Fuzzy Systems

Shakti Nath Mondal ¹, Sankar Kumar Roy ¹

Affiliations
1 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, IN

Abstract

Conventional game theory is concerned with how rational individuals make decisions when they are faced with known payoffs. We present a new methodology for the analysis of random fuzzy payoffs matrix games. The main difficulty then appears in the study of these games is the comparison between the payoff values associated to the strategies of the players because these payoffs are random fuzzy quantities. The purpose of this paper is to introduce a two-person zero-sum matrix game in which the elements of payoff are characterized as random fuzzy variables. Based on expected value operator, an expected minmax equilibrium strategy to the proposed game is defined and the existence of the strategy is proved. Finally, an example is given to illustrate the effectiveness of the proposed game.

Keywords

Matrix Game, Random Fuzzy Variable, Minmax Equilibrium Strategy.

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Prasanta Mula ¹, Sankar Kumar Roy ²

Affiliations
1 Space Navigation Group, ISRO Satellite Centre, Bangalore, IN
2 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, IN

Abstract

Game theory is a collection of mathematical models to study the behaviours of people with interest conflict, and has been applied extensively to economics, sociology, etc. However, in the real world the certainty assumption is not realistic in many occasions. This lack of precision may be modeled by different ways and one can find in the literature several approaches to deal with this vagueness. So the players often lack the information about the elements of the payoff matrix, which leads to the games within the framework of uncertainty theory. The uncertainty of the payoff matrix leads to be a fuzzy variable is the function from credibility space to real numbers. But some situations the payoff elements are behind the uncertainty of fuzzy variable can be represented as a bifuzzy variable is a function from credibility space to fuzzy variables. The aim of this paper is to define bifuzzy matrix game and develop an effective method to find the strategy and value of the bifuzzy matrix game. In two person zero sum matrix game, the elements of the payoff matrix are characterized by biuzzy variables and this type of game are denoted as bifuzzy matrix game. The uncertainty of bifuzzy variable is measured by the bifuzzy measurable function known as chance measure. The bifuzzy matrix game converts into crisp linear programming problem using properties of game theory and bifuzzy set theory. In this method it always assures that player's gain-floor and loss-ceiling have a common and depends upon the confidence level of the decision maker. The decision maker has freedom to choose the appropriate confidence level to get an optimum solution with their strategy of the bifuzzy matrix game. Finally, we give an example for illustrating the usefulness of the theory developed in this paper.

Keywords

Bifuzzy Variable, Bifuzzy Constraint, Expected Value Operator, Chance Measure, Linear Programming.

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Sankar Kumar Roy ¹, Prasanta Mula ², Shakti Nath Mondal ³

Affiliations
1 Department of Applied Mathematics with Oceaniology and Computer Progarnmming, Vidyasagar University, IN
2 ISRO Satellite Centre, Bangalore, IN
3 Department of Applied Mathematics with Oceanology and Computer Programming, V.U. , Midnapore, West Bengal, IN

Abstract

This paper presents the two-person zero-sum game in which the elements of the payoff matrix are characterized by fuzzy variables. This type of game is called credibilistic game. To analyze the credibilistic game, we introduced the concept of credibility measure by credibility theory. Using credibility theory, the fuzzy constraints is converted into equivalent deterministic constraints. Mainly we have discussed the credibilistic game under the light of linear programming, which depends upon the confidence level due to incomplete information. Lastly a numerical example is presented to illustrate the methodology.

Keywords

Fuzzy Variable, Fuzzy Constraint, Credibilistic Game, Credibility Measure, Linear Programming.

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