model L Model S
$ q^{m\ast}_{1i n} $ $ \frac{1-c}{2} $ $ \frac{1-c}{2}+\frac{\gamma(v_{sr}+c(1-v^{2}_{sr}))}{4} $
$ q^{m\ast}_{2i n} $ $ \frac{1-v_{lr}-cv_{lr}-cv^{2}_{lr}}{2(1-v_{lr})} $ $ \frac{1-v_{sr}-cv^{2}_{sr}}{2(1-v_{sr})} $
$ q^{m\ast}_{2i r} $ $ \frac{c(v^{2}_{lr}+2v_{lr}-1)}{2v_{lr}(1-v_{lr})} $ $ \frac{c(v^{2}_{sr}+v_{sr}-1)}{2v_{sr}(1-v_{sr})} $
$ p^{m\ast}_{1i n} $ $ \frac{1+c}{2} $ $ \frac{1+c}{2}+\frac{\gamma(v_{sr}+c(1-v^{2}_{sr}))}{4} $
$ p^{m\ast}_{2i n} $ $ \frac{1+c}{2} $ $ \frac{1+c}{2} $
$ p^{m\ast}_{2i r} $ $ \frac{v+c(1-v_{lr}-v^{2}_{lr})}{2} $ $ \frac{v_{sr}+c(1-v^{2}_{sr})}{2} $
$ \pi^{m\ast}_{2} $ $ \frac{1-2c-6c^2-5c^2v_{lr}-c^2v^{2}_{lr}}{4}+\frac{c^2(1+3v_{lr})}{4v_{lr}(1-v_{lr})} $ $ \frac{1-2c-c^2-3c^2v_{sr}-c^2v^{2}_{sr}}{4}+\frac{c^2}{4v_{sr}(1-v_{sr})} $
$ \pi^{m\ast} $ $ \frac{(1-c)^2}{4}+\gamma(\frac{1-2c-6c^2-5c^2v_{lr}-c^2v^{2}_{lr}}{4}+\frac{c^2(1+3v_{lr})}{4v_{lr}(1-v_{lr})}) $ $ (\frac{1-c}{2}+\frac{r(v_{sr}+c(1-v^{2}_{sr}))}{4})^2+\gamma(\frac{1-2c-c^2-3c^2v_{sr}-c^2v^{2}_{sr}}{4}+\frac{c^2}{4v_{sr}(1-v_{sr})}) $
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