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doi: 10.3934/jimo.2020172

Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution

1. 

School of Business Administration, Hunan University, Changsha, 410082, China

2. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

3. 

No.1 Middle School Attached to Central China Normal University, Wuhan, 430223, China

* Corresponding author: Feimin Zhong

Received  November 2019 Revised  October 2020 Published  December 2020

This paper studies a multi-echelon serial supply chain with negotiations over wholesale prices between successive echelons. Two types of bargaining systems with power structures are compared: one adopts the generalized Kalai-Smorodinsky (KS) solution and the other adopts the generalized Nash solution. Our analyses show that, for any KS bargaining system with a given bargaining power structure, there is a Nash bargaining system with another bargaining power structure, such that the two systems are the same. However under the same power structure, the generalized KS solution results in lower wholesale price and higher total supply chain profit than the Nash solution does. Finally, we characterize the necessary and sufficient condition of the bargaining power structure under which the KS bargaining system Pareto dominates the Nash bargaining system, and the set characterized by such condition does not shrink to an empty set as the number of echelons increases to infinity.

Citation: Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020172
References:
[1]

C. Alós-FerrerJ. García-Segarra and M. Ginés-Vilar, Anchoring on utopia: A generalization of the Kalai-Smorodinsky solution, Economic Theory Bulletin, 6 (2018), 141-155.  doi: 10.1007/s40505-017-0130-7.  Google Scholar

[2]

M. Bennett, When do retail price communications between retailers and manufacturers become RPM?, CRA Competition Memo Vertical Communications and RPM, Last accessed June 30, 2020. Google Scholar

[3]

F. Bernstein and M. Nagarajan, Competition and cooperative bargaining models in supply chains, Foundations and Trends (R) in Technology, Information and Operations Management, 5 (2012), 87-145.  doi: 10.1561/9781601985576.  Google Scholar

[4]

D. BertsimasV. F. Farias and N. Trichakis, The price of fairness, Operations Research, 59 (2011), 17-31.  doi: 10.1287/opre.1100.0865.  Google Scholar

[5]

D. BertsimasV. F. Farias and N. Trichakis, On the efficiency-fairness trade-off, Management Science, 58 (2012), 2234-2250.   Google Scholar

[6]

M. DraganskaD. Klapper and S. B. Villas-Boas, A larger slice or a larger pie? An empirical investigation of bargaining power in the distribution channel, Marketing Science, 29 (2010), 57-74.   Google Scholar

[7]

D. Ertel, Turning negotiation into a corporate capability, Harvard Business Review, 77 (1999), 55-71.   Google Scholar

[8]

J. C. Harsanyi and R. Selten, A generalized Nash solution for two-person bargaining games with incomplete information, Management Science, 18 (1972), 80-106.  doi: 10.1287/mnsc.18.5.80.  Google Scholar

[9]

C. A. IngeneS. Taboubi and G. Zaccour, Game-theoretic coordination mechanisms in distribution channels: Integration and extensions for models without competition, Journal of Retailing, 88 (2012), 476-496.  doi: 10.1016/j.jretai.2012.04.002.  Google Scholar

[10]

G. Iyer and J. M. Villas-Boas, A bargaining theory of distribution channels, Journal of Marketing Research, 40 (2003), 80-100.  doi: 10.1509/jmkr.40.1.80.19134.  Google Scholar

[11]

A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272.   Google Scholar

[12]

E. Kalai and M. Smorodinsky, Other solutions to Nash's bargaining problem, Econometrica: Journal of the Econometric Society, 43 (1975), 513-518.  doi: 10.2307/1914280.  Google Scholar

[13]

C.-L. Li, Quantifying supply chain ineffectiveness under uncoordinated pricing decisions, Operations Research Letters, 36 (2008), 83-88.  doi: 10.1016/j.orl.2007.04.005.  Google Scholar

[14]

W. S. Lovejoy, Bargaining chains, Management Science, 56 (2010), 2282-2301.  doi: 10.1287/mnsc.1100.1251.  Google Scholar

[15]

M. Nagarajan and G. Sosic, Game-theoretic analysis of cooperation among supply chain agents: Review and extensions, European Journal of Operational Research, 187 (2008), 719-745.  doi: 10.1016/j.ejor.2006.05.045.  Google Scholar

[16]

M. Nagarajan and G. Sošić, Coalition stability in assembly models, Operations Research, 57 (2009), 131-145.   Google Scholar

[17]

J. F. Nash, The bargaining problem, Econometrica: Journal of the Econometric Society, 18 (1950), 155-162.  doi: 10.2307/1907266.  Google Scholar

[18]

T. Nguyen, Local bargaining and supply chain instability, Operations Research, 65 (2017), 1535-1545.  doi: 10.1287/opre.2017.1605.  Google Scholar

[19]

M. A. Perles and M. Maschler, The super-additive solution for the Nash bargaining game, International Journal of Game Theory, 10 (1981), 163-193.  doi: 10.1007/BF01755963.  Google Scholar

[20]

M. PervinS. K. Roy and G. Weber, Deteriorating inventory with preservation technology under price- and stock-sensitive demand, Journal of Industrial and Management Optimization, 16 (2020), 1585-1612.  doi: 10.3934/jimo.2019019.  Google Scholar

[21]

M. PervinS. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098.  Google Scholar

[22]

J. Svejnar, Bargaining power, fear of disagreement, and wage settlements: Theory and evidence from US industry, Econometrica: Journal of the Econometric Society, 54 (1986), 1055-1078.   Google Scholar

[23]

W. Thomson, Bargaining and the Theory of Cooperative Games: John Nash and Beyond, Edward Elgar Publishing, Cheltenham, UK / Northampton, MA, USA 2010. Google Scholar

[24]

H. YangQ. YanH. Wan and W. Zhuo, Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer, Journal of Industrial and Management Optimization, 16 (2020), 2723-2741.   Google Scholar

[25]

F. ZhongJ. Xie and J. Jiao, Solutions for bargaining games with incomplete information: General type space and action space, Journal of Industrial and Management Optimization, 14 (2018), 953-966.  doi: 10.3934/jimo.2017084.  Google Scholar

[26]

F. ZhongJ. XieX. Zhao and Z. J. M. Shen, On efficiency of multistage channel with bargaining over wholesale prices, Naval Research Logistics, 63 (2016), 449-459.  doi: 10.1002/nav.21713.  Google Scholar

show all references

References:
[1]

C. Alós-FerrerJ. García-Segarra and M. Ginés-Vilar, Anchoring on utopia: A generalization of the Kalai-Smorodinsky solution, Economic Theory Bulletin, 6 (2018), 141-155.  doi: 10.1007/s40505-017-0130-7.  Google Scholar

[2]

M. Bennett, When do retail price communications between retailers and manufacturers become RPM?, CRA Competition Memo Vertical Communications and RPM, Last accessed June 30, 2020. Google Scholar

[3]

F. Bernstein and M. Nagarajan, Competition and cooperative bargaining models in supply chains, Foundations and Trends (R) in Technology, Information and Operations Management, 5 (2012), 87-145.  doi: 10.1561/9781601985576.  Google Scholar

[4]

D. BertsimasV. F. Farias and N. Trichakis, The price of fairness, Operations Research, 59 (2011), 17-31.  doi: 10.1287/opre.1100.0865.  Google Scholar

[5]

D. BertsimasV. F. Farias and N. Trichakis, On the efficiency-fairness trade-off, Management Science, 58 (2012), 2234-2250.   Google Scholar

[6]

M. DraganskaD. Klapper and S. B. Villas-Boas, A larger slice or a larger pie? An empirical investigation of bargaining power in the distribution channel, Marketing Science, 29 (2010), 57-74.   Google Scholar

[7]

D. Ertel, Turning negotiation into a corporate capability, Harvard Business Review, 77 (1999), 55-71.   Google Scholar

[8]

J. C. Harsanyi and R. Selten, A generalized Nash solution for two-person bargaining games with incomplete information, Management Science, 18 (1972), 80-106.  doi: 10.1287/mnsc.18.5.80.  Google Scholar

[9]

C. A. IngeneS. Taboubi and G. Zaccour, Game-theoretic coordination mechanisms in distribution channels: Integration and extensions for models without competition, Journal of Retailing, 88 (2012), 476-496.  doi: 10.1016/j.jretai.2012.04.002.  Google Scholar

[10]

G. Iyer and J. M. Villas-Boas, A bargaining theory of distribution channels, Journal of Marketing Research, 40 (2003), 80-100.  doi: 10.1509/jmkr.40.1.80.19134.  Google Scholar

[11]

A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272.   Google Scholar

[12]

E. Kalai and M. Smorodinsky, Other solutions to Nash's bargaining problem, Econometrica: Journal of the Econometric Society, 43 (1975), 513-518.  doi: 10.2307/1914280.  Google Scholar

[13]

C.-L. Li, Quantifying supply chain ineffectiveness under uncoordinated pricing decisions, Operations Research Letters, 36 (2008), 83-88.  doi: 10.1016/j.orl.2007.04.005.  Google Scholar

[14]

W. S. Lovejoy, Bargaining chains, Management Science, 56 (2010), 2282-2301.  doi: 10.1287/mnsc.1100.1251.  Google Scholar

[15]

M. Nagarajan and G. Sosic, Game-theoretic analysis of cooperation among supply chain agents: Review and extensions, European Journal of Operational Research, 187 (2008), 719-745.  doi: 10.1016/j.ejor.2006.05.045.  Google Scholar

[16]

M. Nagarajan and G. Sošić, Coalition stability in assembly models, Operations Research, 57 (2009), 131-145.   Google Scholar

[17]

J. F. Nash, The bargaining problem, Econometrica: Journal of the Econometric Society, 18 (1950), 155-162.  doi: 10.2307/1907266.  Google Scholar

[18]

T. Nguyen, Local bargaining and supply chain instability, Operations Research, 65 (2017), 1535-1545.  doi: 10.1287/opre.2017.1605.  Google Scholar

[19]

M. A. Perles and M. Maschler, The super-additive solution for the Nash bargaining game, International Journal of Game Theory, 10 (1981), 163-193.  doi: 10.1007/BF01755963.  Google Scholar

[20]

M. PervinS. K. Roy and G. Weber, Deteriorating inventory with preservation technology under price- and stock-sensitive demand, Journal of Industrial and Management Optimization, 16 (2020), 1585-1612.  doi: 10.3934/jimo.2019019.  Google Scholar

[21]

M. PervinS. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098.  Google Scholar

[22]

J. Svejnar, Bargaining power, fear of disagreement, and wage settlements: Theory and evidence from US industry, Econometrica: Journal of the Econometric Society, 54 (1986), 1055-1078.   Google Scholar

[23]

W. Thomson, Bargaining and the Theory of Cooperative Games: John Nash and Beyond, Edward Elgar Publishing, Cheltenham, UK / Northampton, MA, USA 2010. Google Scholar

[24]

H. YangQ. YanH. Wan and W. Zhuo, Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer, Journal of Industrial and Management Optimization, 16 (2020), 2723-2741.   Google Scholar

[25]

F. ZhongJ. Xie and J. Jiao, Solutions for bargaining games with incomplete information: General type space and action space, Journal of Industrial and Management Optimization, 14 (2018), 953-966.  doi: 10.3934/jimo.2017084.  Google Scholar

[26]

F. ZhongJ. XieX. Zhao and Z. J. M. Shen, On efficiency of multistage channel with bargaining over wholesale prices, Naval Research Logistics, 63 (2016), 449-459.  doi: 10.1002/nav.21713.  Google Scholar

Figure 1.  Generalized KS solution for two players
Figure 2.  The multi-echelon supply chain
Figure 3.  $ n = 3 $ and $ d = 1 $
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