Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals

Abstract

When the game theory is applied to real world problems such as decision making in public and managerial problems, on occasions it is difficult to assess payoffs exactly because of inaccuracy of information and fuzzy understanding of situations by experts. In such cases, games with fuzzy payoffs, in which payoffs are represented as fuzzy numbers, are often considered. In this paper, we consider equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs. We introduce fuzzy goals for payoffs in order to incorporate ambiguity of a player's judgements and assume that the player tries to maximize degrees of attainment of the fuzzy goals. The fuzzy goals for payoffs and the equilibrium solution with respect to the degree of attainment of the 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 of fuzzy payoffs and fuzzy goals are all linear and the fuzzy decision in terms of the intersection is employed, the necessary conditions that pairs of strategies be the equilibrium solutions is obtained. When membership functions of fuzzy payoffs are quadratic functions, those of fuzzy goals are linear, and the fuzzy decision in terms of the convex combination is employed, we also derive the necessary conditions that pairs of strategies be the equilibrium solutions.