An alternative method for solving fuzzy bimatrix game

Abstract


Classical methods of solving bimatrix games assume the fulfilment of the common knowledge clause, according to which the game with all rules is known to the players and each of them knows that all participants are informed about what is known to the other partners in the game, and such a position is preserved until the end of the game, and the results of the decisions made by the players are represented by point, numerical values. There are quite a lot of situations requiring decision making, formalised as a bimatrix game, in which subjective representations of participants about game parameters – values of elements of the payment matrix – are not known to the other party. Besides, these values are approximate due to incompleteness of the information available at the moment of decision making. Thus, two types of non-statistical uncertainties arise: the first is due to ignorance of the specific strategy of the other participant, and the second is due to the inaccurate determination of the values of the elements of the payment matrices, destroying the position of common knowledge. Such situations can be represented as a fuzzy bimatrix game.The paper shows that in such a game, in the general case, players will not be able to find equilibrium strategies, and because of the vagueness of the values of the elements of the payment matrices, there are no conditions for the correct definition of mixed strategies. As a solution, it is proposed to determine the strategies that provide a compromise result that best suits both participants. For this purpose the fuzzy results of possible strategies of a player are represented by an integral fuzzy evaluation of the whole set of strategies of another participant in the form of an equivalent fuzzy set with a triangular belonging function, and the best compromise solution is determined by analysing the areas of intersection of equivalent fuzzy sets.

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About the authors

V. G Chernov

Vladimir State University named after Alexander Grigorievich and Nikolai Grigorievich Stoletov

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