Fuzzy Random Bimatrix Games Based on Possibility and Necessity Measures

Hitoshi Yano

doi:10.20965/jaciii.2021.p1024

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JACIII Vol.25 No.6 pp. 1024-1030

doi: 10.20965/jaciii.2021.p1024

(2021)

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Fuzzy Random Bimatrix Games Based on Possibility and Necessity Measures

Hitoshi Yano

Graduate School of Humanities and Social Sciences, Nagoya City University
1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, Aichi 467-8501, Japan

Received:

June 22, 2021

Accepted:

September 17, 2021

Published:

November 20, 2021

Keywords:

bimatrix games, fuzzy random variables, expectation model, possibility measure, necessity measure

Abstract

In this study, we formulate bimatrix games with fuzzy random payoffs, and introduce equilibrium solution concepts based on possibility and necessity measures. It is assumed that each player has linear fuzzy goals for his/her payoff. To obtain equilibrium solutions based on the possibility and necessity measures, we propose two algorithms in which quadratic programming problems are solved repeatedly until equilibrium conditions are satisfied.

Cite this article as:

H. Yano, “Fuzzy Random Bimatrix Games Based on Possibility and Necessity Measures,” J. Adv. Comput. Intell. Intell. Inform., Vol.25 No.6, pp. 1024-1030, 2021.

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References

[1] M. Larbani, “Non cooperative fuzzy games in normal form: A survey,” Fuzzy Sets and Systems, Vol.160, pp. 3184-3210, 2009.
[2] L. Campos, “Fuzzy linear programming models to solve fuzzy matrix games,” Fuzzy Sets and Systems, Vol.32, pp. 275-289, 1989.
[3] D. Dubois and H. Prade, “Fuzzy Sets and Systems: Theory and Applications,” Academic Press, 1980.
[4] R. R. Yager, “A Procedure for ordering fuzzy numbers in the unit interval,” Information Sciences, Vol.24, pp. 143-161, 1981.
[5] D. F. Li, “A fuzzy multi-objective approach to solve fuzzy matrix games,” The J. of Fuzzy Mathematics, Vol.7, pp. 907-912, 1999.
[6] C. R. Bector, S. Chandra, and V. Vijay, “Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs,” Fuzzy Sets and Systems, Vol.146, pp. 253-269, 2004.
[7] T. Maeda, “On characterization of equilibrium strategy of two person zero-sum game with fuzzy payoffs,” Fuzzy Sets and Systems, Vol.139, pp. 283-296, 2003.
[8] T. Maeda, “Characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff,” J. of Mathematical Analysis and Applications, Vol.251, pp. 885-896, 2000.
[9] C.-L. Li, Q. Zhang, and G.-S. Zhu, “Bimatrix Games on Possibility Space,” 2010 Int. Conf. of Information Science and Management Engineering, pp. 472-475, 2010.
[10] C.-L. Li, “Bimatrix games in necessity space,” 2011 IEEE Int. Conf. on Computer Science and Automation Engineering, pp. 562-566, 2011.
[11] Z. Makó and J. Salamon, “Correlated equilibrium of games in fuzzy environment,” Fuzzy Sets and Systems, Vol.398, pp. 112-127, 2020.
[12] J. Gao, “Uncertain bimatrix game with applications,” Fuzzy Optimization and Decision Making, Vol.12, pp. 65-78, 2013.
[13] B. Liu, “Uncertainty Theory,” 2nd edition, Springer, 2007.
[14] M. Tang and Z. Li, “A novel uncertain bimatrix game with Hurwicz criterion,” Soft Computing, Vol.24, pp. 2441-2446, 2020.
[15] H. Kwakernaak, “Fuzzy random variable – I. definitions and theorems,” Information Sciences, Vol.15, pp. 1-29, 1978.
[16] M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” J. of Mathematical Analysis and Applications, Vol.114, pp. 409-422, 1986.
[17] G. Wang and Y. Zhang, “The theory of fuzzy stochastic processes,” Fuzzy Sets and Systems, Vol.51, pp. 161-178, 1992.
[18] H. Yano, “Multiobjective two-level fuzzy random programming problems with simple recourses and estimated Pareto Stackelberg Solutions,” J. Adv. Comput. Intell. Intell. Inform., Vol.22, No.3, pp. 359-368, doi: 10.20965/jaciii.2018.p0359, 2018.
[19] H. Yano, “Multiobjective Two-level Simple Recourse Programming Problems with Discrete-type LR Fuzzy Random Variables,” Procedia Computer Science, Vol.176, pp. 531-540, 2020.
[20] H. Yano, “Multiobjective Two-Level Simple Recourse Programming Problems with Discrete-Type Fuzzy Random Variables and Optimistic and Pessimistic Pareto Stackelberg Solutions,” 2020 Joint 11th Int. Conf. on Soft Computing and Intelligent Systems and 21st Int. Symp. on Advanced Intelligent Systems (SCIS-ISIS), pp. 52-57, 2020.
[21] M. Sakawa, H. Yano, and I. Nishizaki, “Linear and Multiobjective Programming with Fuzzy Stochastic Extensions,” Springer, 2013.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] M. Larbani, “Non cooperative fuzzy games in normal form: A survey,” Fuzzy Sets and Systems, Vol.160, pp. 3184-3210, 2009.

[2] [2] L. Campos, “Fuzzy linear programming models to solve fuzzy matrix games,” Fuzzy Sets and Systems, Vol.32, pp. 275-289, 1989.

[3] [3] D. Dubois and H. Prade, “Fuzzy Sets and Systems: Theory and Applications,” Academic Press, 1980.

[4] [4] R. R. Yager, “A Procedure for ordering fuzzy numbers in the unit interval,” Information Sciences, Vol.24, pp. 143-161, 1981.

[5] [5] D. F. Li, “A fuzzy multi-objective approach to solve fuzzy matrix games,” The J. of Fuzzy Mathematics, Vol.7, pp. 907-912, 1999.

[6] [6] C. R. Bector, S. Chandra, and V. Vijay, “Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs,” Fuzzy Sets and Systems, Vol.146, pp. 253-269, 2004.

[7] [7] T. Maeda, “On characterization of equilibrium strategy of two person zero-sum game with fuzzy payoffs,” Fuzzy Sets and Systems, Vol.139, pp. 283-296, 2003.

[8] [8] T. Maeda, “Characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff,” J. of Mathematical Analysis and Applications, Vol.251, pp. 885-896, 2000.

[9] [9] C.-L. Li, Q. Zhang, and G.-S. Zhu, “Bimatrix Games on Possibility Space,” 2010 Int. Conf. of Information Science and Management Engineering, pp. 472-475, 2010.

[10] [10] C.-L. Li, “Bimatrix games in necessity space,” 2011 IEEE Int. Conf. on Computer Science and Automation Engineering, pp. 562-566, 2011.

[11] [11] Z. Makó and J. Salamon, “Correlated equilibrium of games in fuzzy environment,” Fuzzy Sets and Systems, Vol.398, pp. 112-127, 2020.

[12] [12] J. Gao, “Uncertain bimatrix game with applications,” Fuzzy Optimization and Decision Making, Vol.12, pp. 65-78, 2013.

[13] [13] B. Liu, “Uncertainty Theory,” 2nd edition, Springer, 2007.

[14] [14] M. Tang and Z. Li, “A novel uncertain bimatrix game with Hurwicz criterion,” Soft Computing, Vol.24, pp. 2441-2446, 2020.

[15] [15] H. Kwakernaak, “Fuzzy random variable – I. definitions and theorems,” Information Sciences, Vol.15, pp. 1-29, 1978.

[16] [16] M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” J. of Mathematical Analysis and Applications, Vol.114, pp. 409-422, 1986.

[17] [17] G. Wang and Y. Zhang, “The theory of fuzzy stochastic processes,” Fuzzy Sets and Systems, Vol.51, pp. 161-178, 1992.

[18] [18] H. Yano, “Multiobjective two-level fuzzy random programming problems with simple recourses and estimated Pareto Stackelberg Solutions,” J. Adv. Comput. Intell. Intell. Inform., Vol.22, No.3, pp. 359-368, doi: 10.20965/jaciii.2018.p0359, 2018.

[19] [19] H. Yano, “Multiobjective Two-level Simple Recourse Programming Problems with Discrete-type LR Fuzzy Random Variables,” Procedia Computer Science, Vol.176, pp. 531-540, 2020.

[20] [20] H. Yano, “Multiobjective Two-Level Simple Recourse Programming Problems with Discrete-Type Fuzzy Random Variables and Optimistic and Pessimistic Pareto Stackelberg Solutions,” 2020 Joint 11th Int. Conf. on Soft Computing and Intelligent Systems and 21st Int. Symp. on Advanced Intelligent Systems (SCIS-ISIS), pp. 52-57, 2020.

[21] [21] M. Sakawa, H. Yano, and I. Nishizaki, “Linear and Multiobjective Programming with Fuzzy Stochastic Extensions,” Springer, 2013.