The general prenucleolus of n-person cooperative fuzzy games

In this paper, we define the concept of the general prenucleolus minimizing the player complaint vector in the lexicographic order over the preimputation set for the cooperative games with fuzzy coalitions. For the general prenucleolus, we get a sufficient condition that the complaints of all players are in equal amount under the case of linear complaint functions concluding the proposed player excess (Sakawa and Nishizaki (1994) [15]). As a result, we can obtain the proposed fuzzy solutions with the corresponding linear complaint functions, such as the equalizer solution (Molina and Tejada (2002) [5]) and Shapley function etc. We prove that the least square general prenucleolus minimizing the variance of the resulting player complaints, is the general prenucleolus. Thus, an optimal solution considered from two aspects of the lexicographic order and the least square criterion, is obtained. In addition, several general prenucleoli are proposed in terms of specific situations.