Solution of fuzzy matrix games: an application of the extension principle

Conventional game theory is concerned with how rational individuals make decisions when they are faced with known payoffs. This article develops a solution method for the two-person zero-sum game where the payoffs are only approximately known and can be represented by fuzzy numbers. Because the payoffs are fuzzy, the value of the game is fuzzy as well. Based on the extension principle, a pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the value of the game at possibility level a. By applying a dual formulation and a variable substitution technique, the pair of two-level mathematical programs is transformed to a pair of ordinary one-level linear programs so they can be manipulated. From different values of a, the membership function of the fuzzy value of the game is constructed. It is shown that the two players have the same fuzzy value of the game. An example illustrates the whole idea of a fuzzy matrix game. (c) 2007 Wiley Periodicals, Inc.