Matrix Games with Payoffs of Intuitionistic Fuzzy Sets and Linear and Nonlinear Programming Methods
In real-life management situations, there is an important kind of competitive decision problems with multiple decision makers (i.e., players). At present, game theory is one of the most effective tools to deal with such a kind of management problems. In t
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Matrix Games with Payoffs of Intuitionistic Fuzzy Sets and Linear and Nonlinear Programming Methods
7.1 Introduction In real-life management situations, there is an important kind of competitive decision problems with multiple decision makers (i.e., players1). At present, game theory is one of the most effective tools to deal with such a kind of management problems. In the classical (or crisp) game theory, we usually assume that payoffs of players are crisp (or numerical) values [1]. In some real game problems, however, players’ payoffs are used to represent their subjective judgments (or opinions) about competitive situations (or outcomes). For example, the decision problem in which two companies try to improve some product’s sales in some target market may be regarded as a game problem. In this scenario, the payoffs of players (i.e., companies) are to represent the company managers’ subjective judgments (or opinions) of the product shares in the target market at various situations. Such subjective judgments may be expressed with terms of linguistic variables such as ‘‘very large’’, ‘‘larger’’, ‘‘medium’’, and ‘‘small’’ as well as ‘‘smaller’’. Obviously, these judgments usually involve some fuzziness or uncertainty due to the bounded rationality of players and behaviour complexity. In this case, the fuzzy set may be used to express the judgments of players. Fuzzy game theory (especially fuzzy matrix games, i.e., two-person zero-sum fuzzy noncooperative finite games) provides an effective tool for solving such a kind of game problems [2–6]. The fuzzy matrix game has been extensively studied and achieved a great success in applications to many competitive (or oppositional/antagonistic) decision problems. Nevertheless, there are always some hesitancy degrees in players’ judgments due to information incompletion and complex factors such as economy, politics, psychology behaviour, and ideology. For instance, two real estate companies are bidding one another. Due to information incompletion and uncertainty, the bidder only estimates from the previous experience or the related expert’s opinion that possibility of winning the bidding at the situation is 60 % and possibility of losing the bidding is 20 %. Whereas, there remains 20 % possibility in which the bidder 1
As stated in Foreword, the terms ‘‘decision maker’’ and ‘‘player’’ may be interchangeably used. However, the term ‘‘player’’ is customarily used in game theory.
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_7, Springer-Verlag Berlin Heidelberg 2014
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7 Matrix Games with Payoffs of Intuitionistic Fuzzy Sets
cannot determine (or judge) whether winning the bidding or not. Namely, there is some hesitancy degree in the estimation of the bidder on the situation or outcome. Such a hesitancy degree affects strategy choice of the bidders. As stated earlier, the fuzzy set characterized by single membership degree only represents two states of win
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