An Effective Methodology for Solving Matrix Games With Fuzzy Payoffs

Abstract

Of the different types of games, the matrix games with fuzzy payoffs have been extensively discussed. Two major kinds of solution methods have been devised. One is the defuzzification approach based on ranking functions. Another is the two-level linear programming method which can obtain membership functions of players' fuzzy values (or gain floor and loss ceiling). These methods cannot always ensure that players' fuzzy/defuzzified values have a common value. The aim of this paper is to develop an effective methodology for solving matrix games with payoffs expressed by trapezoidal fuzzy numbers (TrFNs). In this methodology, we introduce the concept of Alpha-matrix games and prove that players' fuzzy values are always identical, and hereby, any matrix game with payoffs expressed by TrFNs has a fuzzy value, which is also a TrFN. The upper and lower bounds of any Alpha-cut of the fuzzy value and the players' optimal strategies are easily obtained through solving the derived four linear programming problems with the upper and lower bounds of Alpha-cuts of the fuzzy payoffs. In particular, the fuzzy value can be explicitly estimated through solving the auxiliary linear programming with data taken from the 1-cut and 0-cut of the fuzzy payoffs. The proposed method in this paper is illustrated with a real example and compared with other methods to show validity and applicability.