Matrix Games with Goals of Intuitionistic Fuzzy Sets and Linear Programming Method
In the preceding Chaps. 7 – 9 , we discussed three kinds of intuitionistic fuzzy matrix games: matrix games with payoffs of intuitionistic fuzzy sets, matrix games with payoffs of interval-valued intuitionistic fuzzy sets, and matrix games with payoffs o
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Matrix Games with Goals of Intuitionistic Fuzzy Sets and Linear Programming Method
10.1 Introduction In the preceding Chaps. 7–9, we discussed three kinds of intuitionistic fuzzy matrix games: matrix games with payoffs of intuitionistic fuzzy sets, matrix games with payoffs of interval-valued intuitionistic fuzzy sets, and matrix games with payoffs of trapezoidal intuitionistic fuzzy numbers. It is obvious that these intuitionistic fuzzy matrix games only take into consideration uncertainty in players’ payoffs, which are expressed with intuitionistic fuzzy sets and their special form: trapezoidal intuitionistic fuzzy numbers. In reality, however, players may have their goals for the outcome of the game under discussion [1–3]. The goals may be given by players with some uncertainty. As a result, there appears an important type of matrix games with goals expressed by intuitionistic fuzzy sets, which usually are called matrix games with goals of intuitionistic fuzzy sets for short. It is not difficult to see that matrix games with goals of intuitionistic fuzzy sets differ from matrix games with goals of fuzzy sets [1, 2]. The former uses two functions (i.e., the membership and nonmembership functions of the intuitionistic fuzzy set) to express players’ goals while the latter only uses one function (i.e., the membership function of the fuzzy set) to express players’ goals. That is to say, the hesitancy degrees of players’ intuitionistic fuzzy goals may not be equal to 0 while the hesitancy degrees of players’ fuzzy goals are always equal to 0. In this chapter, we will focus on discussing the following special kind of matrix games with goals of intuitionistic fuzzy sets: players may have their goals expressed with intuitionistic fuzzy sets and payoffs of players at every situation are expressed with real numbers rather than intuitionistic fuzzy sets. More specifically, matrix games with goals of intuitionistic fuzzy sets are formulated and auxiliary linear programming models are derived and hereby corresponding method is developed to generate solutions of matrix games with goals of intuitionistic fuzzy sets.
D.-F. Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 308, DOI: 10.1007/978-3-642-40712-3_10, Springer-Verlag Berlin Heidelberg 2014
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Matrix Games with Goals of Intuitionistic Fuzzy Sets
10.2 Formal Representation of Matrix Games with Goals of Intuitionistic Fuzzy Sets and Solutions’ Concepts Let us consider the following matrix games with goals of intuitionistic fuzzy sets in which players may have intuitionistic fuzzy goals and payoffs of players are real numbers rather than intuitionistic fuzzy sets. Namely, as stated earlier, assume that S1 ¼ fa1 ; a2 ; . . .; am g and S2 ¼ fb1 ; b2 ; . . .; bn g are sets of pure strategies for players P1 and P2 , respectively. If player P1 chooses any pure strategy ai 2 S1 (i ¼ 1; 2; . . .; m) and player P2 chooses any pure strategy bj 2 S2 (j ¼ 1; 2; . . .; n), then player P1 gains a payoff
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