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.
