Fuzzy Bimatrix Games with Single and Multiple Objective : introduction to the computational techniques

This paper introduces to the computational techniques of non-cooperative bimatrix games in an uncertain environment. Both single and multiple objective fuzzy-valued bimatrix games are considered theoretically with one numerical example. The presentation is centered on the Nishizaki and Sakawa models. These models are based on the maxmin and minmax principles of the classical matrix game theory. Equivalent nonlinear (possibly quadratic) programming problems are giving optimal solutions. The equilibrium solutions correspond to players maximizing a degree of attainment of the fuzzy goals. Besides the Nash equilibrium, the concept of α-Nash equilibrium supposes Nature be the third Player. The aggregation of all the fuzzy sets in the multiobjective models use the fuzzy decision rule by Bellman and Zadeh. This ‘aggregation by a minimum component’ consists in the intersection of the fuzzy sets, the fuzzy expected payoffs and the fuzzy goals. Numerical examples of two-players nonzero sum games are solved using the MATHEMATICA 7.0.1 software. The numerical solutions are possibly local by using iterative methods. Key–Words: Fuzzy bimatrix game, Single-objective, Multiobjective, Degree of attainment of a fuzzy goal, Quadratic programming