Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism
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Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism issue Wenting Xue 1,2 & Zeshui Xu 1 & Xiao-Jun Zeng 2
# Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The hesitant fuzzy set has been studied as a powerful tool to describe the decision makers’ judgements under uncertain environment and applied to many domains. For solving the matrix games whose payoffs are expressed by the hesitant fuzzy information, the paper proposes the Ambika method of hesitant fuzzy matrix games (HFMGs). In this paper, firstly, the formal representation of HFMGs is established to meet the conditions of two-person finite zero-sum games. Secondly, after a new method of adding elements to the shorter hesitant fuzzy elements (HFEs), i.e. the hesitant fuzzy elements with possibility, is developed to keep the same length of HFEs, a weighting method based on the position of element in the HFEs is proposed. Then the hesitant fuzzy bi-objective nonlinear programming models for both players are established for HFMGs. Thirdly, according to the proposed value and ambiguity indexes, the Ambika method of HFMGs is developed to find the optimal solutions of mixed strategies by solving the converted linear programming models. Finally, as the illustration of the proposed method, a numerical example about how to choose the optimal solutions for a state security department is given in the counter-terrorism issue. Keywords Hesitant fuzzy sets . Ambika method . Hesitant fuzzy matrix games . Counter-terrorism issue
1 Introduction Game theory [1, 2] is a significant branch of mathematics which investigates the interaction between the formulated incentive structures. Game theory studies players’ optimal strategies considering the connection between their predicted behavior and actual behavior. Since it was presented, game theory has been applied to many fields, such as political vote [3], society welfare [4], biological evolutionism [5], online shopping marketing problems [6] and water management [7], and solved many practical problems. In general, games can be roughly divided into cooperative or
* Zeshui Xu [email protected] Wenting Xue [email protected] Xiao-Jun Zeng [email protected] 1
School of Economics and Management, Southeast University, Nanjing 211189, Jiangsu, China
2
Department of Computer Science, University of Manchester, Manchester M13 9PL, UK
non-cooperative ones. The main difference lies in whether it exists a binding agreement among the interacting parties. If it exists, then it is a cooperative game. Otherwise, it is a non-cooperative game. According to players’ knowledge of their opponents, games can be divided into the ones with complete information or incomplete information. The complete information game means that each player has accurate information about other players’ characteristics, strategy space and payoff matrix. Otherwise, it is the incomplete information game. In the incomplete information game, players may encounter with
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