An alternative method for solving fuzzy matrix games

Abstract

The paper presents the solution of the task of finding the best strategies in a conflict situation, which can be formalised by a matrix game. It is assumed, that the initial data — the values of the elements of the payment matrix are given as fuzzy numbers. It is shown, that in the general case of fuzzy input data the conditions for the application of known from the classical theory methods of determining mixed strategies are violated. It is suggested that in fuzzy matrix games players can have at least one pure strategy that provides the best result. It is proposed to find such strategies to consider fuzzy evaluations of the consequences of choosing one or another strategy on the whole set of opposing strategies as an aggregate system of fuzzy sets with its subsequent replacement by an equivalent fuzzy set with a triangular membership function, which is considered as an integral fuzzy evaluation of possible consequences of the choice made. The algorithm for construction of equivalent fuzzy set does not depend on the type of membership functions of fuzzy elements of payment matrix, and its parameters depend on the width of carriers, coordinates of centres of gravity and the type of membership functions of fuzzy elements of payment matrix. The lack of dependence of the algorithm for constructing an equivalent fuzzy set on the type of membership functions allows us to use their different types to simulate different levels of uncertainty of the input data. As a result, the consequences of the selected strategies are represented by fuzzy sets with triangular membership functions, which allows not only to estimate the expected values, but also their level of truth. It is shown, that in the solution considered, there are fuzzy estimates of the equilibrium price of the game as well as the upper and lower price of the game. An important circumstance is that the proposed method for the solution of the fuzzy matrix game does not impose restrictions on the type of membership functions of the elements of the payment matrix used to set, which is an essential difference from the known methods.