Bifuzzy Matrix Game

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.