International Journal of Fuzzy Systems

, Volume 21, Issue 3, pp 908–915 | Cite as

A Linear Programming Approach to Solve Constrained Bi-matrix Games with Intuitionistic Fuzzy Payoffs

  • Jing-Jing An
  • Deng-Feng LiEmail author


In many real games, two players’ payoffs are not exactly opposite and players often have some constraints or preference on their strategies. Such kinds of games are called constrained bi-matrix games (CBGs) for short. Based on dual programming theory, two linear programming models are developed for solving any CBG. Then, a classic example of bi-matrix games called the Rock-scissors-cloth game with considering players’ preference on strategies is used to show the validity of the proposed models and method. Furthermore, we investigate on the CBGs with payoffs represented by intuitionistic fuzzy numbers, which are simply called intuitionistic fuzzy CBGs in which both the ambiguity of the payoffs and the constraints of the strategies are taken into account. At last, the effectiveness of the proposed models and method is demonstrated with a numerical example of the company development strategy choice problem.


Intuitionistic fuzzy numbers (IFNs) Bi-matrix games (BGs) Constrained bi-matrix games (CBGs) Mathematical programming 


  1. 1.
    Babayigit, C., Rocha, P., Das, T.K.: A two-tier matrix game approach for obtaining joint bidding strategies in FTR and energy markets. IEEE Trans. Power Syst. 25(3), 1211–1219 (2010)CrossRefGoogle Scholar
  2. 2.
    Bhurjee, A.K., Panda, G.: Optimal strategies for two-person normalized matrix game with variable payoffs. Oper. Res. 17(2), 547–562 (2017)Google Scholar
  3. 3.
    Liu, T., Deng, Y., Chan, F.: Evidential supplier selection based on DEMATEL and game theory. Int. J. Fuzzy Syst. 20(2), 1–13 (2017)Google Scholar
  4. 4.
    Chen, L., Peng, J., Liu, Z., Zhao, R.: Pricing and effort decisions for a supply chain with uncertain information. Int. J. Prod. Res. 55(1), 264–284 (2017)CrossRefGoogle Scholar
  5. 5.
    Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Knuth, D.E., Papadimitriou, C.H., Tsitsiklis, J.N.: A note on strategy elimination in bimatrix games. Oper. Res. Lett. 7(3), 103–107 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kontogiannis, S.C., Panagopoulou, P.N., Spirakis, P.G.: Polynomial algorithms for approximating Nash equilibria of bimatrix games. Theor. Comput. Sci. 410(17), 1599–1606 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li, D.F.: Linear programming approach to solve interval-valued matrix games. Omega 39(6), 655–666 (2011)CrossRefGoogle Scholar
  9. 9.
    Chakeri, A., Sheikholeslam, F.: Fuzzy Nash equilibriums in crisp and fuzzy games. IEEE Trans. Fuzzy Syst. 21(1), 171–176 (2013)CrossRefGoogle Scholar
  10. 10.
    Liu, S.T., Kao, C.: Matrix games with interval data. Comput. Ind. Eng. 56(4), 1697–1700 (2009)CrossRefGoogle 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(2), 421–429 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nishizaki, I., Sakawa, M.: Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals. Fuzzy Sets Syst. 111(1), 99–116 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Maeda, T.: Characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff. J. Math. Anal. Appl. 251(2), 885–896 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pang, J.H., Zhang, Q.: The equilibrium strategies of bi-matrix games with interval-valued payoffs. Syst. Eng. 25(4), 114–118 (2007)Google Scholar
  15. 15.
    An, J.J., Li, D.F., Nan, J.X.: A mean-area ranking based non-linear programming approach to solve intuitionistic fuzzy bi-matrix games. J. Intell. Fuzzy Syst. 33(1), 563–573 (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    Nan, J.X., Li, D.F., An, J.J.: Solving bi-matrix games with intuitionistic fuzzy goals and intuitionistic fuzzy payoffs. J. Intell. Fuzzy Syst. 33(6), 3723–3732 (2017)CrossRefGoogle Scholar
  17. 17.
    Fei, W., Li, D.F.: Bilinear programming approach to solve interval bimatrix games in tourism planning management. Int. J. Fuzzy Syst. 18(3), 504–510 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Niknam, T.: A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem. Appl. Energy 87(1), 327–339 (2010)CrossRefGoogle Scholar
  19. 19.
    Tsekouras, G.E., Tsimikas, J., Kalloniatis, C.: Interpretability constraints for fuzzy modeling implemented by constrained particle swarm optimization. IEEE Trans. Fuzzy Syst. 26(4), 2348–2361 (2018)CrossRefGoogle Scholar
  20. 20.
    Ghodousian, A., Babalhavaeji, A.: An efficient genetic algorithm for solving nonlinear optimization problems defined with fuzzy relational equations and max-Lukasiewicz composition. Appl. Soft Comput. 69, 475–492 (2018)CrossRefGoogle Scholar
  21. 21.
    Charnes, A.: Constrained games and linear programming. Proc. Natl. Acad. Sci. USA 39(7), 639–641 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Charnes, A., Sorensen, S.: Constrained n-person games. Int. J. Game Theory 3(3), 141–158 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Penn, A.: Generalized lagrange-multiplier method for constrained matrix games. Oper. Res. 19(4), 933–945 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, D.F., Cheng, C.T.: Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers. Int. J. Uncertain. Fuzz. 10(4), 385–400 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, D.F., Hong, F.X.: Solving constrained matrix games with payoffs of triangular fuzzy numbers. Comput. Math. Appl. 64(4), 432–446 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Firouzbakht, K., Noubir, G., Salehi, M.: Constrained bimatrix games in wireless communications. IEEE Trans. Commun. 64(1), 1–11 (2015)Google Scholar
  27. 27.
    Meng, F.Y., Zhan, J.Q.: Two methods for solving constrained bi-matrix games. Open Cybern. Syst. J. 8, 1038–1041 (2014)CrossRefGoogle Scholar
  28. 28.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefzbMATHGoogle Scholar
  29. 29.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8(3), 199–249 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Miyamoto, S.: Remarks on basics of fuzzy sets and fuzzy multisets. Fuzzy Sets Syst. 156(3), 427–431 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Atanassov, K.T.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(1), 343–349 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36(1), 48–49 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Li, D.F.: Decision and Game Theory in Management with Intuitionistic Fuzzy Sets. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  35. 35.
    Nan, J.X., Li, D.F., Zhang, M.J.: A lexicographic method for matrix games with payoffs of triangular intuitionistic fuzzy numbers. Int. J. Comput. Int. Syst. 3(3), 280–289 (2010)CrossRefGoogle Scholar
  36. 36.
    Yu, V.F., Van, L.H., Dat, L.Q.: Analyzing the ranking method for fuzzy numbers in fuzzy decision making based on the magnitude concepts. Int. J. Fuzzy Syst. 19(5), 1–11 (2017)CrossRefGoogle Scholar
  37. 37.
    Varghese, A., Kuriakose, S.: Centroid of an intuitionistic fuzzy number. Notes Intuit. Fuzzy Sets 18(1), 19–24 (2012)zbMATHGoogle Scholar
  38. 38.
    Hung, W.L., Yang, M.S.: Similarity measures of intuitionistic fuzzy sets based on hausdorff distance. Pattern Recognit. Lett. 25(14), 1603–1611 (2004)CrossRefGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Economics and ManagementFuzhou UniversityFuzhouChina

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