Nucleoli as maximizers of collective satisfaction functions

Abstract.

Two preimputations of a given TU game can be compared via the Lorenz order applied to the vectors of satisfactions. One preimputation is `socially 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 order. Here the modified nucleolus is the unique preimputation which lexicographically minimizes the envies between the coalitions, i.e. the differences of excesses. Recently Sudhölter developed this solution concept. Properties of the set of all undominated preimputations, the maximal satisfaction solution, are discussed. A function on the set of preimputations is called collective satisfaction function if it respects the Lorenz order. We prove that both classical nucleoli are unique minimizers of certain `weighted Gini inequality indices', which are derived from some collective satisfaction functions. For the (pre)nucleolus 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.