Matrix Games with Fuzzy Payoffs

  • Tina VermaEmail author
  • Amit Kumar
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 383)


In this chapter, flaws of the existing methods [5, 10, 11, 12, 14] for solving matrix game with fuzzy payoffs (matrix games in which payoffs are represented as fuzzy numbers) are pointed out. To resolve these flaws, a new method (named as Mehar method) is also proposed to obtain the optimal strategies as well as minimum expected gain of Player I and maximum expected loss of Player II for matrix games with fuzzy payoffs. To illustrate the proposed Mehar method, the existing numerical problems of matrix games with fuzzy payoffs are solved by the proposed Mehar method.


  1. 1.
    Bector, C.R., Chandra, S.: Fuzzy Mathematical Programming and Fuzzy Matrix Games. Springer, Berlin (2005)zbMATHGoogle Scholar
  2. 2.
    Campos, L.: Fuzzy linear programming models to solve fuzzy matrix games. Fuzzy Sets Syst. 32, 275–289 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Campos, L., Gonzalez, A.: Fuzzy matrix games considering the criteria of the players. Kybernetes 20, 17–23 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Campos, L., Gonzalez, A., Vila, M.A.: On the use of the ranking function approach to solve fuzzy matrix games in a direct way. Fuzzy Sets Syst. 49, 193–203 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clemente, M., Fernandez, F.R., Puerto, J.: Pareto-optimal security in matrix games with fuzzy payoffs. Fuzzy Sets Syst. 176, 36–45 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems Theory and Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  7. 7.
    Dutta, B., Gupta, S.K.: On Nash equilibrium strat-egy of two person zero sum games with trapezoidal fuzzy payoffs. Fuzzy Inf. Eng. 6, 299–314 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kaufmann, A., Gupta, M.M: Introduction to Fuzzy Arithmetic Theory and Applications. Van Nostrand Publishing Co (1991)Google Scholar
  9. 9.
    Kaufmann, A., Gupta, M.M.: Fuzzy Mathematical Models in Engineering and Management Science. New York, USA (1988)Google Scholar
  10. 10.
    Li, D.F.: Lexicographic method for matrix games with payoffs of triangular fuzzy numbers. Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 16, 371–389 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Li, D.F.: A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers. Eur. J. Oper. Res. 223, 421–429 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, D.F.: An effective methodology for solving matrix games with fuzzy payoffs. IEEE Trans. Cybern. 43, 610–621 (2013)CrossRefGoogle Scholar
  13. 13.
    Li, D.F.: Linear Programming Models and Methods of Matrix Games with Payoffs of Triangular Fuzzy Numbers. Springer, Berlin (2015)Google Scholar
  14. 14.
    Liu, S.T., Kao, C.: Solution of fuzzy matrix games: an application of the extension principle. Int. J. Intell. Syst. 22, 891–903 (2007)CrossRefGoogle Scholar
  15. 15.
    Liu, S.T., Kao, C.: Matrix games with interval data. Comput. Ind. Eng. 56, 1697–1700 (2009)CrossRefGoogle Scholar
  16. 16.
    Maeda, T.: On characterization of equilibrium strategy of two-person zero-sum games with fuzzy payoffs. Fuzzy Sets Syst. 139, 283–296 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoparRupnagarIndia
  2. 2.School of MathematicsThapar Institute of Engineering and TechnologyPatialaIndia

Personalised recommendations