Two preimputations of a given TU game can be compared via the Lorenz o
rder applied to the vectors of satisfactions. One preimputation is 'so
cially more desirable' than the other, if its corresponding vector of
satisfactions Lorenz dominates the satisfaction vector with respect to
the second preimputation. It is shown that the prenucleolus, the anti
-prenucleolus, and the modified nucleolus are maximal in this Lorenz o
rder. Here the modified nucleolus is the unique preimputation which le
xicographically minimizes the envies between the coalitions, i.e. the
differences of excesses. Recently Sudholter developed this solution co
ncept. Properties of the set of all undominated preimputations, the ma
ximal satisfaction solution, are discussed. A function on the set of p
reimputations is called collective satisfaction function if it respect
s the Lorenz order. We prove that both classical nucleoli are unique m
inimizers of certain 'weighted Gini inequality indices', which are der
ived from some collective satisfaction functions. For the (pre)nucleol
us the function proposed by Kohlberg, who characterized the nucleolus
as a solution of a single minimization problem, can be chosen. Finally
, a collective satisfaction function is defined such that the modified
nucleolus is its unique maximizer.