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Equilibrium Solutions for Multiobjective Bimatrix Games Incorporating Fuzzy Goals
Abstract
Equilibrium solutions in terms of an attainment degree of a fuzzy goal for games in fuzzy and multiobjective environments are examined. We introduce a fuzzy goal for a payoff in order to incorporate ambiguity of human judgements and assume that a player tries to maximize his attainment degree of the fuzzy goal. A fuzzy goal for a payoff and the equilibrium solution with respect to an attainment degree of a fuzzy goal are defined. Two basic methods, one by weighting coefficients and the other by a minimum component, are employed to aggregate multiple fuzzy goals. When membership functions are linear functions, the computational methods for the equilibrium solutions are developed. It is shown that hte equilibrium solutions are equal to optimal solutions to mathematical programming problems in both cases. The relations between equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals and the Pareto optimal equilibrium solutions are considered.
When game theory is applied to real world problems such as decision making in public and managerial problems, there are occasions when it is difficult to assess exact payoffs because of inaccuracy in information and uncertainty of describing states. To analyze such situations, games with fuzzy payoffs, in which payoffs are represented as fuzzy numbers, are often employed.In this paper, we consider equilibrium solutions in bimatrix games with fuzzy payoffs. First, we examine the case where there is no information on the preferences of players. The equilibrium solutions are defined from a viewpoint of possibility and necessity, and existence conditions of these solutions are investigated. Second, we examine the case where the preferences of the players are represented by fuzzy goals to the payoffs of the players and consider equilibrium solutions with respect to the attainment of each of their goals. Third, we assume that each player maximized the mean of the fuzzy expected payoff and minimizes its spread, and then consider equilibrium solutions of the games with fuzzy payoffs in which the players optimize these objectives in accordance with their preferences.
In this paper we introduce a concept of equilibrium for a non-cooperative game with fuzzy goals involving fuzzy parameters. This equilibrium is based on Zimmermann's approach for solving linear multiobjective problems with fuzzy goals and the concept of N–S equilibrium introduced by Zhukovskii for a non-cooperative game with payoffs involving unknown parameters in the case of complete ignorance of their behavior. We also provide a theorem of existence of this equilibrium based on the Fan inequality.
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