Stochastic Nash Equilibrium with a Numerical Solution Method

  • Jinwu Gao
  • Yankui Liu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3496)


Recent decades viewed increasing interests in the subject of decentralized decision-making. In this paper, three definitions of stochastic Nash equilibrium, which casts different decision criteria, are proposed for a stochastic decentralized decision system. Then the problem of how to find the stochastic Nash equilibrium is converted to an optimization problem. Lastly, a solution method combined with neural network and genetic algorithm is provided.


decentralized decision system Nash equilibrium stochastic programming neural network genetic algorithm 


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  1. 1.
    Anupindi, R., Bassok, Y., Zemel, E.: A General Framework for the Study of Decentralized Distribution Systems. Manufacturing & Service Operations Management 3, 349–368 (2001)CrossRefGoogle Scholar
  2. 2.
    Bylka, S.: Competitive and Cooperative Policies for the Vendor-buyer System. International Journal of Production Economics 81-82, 533–544 (2003)CrossRefGoogle Scholar
  3. 3.
    Charnes, A., Cooper, W.W.: Chance-constrained Programming. Management Science 6, 73–79 (1959)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Cybenko, G.: Approximations by Superpositions of a Sigmoidal Function. Mathematics of Control, Signals and Systems 2, 183–192 (1989)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Gao, J., Liu, B., Gen, M.: A Hybrid Intelligent Algorithm for Stochastic Multilevel Programming. Transactions of the Institute of Electrical Engineers of Japan 124-C, 1991–1998 (2004)Google Scholar
  6. 6.
    Goldberg, D.E.: Genetic Algorithms in Search. In: Optimization and Machine Learning, Addison-Wesley, MA (1989)Google Scholar
  7. 7.
    Hirasawa, K., Yamamoto, Y., Hu, J., Murata, J., Jin, C.: Optimization of Decentralized Control System Using Nash Equilibrium Concept. Transactions of the Institute of Electrical Engineers of Japan 119-C, 467–473 (1999)Google Scholar
  8. 8.
    Hornik, K., Stinchcombe, M., White, H.: Multilayer Feedforward Networks are Universal Approximators. Neural Networks 2, 359–366 (1989)CrossRefGoogle Scholar
  9. 9.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)Google Scholar
  10. 10.
    Ji, X., Wang, C.: Nash Equilibrium in the Game of Transmission Expansion. In: Proceedings of 2002 IEEE Region 10 Conference on Computer, Communications, Control and Power Engineering, vol. 3, pp. 1768–1771 (2002)Google Scholar
  11. 11.
    Liu, B.: Dependent-chance Programming: A Class of Stochastic Programming. Comput. Math. Appl. 34, 89–104 (1997)CrossRefMATHGoogle Scholar
  12. 12.
    Vriend, N.J.: A Model of Market-making. European Journal of Economic and Social Systems 15, 185–202 (2002)CrossRefGoogle Scholar
  13. 13.
    Wang, H., Guo, M., Efstathiou, J.: A Game-theoretical Cooperative Mechanism Design for a Two-echelon Decentralized Supply Chain. European Journal of Operational Research 157, 372–388 (2004)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jinwu Gao
    • 1
  • Yankui Liu
    • 2
  1. 1.Uncertain Systems Laboratory, School of InformationRenmin University of ChinaBeijingChina
  2. 2.College of Mathematics and Computer ScienceHebei UniversityBaodingChina

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