Fuzzy Models of Antagonistic Games

Abstract

The article is a continuation of a series of works on modeling situations in competitive markets at both micro and macro levels and the development of approaches to finding solutions to the obtained models. The paper proposes a method for solving a certain class of game-theoretic models under conditions of uncertainty. It is substantiated that a significant part of the problems of economic competition can be reduced to a finite matrix game of two players with zero sum, the matrix of winnings of the first player which has a specific form. Given the high degree of uncertainty in modern domestic markets and the need to simplify the current situation in its modeling due to the impossibility of including in the developed model of all real multifaceted relationships, the article considers antagonistic games with fuzzy parameters. It is proposed to look for the solution of the considered class of finite matrix games by reducing them to two dual optimization problems of linear programming with flexible limit constraints. The case is considered when the coefficients in the system of constraints of these models of linear programming are approximated by piecewise-linear membership functions, because they do not raise the question of linearity of the studied models. Using certain linear transformations, the optimization models of linear programming obtained in this work are reduced to models of a special kind, the method of solving which has been developed by other scientists. The essence of this method is that according to the Bellman-Zadeh approach, the resulting fuzzy model is reduced to the decision problem described by the multi-purpose optimization model, the solution of which includes only those alternatives, in such problems are called Pareto effective. Using this method, the fuzzy model obtained in the work is reduced to a "clear" problem of linear programming, some parameters of which are rationally determined by the person making managerial decisions, based on certain limitations obtained by solving two "clear" optimization models with known coefficients. By finding the solution to these dual problems and calculating the mixed strategies of the two players, the person making management decisions will be able to make the right choice among a set of alternative solutions.