MATRIX GAMES WITH LINGUISTIC INFORMATION

Abstract

Decision-making is a non-deterministic process. The antagonistic situations arise in many parts of real life, giving rise to uncertainty and imprecision. Furthermore, it is very common, when real situations are concerned, to deal with data vaguely. Game theory gives a mathematical background for such conflicting events. The imprecision in games can be of several types, such as randomness and fuzziness. The fuzziness in games is handled by fuzzy set theory. From this point of view, fuzzy games investigate and address those real-world issues that can be repre sented by a game with some degree of vagueness. Moreover, human activities and decisions are too ill-defined to be represented by conventional or fuzzy quan titative means. A possible way to solve such a situation is using a fuzzy linguistic approach, which creates the need for computing with words (CW). Beginning with the 2-tuple linguistic model, several computation models are introduced in the literature to deal with CW. Further, game theory is the study of strategic information communication through language in a rigorous and stylized way. However, stylization and rigor cannot eradicate vagueness: there is always a remainder of linguistic vagueness at the heart of the conceptual tools used in game theory. This creates a foundational dilemma for conventional game theory. Therefore in our study, we have addressed the matrix games with linguistic information and have worked towards developing such games to enhance their applicability. This development supports the pro cess of decision analysis in an uncertain qualitative environment. The thesis en titled “Matrix Games with Linguistic Information” comprises six chapters followed by the summary and future scope. The work in this thesis is motivated by the existing computational models that are implemented to resolve the uncertainty pertaining to matrix game problems. A good amount of analysis is done on the existing literature. The primary purpose of the work done in this thesis is to handle the vagueness and impreciseness prevailing in matrix games in the form of fuzzy xiii linguistic variables. The bibliography and the list of publications are provided at the end of the thesis. The introductory Chapter 1 presents an overview of fuzzy games, computing with words, linguistic models, and their elementary applications anticipated in dis tinct decision-making models, followed by their implementation in the game prob lems. Further, this chapter discusses the notion of randomness and fuzziness involved in any qualitative concept. Some basic concepts used throughout the thesis have been defined along with the motivation of the research work. Thus, the current chapter creates a background for this thesis’s work and motivates the work carried out in this thesis. The Chapter 2 entitled, “A methodology for solving bimatrix games under 2- tuple linguistic environment” establishes the basis for a theory of bimatrix games that have payoffs as linguistic 2-tuples. In literature, the matrix games with 2-tuple linguistic information are considered and solved. However, the theory of bimatrix games having 2-tuple linguistic payoffs is pristine and yet to be explored. The use of linguistic 2-tuple variables to represent practical game situations has shown to be a powerful approach. The chapter is based on a research paper entitled, “A methodology for solving bimatrix games under 2-tuple linguistic environment”, published in International Journal of Systems Science: Operations & Logis tics, Taylor & Francis (SCIE, Impact Factor: 4.2). The chapter mentioned above constitutes a game problem with payoffs belong ing to a linguistic term set where all plausible linguistic descriptors provided by ex perts have a symmetric and uniform distribution. It might not be suitable in practi cal life decision problems since the experts may prefer linguistic labels distributed non-uniformly and non-symmetrically. Several computational models are estab lished in literature to deal with unbalanced linguistic data. Therefore, in Chap ter 3 entitled, “Matrix games with unbalanced linguistic information,” we propose a newly constructed methodology to handle matrix games with proportional lin guistic payoffs. The proportional linguistic payoffs consist of two linguistic terms. However, a generalization of the proportional linguistic terms set is provided as a distribution assessment linguistic set. Hence, this chapter also discusses and solves matrix games with payoffs as linguistic distribution assessment. The chap ter is based on the research papers titled, “Matrix games with proportional linguis tic payoffs”, published in Soft Computing, Springer (SCIE, Impact Factor: 4.1) xiv and “Matrix Games with Linguistic Distribution Assessment Payoffs”, published in: Lecture Notes on Data Engineering and Communications Technologies. Springer. Chapter 4 entitled, “A novel cloud model based on multiplicative unbalanced linguistic term set” proposes a novel concept of the cloud model based on a mul tiplicative unbalanced linguistic term set. Further, based on the defined concept, a generalized multiplicative unbalanced linguistic scale function is introduced to map linguistic concepts to numerical intervals. The proposed approach can be perceived as a convenient technique for multiple criteria decision-making (MCDM) problems. Numerical illustrations are presented to discuss the significance of the proposed methodology. The chapter is based on a research paper titled, “A novel cloud model based on multiplicative unbalanced linguistic term set,” published in The Journal of Supercomputing, Springer (SCIE, Impact Factor: 3.3). In Chapter 5 entitled, “Solving Probabilistic Multiplicative Unbalanced Linguistic Games using Linguistic Cloud model”, the concept of probabilistic multiplicative unbalanced linguistic game is proposed in which the players record their expres sion in terms of unbalanced linguistic terms that have probabilities attached to them. The chapter is based on a research paper titled, “Solving Probabilistic Mul tiplicative Unbalanced Linguistic Games using Linguistic Cloud model”, published in The Journal of Supercomputing, Springer (SCIE, Impact Factor: 3.3). Chapter 6 entitled, “The unbalanced 2-tuple intuitionistic fuzzy linguistic num bers and aggregation operators” presents unbalanced 2-tuple intuitionistic fuzzy sets in which the intuitionistic fuzzy linguistic numbers are composed of linguis tic 2-tuples belonging to an unbalanced multiplicative set. The methodology pre sented in this chapter manipulates imprecise and uncertain information in decision making in intuitionistic fuzzy set theory. The chapter is based on the research pa per titled, “The unbalanced 2-tuple intuitionistic fuzzy linguistic numbers and ag gregation operators” (communicated in “International Journal of Fuzzy Systems"). Chapter 6 is followed by the summary of the research work carried out in this thesis. In addition, the future scope of the thesis has been discussed briefly. Finally, the thesis ends with the bibliography and list of publications.