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

When the game theory is applied to real world problems such as decision mak ing in public and managerial problems, on occasions it is difficult to asse ss payoffs exactly because of inaccuracy of information and fuzzy understan ding of situations by experts. In such cases, games with fuzzy payoffs, in which payoffs are represented as fuzzy numbers, are often considered. In th is paper, we consider equilibrium solutions in multiobjective bimatrix game s with fuzzy payoffs. We introduce fuzzy goals for payoffs in order to inco rporate 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 attainme nt of the fuzzy goal are defined. Two basic methods, one by weighting coeff icients and the other by a minimum component, are employed to aggregate mul tiple fuzzy goals. When membership functions of fuzzy payoffs and fuzzy goa ls are all linear and the fuzzy decision in terms of the intersection is em ployed, the necessary conditions that pairs of strategies be the equilibriu m solutions is obtained. When membership functions of fuzzy payoffs are qua dratic functions, those of fuzzy goals are linear, and the fuzzy decision i n terms of the convex combination is employed, we also derive the necessary conditions that pairs of strategies be the equilibrium solutions. (C) 2000 Elsevier Science B.V. All rights reserved.