Bi-matrix Games with Payoffs of Intuitionistic Fuzzy Sets and Bilinear Programming Method
In the preceding Chaps. 7 – 10 , we discussed modeling and solving methods of several kinds of matrix games with intuitionistic fuzzy sets. Obviously, these matrix games are a special case of noncooperative games, i.e., two-person zero-sum finite games.
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Bi-matrix Games with Payoffs of Intuitionistic Fuzzy Sets and Bilinear Programming Method
11.1 Introduction In the preceding Chaps. 7–10, we discussed modeling and solving methods of several kinds of matrix games with intuitionistic fuzzy sets. Obviously, these matrix games are a special case of noncooperative games, i.e., two-person zerosum finite games. In other words, they are a kind of games in which two players are completely antagonistic, i.e., one player wins the other player loses. In a reality, however, it is not always true that players are completely antagonistic. Thus, it is important and useful to study two-person nonzero-sum noncooperative games in normal form. Bi-matrix games are one of important kinds of the above two-person nonzero-sum noncooperative finite games [1, 2]. In this chapter, we will focus on studying bi-matrix games in which the payoffs of players are expressed with intuitionistic fuzzy sets, which are called bi-matrix games with payoffs of intuitionistic fuzzy sets for short. Specifically, we will propose a total order relation (or ranking method) of intuitionistic fuzzy sets based on the equivalent relation between intuitionistic fuzzy sets and interval-valued fuzzy sets and hereby introduce the concepts of solutions of bi-matrix games with payoffs of intuitionistic fuzzy sets and parametric bi-matrix games. It is proven that any bi-matrix game with payoffs of intuitionistic fuzzy sets has at least one satisfying Nash equilibrium solution, which is equivalent to a Nash equilibrium solution of the corresponding parametric bi-matrix game. The latter can be obtained through solving an auxiliary parametric bilinear programming model. Clearly, bi-matrix games with payoffs of intuitionistic fuzzy sets are a general form of the matrix games with payoffs of intuitionistic fuzzy sets as discussed in Chap. 7.
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_11, Springer-Verlag Berlin Heidelberg 2014
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Bi-matrix Games with Payoffs of Intuitionistic Fuzzy
11.2 The Defuzzification Ranking Method of Intuitionistic Fuzzy Sets and Bi-matrix Games 11.2.1 The Defuzzification Function of Intuitionistic Fuzzy Sets and Properties As stated earlier, when the intuitionistic fuzzy set is used to model management problems, specially decision making problems and game problems, the comparison or ranking order of intuitionistic fuzzy sets is very important. In Sect. 2.2.2, we reviewed the scoring function ranking method based on score and accuracy functions of intuitionistic fuzzy sets. Such a ranking method is essentially a kind of lexicographic methods. Thus, in order to facilitate the sequent discussions, we propose a new ranking method based on a defuzzification function as follows. Noticing that an intuitionistic fuzzy set C ¼ hlC ; tC i is mathematically equivalent to the interval-valued fuzzy set ½lC ; 1 tC (or ½lC ; lC þ pC ). Inspired by the ranking order of interval-value
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