On solving two-person zero-sum fuzzy matrix games via linear programming approach

Document Type: Research Paper

Author

Department of Operations Research, Institute of Statistical Studies and Research, Cairo University, Giza, Egypt.

Abstract

In this paper, a two-person zero-sum matrix game with fuzzy numbers payoff is introduced. Using the fuzzy number comparison introduced by Rouben's method (1991), the fuzzy payoff is converted into the corresponding deterministic payoff. Then, for each player, a linear programming problem is formulated. Also, a solution procedure for solving each problem is proposed. Finally, a numerical example is given for illustration.    

Keywords

Main Subjects


[1]     Baldwin, J. F., & Guild, N. C. F. (1979). Comparison of fuzzy sets on the same decision space. Fuzzy sets and systems2(3), 213-231.

[2]      Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management science17(4), B-141.

[3]     Campos, L. (1989). Fuzzy linear programming models to solve fuzzy matrix games. Fuzzy sets and systems32(3), 275-289.

[4]      Cevikel, A. C., & Ahlatcioglu, M. (2009). A linear interactive solution concept for fuzzy multiobjective games. European journal of pure and applied mathematics3(1), 107-117.

[5]     Chen, Y. W., & Larbani, M. (2006). Two-person zero-sum game approach for fuzzy multiple attribute decision making problems. Fuzzy sets and systems157(1), 34-51.

[6]      Dhingra, A. K., & Rao, S. S. (1995). A cooperative fuzzy game theoretic approach to multiple objective design optimization. European journal of operational research83(3), 547-567.

[7]     Dubois, D., & Prade, H., (1980). Possibility theory: an approach to computerized processing of uncertainty. Boston, MA: Springer.

[8]     Ein-Dor, L., & Kanter, I. (2001). Matrix games with nonuniform payoff distributions. Physica A: statistical mechanics and its applications302(1-4), 80-88.

[9]     El–Shafei, K. M. M. (2007). An interactive approach for solving Nash cooperative continuous static games (NCCSG). International journal of contemporary mathematical sciences2, 1147-1162.

[10]   Espin, R., Fernandez, E., & Mazcorro, G., (2007). A fuzzy approach to cooperative person games. European journal of operational research, 176(3), 1735- 1751.

[11]  Fortemps, P., & Roubens, M. (1996). Ranking and defuzzification methods based on area compensation. Fuzzy sets and systems82(3), 319-330.

[12]  Gogodze, J. (2018). Using a two-person zero-sum game to solve a decision-making problem. Pure and applied mathematics journal7(2), 11.

[13]  Kaufmann, A. (1975). Introduction to the theory of fuzzy subsets (Vol. 2). Academic Pr.

[14]  Kaufmann, A., & Gupta, M. M. (1988). Fuzzy mathematical models in engineering and management science. Elsevier Science Inc.

[15]  Khalifa, H. A., & Zeineldin, R. A. (2015). An interactive approach for solving fuzzy cooperative continuous static games. International journal of computer applications975, 8887.

[16]  Kumar, S. (2016). Max-min solution approach for multi-objective matrix game with fuzzy goals. Yugoslav journal of operations research26(1).

[17]  Li, F. D., & Hong, X. F. (2012). Solving constrained matrix games with payoffs of triangular fuzzy numbers. Computers & mathematics with applications, 64(4), 432- 446.

[18]  Nakamura, K. (1986). Preference relations on a set of fuzzy utilities as a basis for decision making. Fuzzy sets and systems, 20(2), 147-162.

[19]  Nan, J. X., Li, D. F., & Zhang, M. J. (2010). A lexicographic method for matrix games with payoffs of triangular intuitionistic fuzzy numbers. International journal of computational intelligence systems3(3), 280-289.

[20]   Navidi, H., Amiri, A. H., & Kamranrad, R. (2014). Multi responses optimization through game theory approach. International journal of industrial engineering & production research, 25(3), 215-224.

[21]   Osman, M. S., El-Kholy, N. A., & Soliman, E. I. (2015). A recent approach to continuous time open loop stackelberg dynamic game with min-max cooperative and noncooperative followers. European scientific journal, ESJ11(3).

[22]   Parthasarathy, T., & Raghavan, T. E. S. (1971). Some topics in two-person games. New York: American Elsevier Publishing Company.

[23]  Pęski, M. (2008). Comparison of information structures in zero-sum games. Games and economic behavior62(2), 732-735.

[24]  Roubens, M. (1990). Inequality constraints between fuzzy numbers and their use in mathematical programming. In Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty (pp. 321-330). Springer, Dordrecht.

[25]  Roy, S. K., Biswal, M. P., & Tiwari, R. N. (2000). Cooperative fuzzy game theoretic approach to some multi-objective linear programming problems. Journal of fuzzy mathematics8(3), 635-644.

[26]  Sahoo, L. (2017). An approach for solving fuzzy matrix games using signed distance method. Journal of information and computing science12(1), 73-80.

[27]   Seikh, M. R., Nayak, P. K., & Pal, M. (2015). An alternative approach for solving fuzzy matrix games. International journal of mathematics and soft computing5(1), 79-92.

[28]  Selvakumari, K., & Lavanya, S. (2015). An approach for solving fuzzy game problem. Indian journal of science and technology8, 1-6.

[29]  Takahashi, S. (2008). The number of pure Nash equilibria in a random game with nondecreasing best responses. Games and economic behavior63(1), 328-340.

[30]  Thirucheran, M., Meena, R. E., & Lavanya, S.  (2017). A new approach for solving fuzzy game problem. International journal of pure and applied mathematics, 114(6), 67- 75.

[31]  Xu, J. (1998). Zero sum two-person game with grey number payoff matrix in linear programming. The journal of grey system10(3), 225-233.

[32]  Xu, J., & Yao, L. (2010). A class of two-person zero-sum matrix games with rough payoffs. International journal of mathematics and mathematical scienceshttp://dx.doi.org/10.1155/2010/404792.

[33]   Zadeh, L. A. (1965). Fuzzy sets. Information and control8(3), 338-353.

[34]  Zhao, R., Govind, R., & Fan, G. (1992). The complete decision set of the generalized symmetrical fuzzy linear programming problem. Fuzzy sets and systems51(1), 53-65.

[35]  Shaocheng, T. (1994). Interval number and fuzzy number linear programmings. Fuzzy sets and systems66(3), 301-306.