Methods for Solving Non-cooperative Games with Fuzzy Payoffs

Abstract

In this thesis, flaws of some of the existing methods, published in last ten years, for solving matrix games with interval/fuzzy/intuitionistic fuzzy payoffs, constrained matrix games with fuzzy payoffs and bimatrix games with intuitionistic fuzzy payoffs are pointed out. To resolve flaws of the existing methods, new methods are also proposed. The chapter wise summary of the thesis is as follows: Chapter 1 In this chapter, flaws of the existing methods [37,68,69,79,84,100] for solving matrix games (or two person zero sum games) with interval payoffs (matrix games in which payoffs are represented by intervals) are pointed out. To resolve these flaws, a new method (named as Gaurika method) is also proposed to obtain the optimal strategies as well as minimum expected gain of Player I and maximum expected loss of Player II for matrix games with interval payoffs. To illustrate the proposed Gaurika method, some existing numerical problems of matrix games with interval payoffs are solved by the proposed Gaurika method. Chapter 2 In this chapter, flaws of the existing methods [36,65,70,71,83] for solving matrix game with fuzzy payoffs (matrix games in which payoffs are represented as fuzzy numbers) are pointed out. Also, to resolve these flaws, a new method (named as Mehar method) is proposed to obtain the optimal strategies as well as minimum expected gain of Player I and maximum expected loss of Player II for matrix games with fuzzy payoffs. To illustrate the proposed Mehar method, an existing numerical problem of matrix games with fuzzy payoffs are solved by the proposed Mehar method. Chapter 3 In this chapter, flaws of the existing methods [64,74-76] for solving constrained matrix games with fuzzy payoffs (constrained matrix games in which payoffs are represented by fuzzy numbers) are pointed out. Also, to resolve these flaws, a new method (named as Vaishanvi method) is proposed to obtain the optimal strategies as well as minimum expected gain of Player I and maximum expected loss of Player II for constrained matrix games with fuzzy payoffs. To illustrate the proposed Vaishnavi method, some existing numerical problems of constrained matrix games with fuzzy payoffs are solved by the proposed Vaishnavi method. Chapter 4 In this chapter, flaws of the existing methods [78,95-97] for solving matrix games with intuitionistic fuzzy payoffs (matrix games in which payoffs are represented by intuitionistic fuzzy numbers) are pointed out. Also, to resolve these flaws, new methods (named as Ambika method-I, Ambika method-II, Ambika method-III and Ambika method-IV) are proposed to obtain the optimal strategies as well as minimum expected gain of Player I and maximum expected loss of Player II for matrix games with intuitionistic fuzzy payoffs. To illustrate proposed Ambika methods, some existing numerical problems of matrix games with intuitionistic fuzzy payoffs are solved by proposed Ambika methods. Chapter 5 Li and Yang [80] pointed out that there is no method in the literature for solving such bimatrix games or two person non-zero sum games (matrix games in which gain of one player is not equal to the loss of other player) in which payoffs are represented by intuitionistic fuzzy numbers and proposed a method for the same. In this chapter, it is pointed out that Li and Yang have considered some mathematical incorrect assumptions in their proposed method. For resolving the shortcomings of Li and Yang’s method, a new method (named as Mehar method) is proposed. Also, the exact optimal solution of the numerical problem, solved by Li and Yang by their proposed method, is obtained by the proposed Mehar method. Chapter 6 In this chapter, future work has been suggested.