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On Solving Bimatrix Games with Triangular Fuzzy Payoffs

  • Subrato Chakravorty
  • Debdas GhoshEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 253)

Abstract

The aim of this paper is to introduce the concept of bimatrix fuzzy games. The fuzzy games are defined by payoff matrices constructed using triangular fuzzy numbers. The bimatrix fuzzy game discussed in this paper is different from the one given by Maeda and Cunlin in respect that it is not a zero-sum game and two different payoff matrices are provided for the two players. Three kinds of Nash equilibriums are introduced for fuzzy games, and their existence conditions are studied. A solution method for bimatrix fuzzy games is given using crisp parametric bimatrix games. Finally, a numerical example is discussed to support the model described in the paper.

Keywords

Bimatrix games Nash equilibrium Fuzzy set theory Fuzzy games Non-cooperative games 

References

  1. 1.
    Nash, J.F.: Non-cooperativegames. In: Annals of Mathematics, Second Series, vol. 54, no. 2, pp. 286–295 (1951)Google Scholar
  2. 2.
    Butnariu, D.: Fuzzy games: a description of the concept. Fuzzy Sets Syst. 1(3), 181–192 (1978). JulyMathSciNetCrossRefGoogle Scholar
  3. 3.
    Butnariu, D.: Stability and Shapley value for an n-persons fuzzy game. Fuzzy Sets Syst. 4(1), 63–72 (1980). JulyMathSciNetCrossRefGoogle Scholar
  4. 4.
    Campos, L.: Fuzzylinear programming models to solve fuzzy matrix games. FuzzySets Syst. 32, 27589 (1989)Google Scholar
  5. 5.
    Yager, R.: Ranking fuzzysubsets over the unit interval. In: Proceedings of the CDC, pp. 1435–1437 (1978)Google Scholar
  6. 6.
    Li, D.F.: A fuzzy multiobjective approach to solve fuzzy matrix games. J. Fuzzy Math. 7, 90712 (1999)MathSciNetGoogle Scholar
  7. 7.
    Li, D.F.: A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers. Eur. J. Oper. Res. 223(2), 421–429 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sakawa, M., Nishizaki, I.: Max-min solutions for fuzzy multiobjective matrix games. FuzzySets Syst. 61, 26575 (1994)zbMATHGoogle Scholar
  9. 9.
    Bector, C.R., Chandra, S.: On duality in linear programming under fuzzy environment. FuzzySets Syst. 125, 31725 (2002)MathSciNetGoogle Scholar
  10. 10.
    Vijay, V., Chandra, S., Bector, C.R.: Matrix games with fuzzy goals and fuzzy pay offs. Omega 33, 425429 (2005)CrossRefGoogle Scholar
  11. 11.
    Kacher, F., Larbani, M.: Existence of equilibrium solution for a non-cooperative game with fuzzy goals and parameters. Fuzzy Sets Syst. 159(2), 164–176 (2008). JanuaryMathSciNetCrossRefGoogle Scholar
  12. 12.
    Larbani, M.: Non cooperative fuzzy games in normal form: a survey. Fuzzy Sets Syst. 160(22), 3184–3210 (2009). NovemberMathSciNetCrossRefGoogle Scholar
  13. 13.
    Maeda, T.: On characterization of equilibrium strategy of two person zero-sum games with fuzzy pay offs. FuzzySets Syst. 139, 28396 (2003)Google Scholar
  14. 14.
    Maeda, T.: On characterization of equilibrium strategy of bimatrix games with fuzzy pay offs. J. Math. Anal. Appl. 251, 885896 (2000)CrossRefGoogle Scholar
  15. 15.
    Cunlin, L., Qiang, Z.: Nash equilibrium strategy for fuzzy non-cooperative games. FuzzySets Syst. 176, 4655 (1976)zbMATHGoogle Scholar
  16. 16.
    Dutta, B., Gupta, S.K.: On nash equilibrium strategy of two-person zero-sum games with trapezoidal fuzzy payoffs. Fuzzy Inf. Eng. 6, 299–314 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zadeh, L.A.: Fuzzy Sets. Inf. Control 8(3), 338–352 (1968)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology (BHU)VaranasiIndia

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