A method to solve two-player zero-sum matrix games in chaotic environment

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Hamiden Abd El- Wahed Khalifa
Pavan Kumar


This research article proposes a method for solving the two-player zero-sum matrix games in chaotic environment. In a fast growing world, the real life situations are characterized by their chaotic behaviors and chaotic processes. The chaos variables are defined to study such type of problems. Classical mathematics deals with the numbers as static and less value-added, while the chaos mathematics deals with it as dynamic evolutionary, and comparatively more value-added. In this research article, the payoff is characterized by chaos numbers. While the chaos payoff matrix converted into the corresponding static payoff matrix. An approach for determining the chaotic optimal strategy is developed. In the last, one solved example is provided to explain the utility, effectiveness and applicability of the approach for the problem.

Abbreviations: DM= Decision Maker; MCDM = Multiple Criteria Decision Making; LPP = Linear Programming Problem; GAMS= General Algebraic Modeling System.


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Author Biography

Pavan Kumar, Department of Mathematics, School of Advanced Science and Languages, VIT Bhopal University

Asst Professor, Department of Mathematics, VIT University, Bhopal.


Bandyopadhyay, S., Nayak, P. K., & Pal, M. (2013) Solution of matrix game with triangular intuitionistic fuzzy pay-off using score function. Open Journal of Optimization, 2(1), 9-15. DOI: 10.4236/ojop.2013.21002

Bandyopadhyay, S., & Nayak, P. K. (2013) Matrix games with trapezoidal fuzzy payoff. International Journal of Engineering Research and Development, 5, 21- 29.

Bector, C. R., & Chandra, S., Vijay, (2004) Duality in linear programming with fuzzy parameters and matrix games with fuzzy payoffs. Fuzzy Sets and Systems, 146, 253-269. DOI: 10.1016/S0165-0114(03)00260-4

Bector, C. R., & Chandra, S., Vijay, (2004a) Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy Optimization and Decision Making, 3(3), 255-269. DOI: 10.1023/B:FODM.0000036866.18909.f1

Bellman, R. E., & Zadeh, L. A. (1970) Decision making in a fuzzy environment. Management Science, 17(4), 141-164. DOI: 10.1287/mnsc.17.4.B141

Berg, J., & Engel, A. (1998) Matrix games, mixed strategies, and statistical mechanics. Physical Review Letters, 81(22), 4999-5002. DOI: 10.1103/PhysRevLett.81.4999

Bhuiyan, B. A. (2018) An overview of game theory and some applications. Philosophy and Progress, 59(1-2), 111-128. DOI: 10.3329/pp.v59i1-2.36683

Campos, L. (1989) Fuzzy linear programming models to solve fuzzy matrix games. Fuzzy Sets and Systems, 32(3), 275-289. DOI: 10.1016/0165-0114(89)90260-1

Camona, G., & Carvalho, L. (2016) Repeated two-person zero-sum games with unequal discounting and private monitoring. Journal of Mathematical Economics, 63(C), 131-138. Available: http://www.sciencedirect.com/science/article/pii/S0304406816000136. DOI: 10.1016/j.jmateco.2016.02.005

Chen, Y- W., & Larboni, M. (2006) Two-person zero-sum game approach for fuzzy multiple attribute decision-making problem. Fuzzy Sets and Systems, 157(1), 34-51. DOI: 10.1016/j.fss.2005.06.004

Deng, X., Jiang, W., & Zhang, J. (2017) Zero-sum matrix game with payoffs of Dempster- Shafer belief structures and its applications on sensors. Sensors, 17(4), 922. DOI: 10.3390/s17040922.

Dhingra, A. K., & Rao, S. S. (1995) A cooperative fuzzy game theoretic approach to multiple objective design optimizations. European Journal of Operational Research, 83(3, p. 547-567. DOI: 10.1016/0377-2217(93)E0324-Q

Dubois, D., & Prade, H. (1980) Possibility Theory: An approach to computerized processing of uncertainty, Plenum, New York.

Gutzwiller, M. C. (1991) Chaos in classical and quantum mechanics. Interdisciplinary Applied Mathematics, Springer, New York, NY.

Ketata, C., Satish, M. G., & Islam, M. R. (2006) Chaos numbers. In: International Conference On Computational Intelligence For Modeling Control And Automation (CIMCA), IEEE, Proceedings, Sydney, Australia. DOI: 10.1109/CIMCA.2006.132

Khalifa, H. A. (2019) An approach for solving two- person zero- sum matrix games in neutrosophic environment. Journal of Industrial and Systems Engineering, 12(2), 186-198. Available: http://www.jise.ir/article_87288.html

Krishnaveni, G., & Ganesan, K. (2018) New approach for the solution of two person zero sum fuzzy games. International Journal of Pure and Applied Mathematics, 119(9), 405-414. Available: https://acadpubl.eu/jsi/2018-119-9/articles/9/39.pdf

Kumar, S., Chopra, R., & Saxena, R. R. (2016) A fast approach to solve matrix games with payoffs of trapezoidal fuzzy numbers. Asia-Pacific Journal of Operational Research, 33(06), 1650047. DOI: 10.1142/S0217595916500470

Li, F. D., & Hong, X. F. (2012) Solving constrained matrix games with payoffs of triangular fuzzy numbers. Computers & Mathematics with Applications, 64(4), 432-448. DOI: 10.1016/j.camwa.2011.12.009

Liu, B. (2002) Theory and practice of uncertain programming, Physica, New York, NY, USA.

Luce, R., & Raiffa, H. (1957) Games and decision, John Wiley & Sons, New York, NY, USA.

Nan, R., 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 Systems, 3(3), 280-289. DOI: 10.1080/18756891.2010.9727699

Pandey, D., & Kumar, S. (2010) Modified approach to multiobjective matrix game with vague payoffs. Journal of International Academy of Physical of Sciences, 14(2), 149-157.

Parthasarathy, T., & Raghavan, E. T. (2010) Some topics in two-person games. American Elsevier Publishing, NEW York, NY, USA, 1971.

Peitgen, H- O., Jurgens, H., & Saupe, D. (2004) Chaos and fractals, 2nd Edition, Springer, New York, NY.

Peski, M. (2008) Comparison of information structures in zero- sum games. Games and Economic Behavior, 62(2), 732-735. DOI: 10.1016/j.geb.2007.06.004. Available: https://www.sciencedirect.com/science/article/abs/pii/S0899825607001169.

Sahoo, L. (2015) An interval parametric technique for solving fuzzy matrix games. Elixir Applied Mathematics, 93, 39392-39397. Available: https://www.elixirpublishers.com

SAHOO, L. (2017) An approach for solving fuzzy matrix games using signed distance method. Journal of Information and Computing Science, 12(1), 073-080. Available: https://pdfs.semanticscholar.org/149a/eddfb9fd64ebf522fd424c404071d41c49ec.pdf

Sakawa, M., & Nishizaki, I. (1994) Max-min solutions for fuzzy multiobjective matrix games. Fuzzy Sets and Systems, 67(1), 53-69. DOI: 10.1016/0165-0114(94)90208-9. Available: https://www.sciencedirect.com/science/article/abs/pii/0165011494902089.

Seikh, M. R., Nayak, K. P., & Pal, M. (2013) An alternative approach for solving fuzzy matrix games. International Journal of Mathematics and Soft Computing, 5(1), 79-92.

Selvakumari, K., & Lavanya, S. (2015) An approach for solving fuzzy game problem. Indian Journal of Science and Technology, 8(15), 1-6. DOI: 10.17485/ijst/2015/v8i15/56807

Strogatz, S. H. (2001) Nonlinear dynamic and chaos: with applications to physics, biology, chemistry, and engineering, Preseus Books Group, New York, NY.

Takahashi, S. (2008) The number of pure nash equilibria in a random game with nondecreasing best responses. Games and Economic Behavior, 63(1), 328-340. DOI: 10.1016/j.geb.2007.10.003

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. Available: https://acadpubl.eu/jsi/2017-114-5/articles/2/8.pdf

Vijay, V., Chandra, S., & Bector, C. R. (2005) Matrix games with fuzzy goals and fuzzy payoffs. Omega, 33(5), 425-429. DOI: 10.1016/j.omega.2004.07.007

Vijay, V., Mehra, A., Chandra, S., & Bector, C. R. (2007) Fuzzy matrix games via a fuzzy relation approach. Fuzzy Optimization and Decision Making, 6, 299-314. DOI: 10.1007/s10700-007-9015-9

Von Neumann, J., & Morgenstern, D. (1944) The theory of games in economic behavior, Wiley, New York.

Xu, J. (1998) Zero sum two-person game with grey number payoff matrix in linear programming. The Journal of Grey System, 10(3), 225-233. DOI:

Xu, J., & Yao, L. (2010) A class of two-person zero-sum matrix games with rough payoffs. International Journal of Mathematics and Mathematical Sciences, 22. DOI: 10.1155/2010/404792

Zadeh, L. A. (1965) Fuzzy sets. Information Control, 8(3), 338- 353.

Zhao, R., Govind, R., & Fan, G. (1992) The complete decision set of the generalized symmetrical fuzzy linear programming problem. Fuzzy Sets and Systems, 51(1), 53-65. DOI: 10.1016/0165-0114(92)90075-F