# American Institute of Mathematical Sciences

January  2022, 18(1): 635-654. 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  January 2022 Early access  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 and Management Optimization, 2022, 18 (1) : 635-654. doi: 10.3934/jimo.2020172
##### References:
 [1] C. Alós-Ferrer, J. 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. [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. [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. [4] D. Bertsimas, V. F. Farias and N. Trichakis, The price of fairness, Operations Research, 59 (2011), 17-31.  doi: 10.1287/opre.1100.0865. [5] D. Bertsimas, V. F. Farias and N. Trichakis, On the efficiency-fairness trade-off, Management Science, 58 (2012), 2234-2250. [6] M. Draganska, D. 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. [7] D. Ertel, Turning negotiation into a corporate capability, Harvard Business Review, 77 (1999), 55-71. [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. [9] C. A. Ingene, S. 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. [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. [11] A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272. [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. [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. [14] W. S. Lovejoy, Bargaining chains, Management Science, 56 (2010), 2282-2301.  doi: 10.1287/mnsc.1100.1251. [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. [16] M. Nagarajan and G. Sošić, Coalition stability in assembly models, Operations Research, 57 (2009), 131-145. [17] J. F. Nash, The bargaining problem, Econometrica: Journal of the Econometric Society, 18 (1950), 155-162.  doi: 10.2307/1907266. [18] T. Nguyen, Local bargaining and supply chain instability, Operations Research, 65 (2017), 1535-1545.  doi: 10.1287/opre.2017.1605. [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. [20] M. Pervin, S. 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. [21] M. Pervin, S. 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. [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. [23] W. Thomson, Bargaining and the Theory of Cooperative Games: John Nash and Beyond, Edward Elgar Publishing, Cheltenham, UK / Northampton, MA, USA 2010. [24] H. Yang, Q. Yan, H. 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. [25] F. Zhong, J. 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. [26] F. Zhong, J. Xie, X. 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.

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##### References:
 [1] C. Alós-Ferrer, J. 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. [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. [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. [4] D. Bertsimas, V. F. Farias and N. Trichakis, The price of fairness, Operations Research, 59 (2011), 17-31.  doi: 10.1287/opre.1100.0865. [5] D. Bertsimas, V. F. Farias and N. Trichakis, On the efficiency-fairness trade-off, Management Science, 58 (2012), 2234-2250. [6] M. Draganska, D. 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. [7] D. Ertel, Turning negotiation into a corporate capability, Harvard Business Review, 77 (1999), 55-71. [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. [9] C. A. Ingene, S. 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. [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. [11] A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272. [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. [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. [14] W. S. Lovejoy, Bargaining chains, Management Science, 56 (2010), 2282-2301.  doi: 10.1287/mnsc.1100.1251. [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. [16] M. Nagarajan and G. Sošić, Coalition stability in assembly models, Operations Research, 57 (2009), 131-145. [17] J. F. Nash, The bargaining problem, Econometrica: Journal of the Econometric Society, 18 (1950), 155-162.  doi: 10.2307/1907266. [18] T. Nguyen, Local bargaining and supply chain instability, Operations Research, 65 (2017), 1535-1545.  doi: 10.1287/opre.2017.1605. [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. [20] M. Pervin, S. 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. [21] M. Pervin, S. 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. [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. [23] W. Thomson, Bargaining and the Theory of Cooperative Games: John Nash and Beyond, Edward Elgar Publishing, Cheltenham, UK / Northampton, MA, USA 2010. [24] H. Yang, Q. Yan, H. 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. [25] F. Zhong, J. 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. [26] F. Zhong, J. Xie, X. 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.
Generalized KS solution for two players
The multi-echelon supply chain
$n = 3$ and $d = 1$
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