A novel equilibrium solution concept for intuitionistic fuzzy bi-matrix games considering proportion mix of possibility
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ORIGINAL PAPER
A novel equilibrium solution concept for intuitionistic fuzzy bi-matrix games considering proportion mix of possibility and necessity expectations Imran Khan1 • Aparna Mehra2 Received: 7 January 2019 / Accepted: 19 April 2019 Ó Springer Nature Switzerland AG 2019
Abstract This paper considers a class of bi-matrix games involving two players with their payoffs matrices having entries from the set of trapezoidal intuitionistic fuzzy numbers. We generalize the notion of Nash equilibrium for such a class of games by modeling the variations in the proportion of the actual realization of the expected values from the game by the two players in terms of the two matrices a and b depending upon the subjectivity of the respective players. Using the intuitionistic fuzzy measure approach, a convex combination of the possibility and the necessity measures, we introduce the notion of ða; bÞ-intuitionistic fuzzy measure equilibrium solutions for games in this class. A methodology is developed to extract the proposed equilibrium solution of an intuitionistic fuzzy bi-matrix game by solving an equivalent quadratic programming problem with linear constraints. The viability of the proposed concept is depicted through two real-life examples of decision-making on marketing strategies to magnetize the preference degrees of the customers. Conclusively, a comparison is drawn between the proposed solution concept with a few of the proximate equilibrium solutions concepts existing for such a class of games. Keywords Bi-matrix game Trapezoidal intuitionistic fuzzy numbers Possibility and necessity expectations Intuitionistic fuzzy measure expectation Intuitionistic fuzzy measure equilibrium solution
1 Introduction A bi-matrix game is a two-person non-cooperative non-zero sum matrix game in which each player has a finite strategies sets to choose from to play the game. Two matrices, called their respective payoffs matrices, are used to explain the payoffs of players. The aim is to obtain a Nash-equilibrium solution of a game which is defined typically in the sense of optimizing a utility function involving the payoffs matrices of two players. & Imran Khan [email protected] Aparna Mehra [email protected] 1
Department of Management Studies, Rukmini Devi Institute of Advanced Studies, Madhuban Chowk, Rohini, Delhi 110085, India
2
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India
The matrix and the bi-matrix games have been extensively studied and successfully applied in a variety of areas. The pioneer work of Campos (1989) laid the foundation of modeling fuzzy matrix games with fuzzy payoffs applying the linear programming models. Several new models describing matrix games with fuzzy payoffs have emerged since then. Bector et al. (2004) use a suitable defuzzification function to develop duality results for linear programming with fuzzy parameters and apply them to solve the matrix games with fuzzy payoffs. Ba
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