Matrix Games with Payoffs of Interval-Valued Intuitionistic Fuzzy Sets and Linear and Nonlinear Programming Methods
In the preceding Chap. 7, we discussed the concepts of solutions of matrix games with payoffs of intuitionistic fuzzy sets and methods. It is easy to see that how to estimate players’ payoffs at any situations is a key of applying the aforementioned intui
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Matrix Games with Payoffs of Interval-Valued Intuitionistic Fuzzy Sets and Linear and Nonlinear Programming Methods
8.1 Introduction In the preceding Chap. 7, we discussed the concepts of solutions of matrix games with payoffs of intuitionistic fuzzy sets and methods. It is easy to see that how to estimate players’ payoffs at any situations is a key of applying the aforementioned intuitionistic fuzzy matrix game theory and methods to solve real competitive management problems. Namely, choosing adequate intuitionistic fuzzy sets to represent players’ payoffs is an important problem. It is easy to see that constructing intuitionistic fuzzy sets is to determine their membership degrees and nonmembership degrees. In real-life management problems, however, it may not be easy to identify exact values for the membership and nonmembership degrees of intuitionistic fuzzy sets (i.e., players’ payoffs) due to the complexity and diversity of game environments and incompleteness and uncertainty of information. Thus, players’ payoffs seem to be suitably expressed with interval-valued intuitionistic fuzzy sets, which are characterized by membership and nonmembership functions, whose values are intervals rather than real numbers (i.e., exact values). For instance, in a ground bidding, some real estate company may provide the prior estimation on its bidding situations (or outcomes) as follows: the possibility of winning the bidding is at least 70% whereas at most 80 % and the possibility of losing the bidding is between 10 and 15%. In this case, there is an uncertainty’s range being from 5 to 20% in which the real estate company cannot judge whether wining the bidding or not. In other words, there is some hesitancy degree in the real estate company’s judgment on the bidding outcomes. Then, studying matrix games with payoffs of interval-valued intuitionistic fuzzy sets is of important values in theory and practice. Up to now, as far as we know, there exists very little investigation on matrix games with payoffs of interval-valued intuitionistic fuzzy sets. Therefor, this chapter will focus on discussing matrix games with payoffs of interval-valued intuitionistic fuzzy sets, which sometimes are simply called interval-valued intuitionistic fuzzy matrix games [1]. Specifically, in this chapter, we will formulate matrix games with payoffs of interval-valued intuitionistic fuzzy sets and propose their solution concepts. Hereby, solving 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_8, Springer-Verlag Berlin Heidelberg 2014
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8 Matrix Games with Payoffs of Interval-Valued Intuitionistic Fuzzy Sets
strategies of two players is converted into solving a pair of auxiliary nonlinear multiobjective programming models, which are further transformed into solving a pair of primal–dual linear programming models or nonlinear programming models.
8.2 Formal Representation of Matrix Games with Payoffs of Interval-Valued Intuition
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