Solving constrained matrix games with payoffs of triangular fuzzy numbers

Abstract

Constrained matrix games with payoffs of triangular fuzzy numbers (TFNs) are a type of matrix games with payoffs expressed by TFNs and sets of players’ strategies which are constrained. So far as we know, no study have yet been attempted for constrained matrix games with payoffs of TFNs since there is no effective way to simultaneously incorporate the payoffs’ fuzziness and strategies’ constraints into classical and/or fuzzy matrix game methods. The aim of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of TFNs. In this methodology, we introduce the concepts of Alpha-constrained matrix games for constrained matrix games with payoffs of TFNs and the values. By the duality theorem of linear programming, it is proven that players’ gain-floor and loss-ceiling always have a common interval-type value and hereby any Alpha-constrained matrix game has an interval-type value. Moreover, using the representation theorem for the fuzzy set, it is proven that any constrained matrix game with payoffs of TFNs always has a TFN-type fuzzy value. The auxiliary linear programming models are derived to compute the lower and upper bounds of the interval-type value and optimal strategies of players for any Alpha-constrained matrix game. In particular, the mean and the lower and upper limits of the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs can be directly obtained through solving the derived three linear programming models with data taken from only 1-cut and 0-cut of payoffs. Hereby the TFNtype fuzzy value of any constrained matrix game with payoffs of TFNs are easily computed. The proposed method in this paper is compared with other methods and its validity and applicability are illustrated with a numerical example.