Bi-matrix Games with Payoffs of Intuitionistic Fuzzy Sets and Bilinear Programming Method

Abstract

AbstarctIn 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. 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.