Abstract
A bi-matrix game with fuzzy goal is shown to be equivalent to a (crisp) non-linear programming problem in which the objective as well as all constraint functions are linear except two constraint functions, which are quadratic. This equivalence is further extended to bi-matrix games with fuzzy pay-offs, as well as to bi-matrix games with fuzzy goals and fuzzy payoffs, whose equilibrium strategies are conceptualized by employing a suitable ranking (defuzzification) function.
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References
Bector, C.R.and S.Chandra.(2002).''On Duality in Linear Programming under Fuzzy Environment,'' Fuzzy Sets and Systems 125,317-325.
Bector, C.R.,S.Chandra,and Vidyottama Vijay.(2004a).''Matrix Games with Fuzzy Goals and Fuzzy Linear Programming Duality,''Fuzzy Optimization and Decision Making 3,255-269.
Bector, C.R.,S.Chandra,and Vidyottama Vijay.(2004b).''Duality in Linear Programming with Fuzzy Parameters and Matrix Games with Fuzzy Pay-o.s,''Fuzzy Sets and Systems 146,253-269.
Vijay, Vidyottama,S.Chandra,and C.R. Bector.(2004c).''Matrix Games with Fuzzy Goals and Fuzzy Pay-o.s,''Omega.
Campos,L.(1989).''Fuzzy Linear Programming Models to Solve Fuzzy Matrix Games,''Fuzzy Sets and Systems 32,275-289.
Dubois,D.and H.Prade.(1983).''Ranking Fuzzy Numbers in the Setting of Possibility Theory,''In-formation Sciences 30,183-224.
Inuiguchi,M.,H.Ichihashi,and Y.Kume.(1992a).''Relationship Between Modality Constrained Pro-gramming Problems and Various Fuzzy Mathematical Programming Problems,''Fuzzy Sets and Systems 49,243-259.
Inuiguchi,M.,H.Ichihashi,and Y.Kume.(1992b).''Some Properties of Extended Fuzzy Preference Relations using Modalities,''Information Sciences 61,187-209.
Inuiguchi,M.,H.Ichihashi and Y.Kume.(1993).''Modality Constrained Programming Problems:A Uni ed Approach to Fuzzy Mathematical Programming Problems in the Setting of Possibility Theory,'' Information Sciences 67,93-126.
Inuiguchi,M.,J.Ramik, T.Tanino,and M.Vlach.(2003).''Satis cing Solutions and Duality in Interval and Fuzzy Linear Programming,''Fuzzy Sets and Systems 135,151-177.
Klir, G.J.and B.Yuan.(1995).Fuzzy Sets and Fuzzy Logic:Theory and Applications.N.J.(USA): Prentice-Hall Inc.
Mangasarian, O.L.and H.Stone.(1964).''Two-Person Non-Zero Sum Games and Quadratic Pro-gramming,''Journal of Mathematical Analysis and Applications 9,348-355.
Maeda,T.(2000).''Characterization of the Equilibrium Strategy of the Bi-matrix Game with Fuzzy Payo.,''Journal of Mathematical Analysis and Applications 251,885-896.
Nash, J.F.(1950).''Equilibrium Points in n-Person Games,''Proceedings of the National Academy of Sciences of the USA36,48-49.
Nash, J.F.(1951).''Non-Cooperative Games,''Annals of Mathematics,54(2),286-295.
Nishizaki,I.and M.Sakawa.(2001).Fuzzy and Multiobjective Games for Conflict Resolution.Heidelberg: Physica-Verlag.
Nishizaki,I.and M.Sakawa.(1995).''Equilibrium Solution for Multiobjective Bimatrix Games incor-porating Fuzzy Goals,''Journal of Optimization theory and Applications 86(2),433-457.
Owen,G.(1995).Game Theory.San Diego: Academic Press.
Van De Panne, C.(1966).''Programming with a Quadratic Constraint,''Management Sciences 12, 798-815.
Yager, R.R.(1978).''Ranking Fuzzy Subsets over the Unit Interval,''Proceeding CDC.,1435-1437.
Zimmermann, H.-J.(1978).''Fuzzy Programming and Linear Programming with Several Objective Function,''Fuzzy Sets and Systems 1,45-55.
Zimmermann, H.-J.(1991).Fuzzy Set Theory-Its Application,2 nd Editions.Dordercht: Kluwer Academic Publishers.
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Vidyottama, V., Chandra, S. & Bector, C. Bi-matrix Games with Fuzzy Goals and Fuzzy. Fuzzy Optimization and Decision Making 3, 327–344 (2004). https://doi.org/10.1007/s10700-004-4202-4
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DOI: https://doi.org/10.1007/s10700-004-4202-4