## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 824790, 15 pages.

### Characterization of the Equilibrium Strategy of Fuzzy Bimatrix Games Based on $L\text{-}R$ Fuzzy Variables

Cun-lin Li

#### Abstract

This paper deals with bimatrix games in uncertainty environment based on several types of ordering, which Maeda proposed. But Maeda’s models was just made based on symmetrical triangle fuzzyvariable. In this paper, we generalized Maeda’s model to the non-symmetrical environment. In other words, we investigated the fuzzy bimatrix games based on nonsymmetrical $L\text{-}R$ fuzzy variables. Then the pseudoinverse of a nonconstant monotone function was given and the concept of crisp parametric bimatrix games was introduced. At last, the existence condition of Nash equilibrium strategies of the fuzzy bimatrix games is proposed and (weak) Pareto equilibrium of the fuzzy bimatrix games was obtained through the Nash equilibrium of the crisp parametric bimatrix.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 824790, 15 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357180249

Digital Object Identifier
doi:10.1155/2012/824790

Mathematical Reviews number (MathSciNet)
MR2935533

Zentralblatt MATH identifier
1244.91006

#### Citation

Li, Cun-lin. Characterization of the Equilibrium Strategy of Fuzzy Bimatrix Games Based on $L\text{-}R$ Fuzzy Variables. J. Appl. Math. 2012, Special Issue (2012), Article ID 824790, 15 pages. doi:10.1155/2012/824790. https://projecteuclid.org/euclid.jam/1357180249

#### References

• J. F. Nash, Jr., “Equilibrium points in $n$-person games,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 48–49, 1950.
• J. Nash, “Non-cooperative games,” Annals of Mathematics, vol. 54, pp. 286–295, 1951.
• D. Butnariu, “Fuzzy games: a description of the concept,” Fuzzy Sets and Systems, vol. 1, no. 3, pp. 181–192, 1978.
• L. Campos, “Fuzzy linear programming models to solve fuzzy matrix games,” Fuzzy Sets and Systems, vol. 32, no. 3, pp. 275–289, 1989.
• R. Yager, “A procedure for ordering fuzzy subsets of the unit interval,” Information Sciences, vol. 24, no. 2, pp. 143–161, 1981.
• I. Nishizaki and M. Sakawa, Fuzzy and Multi-Objective Games for Conflict Resolution, Physica, New York, NY, USA, 2001.
• I. Nishizaki and M. Sakawa, “Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals,” Fuzzy Sets and Systems, vol. 111, no. 1, pp. 99–116, 2000.
• C. R. Bector, S. Chandra, and V. Vidyottama, “Matrix games with fuzzy goals and fuzzy linear programming duality,” Fuzzy Optimization and Decision Making, vol. 3, no. 3, pp. 255–269, 2004.
• C. R. Bector, S. Chandra et al., Fuzzy Mathematical Programming and Fuzzy Matrix Games, Springer, Berlin, Germany, 2005.
• V. Vijay and C. R. Bector, “Bi-matrix games with fuzzy goals and fuzzy payoffs,” Fuzzy Optimization and Decision Making, vol. 3, pp. 327–344, 2004.
• V. Vijay, S. Chandra, and C. R. Bector, “Matrix games with fuzzy goals and fuzzy payoffs,” Omega, vol. 33, no. 5, pp. 425–429, 2005.
• M. Takashi, “On characterization of equilibrium strategy of two-person zero-sum games with fuzzy payoffs,” Fuzzy Sets and Systems, vol. 139, no. 2, pp. 283–296, 2003.
• M. Takashi, “Characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff,” Journal of Mathematical Analysis and Applications, vol. 251, no. 2, pp. 885–896, 2000.
• J. Ramík and J. \'Imánek, “Inequality relation between fuzzy numbers and its use in fuzzy optimization,” Fuzzy Sets and Systems, vol. 16, no. 2, pp. 123–138, 1985.
• L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning,” Information Sciences, vol. 8, pp. 199–249, 1975.
• M. Sakawa and H. Yano, “Feasibility and Pareto optimality for multiobjective nonlinear programming problems with fuzzy parameters,” Fuzzy Sets and Systems, vol. 43, no. 1, pp. 1–15, 1991.