# American Institute of Mathematical Sciences

July  2018, 14(3): 953-966. doi: 10.3934/jimo.2017084

## Solutions for bargaining games with incomplete information: General type space and action space

 1 School of Business Administration, Hunan University, Changsha 410082, China 2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 3 School of Economics and Management, Northwest University, Xi'an 710127, China

* Corresponding author

Received  March 2016 Revised  September 2016 Published  July 2018 Early access  September 2017

Fund Project: This work has been supported by the National Natural Science Foundation of China under Projects Nos. 71210002 and 71671099. The authors are grateful to the anonymous referees for their constructive comments and suggestions.

A Nash bargaining solution for Bayesian collective choice problem with general type and action spaces is built in this paper. Such solution generalizes the bargaining solution proposed by Myerson who uses finite sets to characterize the type and action spaces. However, in the real economics and industries, types and actions can hardly be characterized by a finite set in some circumstances. Hence our generalization expands the applications of bargaining theory in economic and industrial models.

Citation: Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2018, 14 (3) : 953-966. doi: 10.3934/jimo.2017084
##### References:
 [1] X. Brusset and P. J. Agrell, Intrinsic impediments to category captainship collaboration, Journal of Industrial and Management Optimization, 13 (2017), 113-133.  doi: 10.3934/jimo.2016007.  Google Scholar [2] W. S. Chang, B. Chen and T. C. Salmon, An investigation of the average bid mechanism for procurement auctions, Management Science, 61 (2015), 1237-1254.  doi: 10.1287/mnsc.2013.1893.  Google Scholar [3] 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 [4] B. Holmström and R. B. Myerson, Efficient and durable decision rules with incomplete information, Econometrica, 51 (1983), 1799-1819.   Google Scholar [5] M. Huang, X. Qian, S. C. Fang and X. Wang, Winner determination for risk aversion buyers in multi-attribute reverse auction, Omega, 59 (2016), 184-200.  doi: 10.1016/j.omega.2015.06.007.  Google Scholar [6] E. Kalai and M. Smorodinsky, Other solutions to Nash's bargaining problem, Econometrica, 43 (1975), 513-518.  doi: 10.2307/1914280.  Google Scholar [7] T. Kruse and P. Strack, Optimal stopping with private information, Journal of Economic Theory, 159 (2015), 702-727.  doi: 10.1016/j.jet.2015.03.001.  Google Scholar [8] R. B. Myerson, Incentive compatibility and the bargaining problem, Econometrica, 47 (1979), 61-73.  doi: 10.2307/1912346.  Google Scholar [9] R. B. Myerson, Cooperative games with imcomplete information, International Journal of Game Theory, 13 (1984), 69-96.  doi: 10.1007/BF01769817.  Google Scholar [10] R. B. Myerson, Two-person bargaining problems with incomplete information, Econometrica, 52 (1984), 461-487.  doi: 10.2307/1911499.  Google Scholar [11] J. F. Nash, The bargaining problem, Econometrica, 18 (1950), 155-162.  doi: 10.2307/1907266.  Google Scholar [12] Ö. Özer and W. Wei, Strategic commitments for an optimal capacity decision under asymmetric forecast information, Management Science, 52 (2006), 1238-1257.   Google Scholar [13] 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 [14] H. L. Royden and P. Fitzpatrick, Real Analysis, 3$^{ed}$ edition, Macmillan, New York, 1988.   Google Scholar [15] F. Weidner, The generalized Nash bargaining solution and incentive compatible mechanisms, International Journal of Game Theory, 21 (1992), 109-129.  doi: 10.1007/BF01245455.  Google Scholar

show all references

##### References:
 [1] X. Brusset and P. J. Agrell, Intrinsic impediments to category captainship collaboration, Journal of Industrial and Management Optimization, 13 (2017), 113-133.  doi: 10.3934/jimo.2016007.  Google Scholar [2] W. S. Chang, B. Chen and T. C. Salmon, An investigation of the average bid mechanism for procurement auctions, Management Science, 61 (2015), 1237-1254.  doi: 10.1287/mnsc.2013.1893.  Google Scholar [3] 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 [4] B. Holmström and R. B. Myerson, Efficient and durable decision rules with incomplete information, Econometrica, 51 (1983), 1799-1819.   Google Scholar [5] M. Huang, X. Qian, S. C. Fang and X. Wang, Winner determination for risk aversion buyers in multi-attribute reverse auction, Omega, 59 (2016), 184-200.  doi: 10.1016/j.omega.2015.06.007.  Google Scholar [6] E. Kalai and M. Smorodinsky, Other solutions to Nash's bargaining problem, Econometrica, 43 (1975), 513-518.  doi: 10.2307/1914280.  Google Scholar [7] T. Kruse and P. Strack, Optimal stopping with private information, Journal of Economic Theory, 159 (2015), 702-727.  doi: 10.1016/j.jet.2015.03.001.  Google Scholar [8] R. B. Myerson, Incentive compatibility and the bargaining problem, Econometrica, 47 (1979), 61-73.  doi: 10.2307/1912346.  Google Scholar [9] R. B. Myerson, Cooperative games with imcomplete information, International Journal of Game Theory, 13 (1984), 69-96.  doi: 10.1007/BF01769817.  Google Scholar [10] R. B. Myerson, Two-person bargaining problems with incomplete information, Econometrica, 52 (1984), 461-487.  doi: 10.2307/1911499.  Google Scholar [11] J. F. Nash, The bargaining problem, Econometrica, 18 (1950), 155-162.  doi: 10.2307/1907266.  Google Scholar [12] Ö. Özer and W. Wei, Strategic commitments for an optimal capacity decision under asymmetric forecast information, Management Science, 52 (2006), 1238-1257.   Google Scholar [13] 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 [14] H. L. Royden and P. Fitzpatrick, Real Analysis, 3$^{ed}$ edition, Macmillan, New York, 1988.   Google Scholar [15] F. Weidner, The generalized Nash bargaining solution and incentive compatible mechanisms, International Journal of Game Theory, 21 (1992), 109-129.  doi: 10.1007/BF01245455.  Google Scholar
 [1] 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, 2020  doi: 10.3934/jimo.2020172 [2] Wai-Ki Ching, Jia-Wen Gu, Harry Zheng. On correlated defaults and incomplete information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 889-908. doi: 10.3934/jimo.2020003 [3] Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219 [4] Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379 [5] Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096 [6] Narges Torabi Golsefid, Maziar Salahi. Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021032 [7] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [8] Chady Ghnatios, Guangtao Xu, Adrien Leygue, Michel Visonneau, Francisco Chinesta, Alain Cimetiere. On the space separated representation when addressing the solution of PDE in complex domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 475-500. doi: 10.3934/dcdss.2016008 [9] Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085 [10] Miquel Oliu-Barton. Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights. Journal of Dynamics & Games, 2019, 6 (4) : 259-275. doi: 10.3934/jdg.2019018 [11] Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061 [12] Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511 [13] Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021104 [14] Yoshitsugu Kabeya. A unified approach to Matukuma type equations on the hyperbolic space or on a sphere. Conference Publications, 2013, 2013 (special) : 385-391. doi: 10.3934/proc.2013.2013.385 [15] Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28 [16] Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261 [17] Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007 [18] Zhiping Zhou, Yao Yin, Mi Zhou, Hao Cheng, Panos M. Pardalos. Equity-based incentive to coordinate shareholder-manager interests under information asymmetry. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021167 [19] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [20] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

2020 Impact Factor: 1.801